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w(F)j FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)124 b(16)830 4342 y(4.1.3)98 b(Display)20 b(Functions)f(for)h(epipolar)f(geometry)78 b(.)41 b(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)124 b(16)639 4442 y(4.2)86 b(An)20 b(Ov)o(ervie)n(w)f(of)h(methods)f(to)i(estimate)f FL(F)i FQ(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)124 b(18)639 4541 y(4.3)86 b(Homogeneous)18 b(and)h(Projecti)n(v)o (e)g(coordinates)69 b(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)124 b(18)830 4641 y(4.3.1)98 b(Function)19 b FO(torr)p 1617 4641 V 29 w(skew)p 1846 4641 V 29 w(sym)72 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)124 b(21)639 4741 y(4.4)86 b(The)20 b(Fundamental)e(Matrix)58 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(21)639 4840 y(4.5)86 b(Linear)20 b(Methods:)k FL(O)1479 4852 y FK(1)1521 4840 y FQ(,)c FL(O)1634 4852 y FK(2)1719 4840 y FQ(.)41 b(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)124 b(22)830 4940 y(4.5.1)98 b(Ordinary)18 b(Least)j(Squares)f(Re)o (gression:)k FL(O)2389 4952 y FK(1)2466 4940 y FQ(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(23)1926 5255 y(1)p eop %%Page: 2 4 2 3 bop 515 232 a FJ(CONTENTS)2388 b FQ(2)830 523 y(4.5.2)98 b(Orthogonal)18 b(Least)i(Squares)g(Re)o(gression:)25 b FL(O)2468 535 y FK(2)2590 523 y FQ(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)124 b(24)830 623 y(4.5.3)98 b(Linear)19 b(Estimation:)25 b FO(torr)p 1947 623 25 4 v 29 w(ls)c FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(26)830 722 y(4.5.4)98 b(Linear)19 b(Estimation)h(of)g FL(F)p FQ(,)g FO(torr)p 2110 722 V 30 w(estf)68 b FQ(.)42 b(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(26)830 822 y(4.5.5)98 b(The)20 b(Shortcomings)e(of)i(the)g(Linear)f(Methods)59 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(26)830 922 y(4.5.6)98 b(Imposing)18 b(the)j(cubic)e(constraint)g FI(det)q(\()p FL(F)p FI(\))k(=)g(0)34 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)124 b(27)830 1021 y(4.5.7)98 b(In)m(v)n(ariant)18 b(linear)i(\002tting)j(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(28)830 1121 y(4.5.8)98 b(Imposition)18 b(of)i(the)h(quadratic)d(constraint)h FH(f)2416 1091 y FG(2)2407 1141 y(1)2472 1121 y FI(+)f FH(f)2605 1091 y FG(2)2596 1141 y(2)2660 1121 y FI(+)g FH(f)2793 1091 y FG(2)2784 1141 y(4)2848 1121 y FI(+)g FH(f)2981 1091 y FG(2)2972 1141 y(5)3041 1121 y FI(=)23 b FH(K)96 b FQ(29)830 1220 y(4.5.9)i(Imposing)18 b(Linear)i (Constraints)g(on)g FL(F)48 b FQ(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)g(.)h(.)124 b(30)639 1320 y(4.6)86 b(Bookstein)20 b(function:)j FO(torr)p 1718 1320 V 29 w(estf)p 1947 1320 V 30 w(bookstein)44 b FQ(.)d(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)124 b(30)639 1420 y(4.7)86 b(Iterati)n(v)o(ely)19 b(re)n(weighted)f(Least)j(Squares:)k FL(S)2142 1432 y FK(1)2183 1420 y FQ(,)c FL(S)2278 1432 y FK(2)2341 1420 y FQ(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(31)830 1519 y(4.7.1)98 b(Error)19 b(function:)k FO(torr)p 1822 1519 V 29 w(errf2)58 b FQ(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(33)830 1619 y(4.7.2)98 b(W)-7 b(eight)20 b(function:)k FO(torr)p 1885 1619 V 29 w(grad)p 2114 1619 V 29 w(f)29 b FQ(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(33)830 1719 y(4.7.3)98 b(Sampson)19 b(function:)24 b FO(torr)p 1952 1719 V 29 w(estf)p 2181 1719 V 29 w(bookstein)p 2660 1719 V 28 w(sampson)55 b FQ(.)42 b(.)124 b(33)639 1818 y(4.8)86 b(P)o(arameterised)19 b(Descent)h(Methods:)25 b FL(N)2023 1830 y FK(1)2064 1818 y FQ(-)p FL(N)2167 1830 y FK(3)2279 1818 y FQ(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)124 b(33)639 1918 y(4.9)86 b(Constrained)19 b(Estimation)h(of)g FL(F)60 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(35)830 2017 y(4.9.1)98 b(Constrained)14 b(Estimator)h(function:)21 b FO(torr)p 2380 2017 V 29 w(nonlinf)p 2759 2017 V 28 w(mincon2x2)1142 2117 y FQ(36)639 2217 y(4.10)44 b(Thoughts)19 b(on)g(T)-6 b(esting)20 b(Estimators)80 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(36)830 2316 y(4.10.1)56 b(T)-6 b(est)21 b(script:)k FO(torr)p 1692 2316 V 29 w(evalFsc)h FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(36)515 2499 y FM(5)82 b(Rob)n(ust)21 b(Estimation)f(of)g FL(F)1826 b FM(38)639 2599 y FQ(5.1)86 b(Introduction)39 b(.)i(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(38)639 2698 y(5.2)107 b(Random)19 b(Sampling)g(Algorithms)33 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(40)830 2798 y(5.2.1)98 b(Se)n(v)o(en)19 b(Point)h(Function:)k FO(torr)p 2061 2798 V 30 w(F)p 2141 2798 V 29 w(constrained)p 2720 2798 V 28 w(fit)71 b FQ(.)41 b(.)g(.)h(.)124 b(42)830 2897 y(5.2.2)100 b(M)t(A)t(P)t(S)t(A)q(C)24 b(Function:)g FO(torr)p 2029 2897 V 29 w(mapsac)p 2358 2897 V 29 w(F)96 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(43)830 2997 y(5.2.3)98 b(Least)20 b(Median)g(Estimator)73 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)124 b(44)639 3097 y(5.3)86 b(Maximum)18 b(Lik)o(elihood)g (Estimation)h(in)h(the)g(Presence)f(of)g(Outliers:)27 b(M)t(A)t(P)t(S)t(A)q(C)49 b(44)639 3196 y(5.4)86 b(The)20 b(rob)n(ust)g(estimators:)27 b(M)t(A)t(P)t(S)t(A)q(C)73 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) g(.)h(.)124 b(45)639 3296 y(5.5)86 b(Standard)19 b(De)n(viation)59 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(47)515 3479 y FM(6)103 b(Rematching)2206 b(48)515 3661 y(7)82 b(Self)21 b(Calibration,)e(establishing)h(a)h(pr)o(ojecti)o(v)o(e)d (frame)962 b(49)639 3761 y FQ(7.1)86 b(Reco)o(v)o(ery)19 b(of)h(Projection)f(Matrices)58 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(49)830 3861 y(7.1.1)98 b FL(P)p FQ(,)p FL(P)1247 3824 y FF(0)1291 3861 y FQ(from)19 b FL(F)61 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(50)639 3960 y(7.2)86 b(Reco)o(v)o(ery)19 b(of)h(Projecti)n(v)o(e)f (Structure)51 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(50)830 4060 y(7.2.1)98 b(Quick)20 b(triangulation)e(function)g FO(torr)p 2267 4060 V 29 w(triangulate)61 b FQ(.)41 b(.)g(.)g(.)h(.)124 b(51)639 4159 y(7.3)86 b(Correction)19 b(of)h(the)g(matches)28 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(51)830 4259 y(7.3.1)98 b(Correction)19 b(function)f FO(torr)p 1974 4259 V 29 w(correctx4F)30 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(52)830 4359 y(7.3.2)98 b(T)-6 b(esting)15 b(the)g(tw)o(o)h(vie)n(w)f (match)f(correction:)21 b FO(torr)p 2593 4359 V 29 w(test)p 2822 4359 V 30 w(correct)p 3202 4359 V 28 w(sc)45 b FQ(52)639 4458 y(7.4)86 b(Self)21 b(Calibration)47 b(.)41 b(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)g(.)h(.)124 b(53)830 4558 y(7.4.1)98 b(Reco)o(v)o(ery)18 b(of)i FL(C)65 b FQ(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(53)830 4658 y(7.4.2)98 b(Sturm)20 b(Self)g(Calibration)f(function)70 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(54)830 4757 y(7.4.3)98 b(Reco)o(v)o(ery)18 b(of)i FL(R)h FQ(and)e FL(t)51 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(54)830 4857 y(7.4.4)98 b(Function)19 b(for)g FL(R)i FQ(and)e FL(t)q FQ(;)h FO(torr)p 2048 4857 V 30 w(linear)p 2378 4857 V 28 w(EtoPX)64 b FQ(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(55)830 4956 y(7.4.5)98 b(Non-linear)18 b(Optimization)h(of)h FL(G)54 b FQ(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)h(.)124 b(56)p eop %%Page: 3 5 3 4 bop 515 232 a FJ(CONTENTS)2388 b FQ(3)830 523 y(7.4.6)98 b(Non)20 b(Linear)f(minimization)g(of)h FL(g)55 b FQ(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(56)639 623 y(7.5)86 b(T)-6 b(esting)20 b(Self)h(Calibration)e FO(torr)p 1848 623 25 4 v 29 w(test)p 2077 623 V 29 w(calib)p 2356 623 V 29 w(sc)48 b FQ(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)124 b(56)830 722 y(7.5.1)98 b(Displaying)19 b(Structure,)g FO(torr)p 2032 722 V 29 w(display)p 2411 722 V 29 w(structure)79 b FQ(.)41 b(.)g(.)h(.)124 b(59)830 822 y(7.5.2)98 b(An)20 b(e)o(xample)f(script)h(for)g(3D)g(structure)f(generation)25 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(59)515 1005 y FM(8)82 b(Generating)19 b(Synthetic)h(Data)1710 b(61)830 1104 y FQ(8.0.3)98 b(Synthetic)19 b(T)-7 b(w)o(o)21 b(vie)n(w)f(match)f (function)63 b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)124 b(61)830 1204 y(8.0.4)98 b(A)21 b(Script)f(to)g(generate)f(and)h (display)f(synthetic)h(matches)61 b(.)42 b(.)f(.)g(.)g(.)h(.)124 b(62)515 1386 y FM(9)82 b(The)21 b FO(Torr)p 1003 1386 V 30 w(tool)f FM(GUI)1891 b(63)639 1486 y FQ(9.1)86 b(Example)31 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(63)639 1586 y(9.2)86 b(Manual)20 b(Addition)f(of)g(Matches)51 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)g(.)h(.)124 b(65)515 1768 y FM(10)40 b(Conclusion)21 b(and)f(Futur)o(e)g(W)-6 b(ork)1619 b(66)515 1951 y(A)64 b(Deri)o(v)o(ation)20 b(of)f(the)i(Fundamental)f(Matrix)1292 b(67)515 2134 y(B)69 b(Singular)21 b(V)-8 b(alue)20 b(Decomposition)g (and)g(Least)h(Squar)o(es)910 b(69)515 2316 y(C)64 b(Orthogonal)19 b(Regr)o(ession\227af\002ne)g(case)1389 b(71)515 2499 y(D)64 b(V)-8 b(ariance)20 b(of)g(r)o(esiduals)1914 b(73)639 2599 y FQ(D.1)68 b(Bias)22 b(in)e(Linear)f(Estimation)37 b(.)k(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)g(.)h(.)124 b(76)p eop %%Page: 4 6 4 5 bop 515 1146 a FE(Chapter)44 b(1)515 1561 y FN(Intr)l(oduction)515 1993 y FQ(This)23 b(document)e(is)j(slightly)f(dif)n(ferent)e(from)h(a) i(con)m(v)o(entional)c(book)i(in)h(that)g(it)h(is)g(a)f(li)n(ving)g (web)515 2092 y(document)16 b(rather)i(than)g(a)h(\002x)o(ed)f(entity) -5 b(.)23 b(Publishing)17 b(a)i(con)m(v)o(entional)d(book)h(is)i(not)f (so)h(bene\002cial)515 2192 y(in)i(a)g(\002eld)g(which)f(is)i(changing) d(as)i(f)o(ast)h(as)f(computer)e(vision.)26 b(W)m(ith)c(the)e(spread)g (of)h(the)g(Internet)515 2291 y(it)26 b(is)h(possible)f(to)g(produce)e (a)j(book)d(that)j(e)n(v)n(olv)o(es)e(with)h(the)g(\002eld)g(\(time)g 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b(=)e(\()p FH(\016)s(x;)14 b(\016)s(y)s FI(\))h FL(N)2221 3209 y FC(\022)2324 3276 y FH(\016)s(x)2326 3375 y(\016)s(y)2453 3209 y FC(\023)3220 3326 y FQ(\(2.2\))515 3526 y(where)1458 3657 y FL(N)p FI(\()p FH(x;)f(y)s FI(\))23 b(=)1836 3540 y FC(\024)1958 3606 y FH(I)2001 3575 y FG(2)1994 3626 y FB(x)2159 3606 y FH(I)2195 3618 y FB(x)2237 3606 y FH(I)2273 3618 y FB(y)1921 3705 y FH(I)1957 3717 y FB(x)2000 3705 y FH(I)2036 3717 y FB(y)2196 3705 y FH(I)2239 3675 y FG(2)2232 3726 y FB(y)2355 3540 y FC(\025)2413 3657 y FH(:)784 b FQ(\(2.3\))515 3836 y(The)22 b(tw)o(o)h(eigen)m(v)n (alues)e(of)h FL(N)h FQ(are)g(proportional)c(to)k(the)g(principal)e (curv)n(atures)g(of)i FL(N)g FQ(and)f(func-)515 3936 y(tions)e(of)g(them)g(ha)n(v)o(e)g(the)g(property)e(of)i(rotational)f (in)m(v)n(ariance.)k(It)e(is)g(sho)n(wn)e([18)o(])i(that)f(when)g(the) 515 4035 y(trace)j(of)f(the)h(matrix)g(is)h(lar)o(ge)e(there)g(is)i(an) f(edge)f(and)h(when)f(the)h(determinant)e(is)j(lar)o(ge)e(there)h(is) 515 4135 y(an)d(edge)f(or)h(a)h(corner)1172 4105 y Fz(1)1205 4135 y FQ(.)k(A)c(corner)e(strength)g(signal)h(is)1289 4291 y FI(\010\()p FH(x;)14 b(y)s FI(\))23 b(=)g FA(j)p FL(N)p FI(\()p FH(x;)14 b(y)s FI(\))p FA(j)19 b(\000)f FH(\024)p FQ(T)m(race)2301 4256 y FG(2)2338 4291 y FL(N)p FI(\()p FH(x;)c(y)s FI(\))615 b FQ(\(2.4\))515 4447 y(where)25 b FH(\024)33 b FI(=)f(0)p FH(:)p FI(04)25 b FQ(is)i(routinely)d(used) 1682 4417 y Fz(2)1715 4447 y FQ(.)42 b(Corners)25 b(are)g(de\002ned)g (at)h(the)g(local)f(maxima)g(of)g(tw)o(o)515 4546 y(dimensional)18 b(quadratic)g(patches)i(\002tted)g(to)g FI(\010\()p FH(x;)14 b(y)s FI(\))p FQ(,)21 b(resulting)e(in)h(sub-pix)o(el)e(accurac)o(y)-5 b(,)17 b(in)j(v)o(er)n(-)515 4646 y(sion)g(1)g(subpix)o(el)f(accurac)o (y)g(is)i(not)e(implemented.)p 515 4688 1146 4 v 605 4744 a Fy(1)634 4767 y FP(The)25 b(term)g Fx(corner)h FP(is)e(used,)j(e)n(v)o(en)f(though)g(this)f(is)g(truly)h(an)f (interesting)j(point)e(operator)g(partly)l(,)i(for)d(historical)515 4846 y(reasons)605 4904 y Fy(2)634 4927 y FP(Harris)i(originally)j (used)d(the)g(determinant)j(di)n(vided)e(by)f(the)g(trace,)j(b)o(ut)d (he)g(discarded)i(this)e(measure)g(to)g(a)o(v)o(oid)515 5006 y(potential)20 b(di)n(vision)f(by)e(zero.)k(The)c(v)n(alue)i Fw(\024)g Fv(=)h(0)p Fw(:)p Fv(04)d FP(w)o(as)g(empirically)j(arri)n(v) o(ed)f(at)f(as)f(it)g(ga)o(v)o(e)h(the)f(best)h(result.)1926 5255 y FQ(7)p eop %%Page: 8 10 8 9 bop 515 232 a FJ(CHAPTER)21 b(2.)45 b(FEA)-9 b(TURE)21 b(DETECTION)1462 b FQ(8)515 523 y FD(2.1)119 b(Sub)31 b(Pixel)f(Accuracy)515 709 y FQ(Gi)n(v)o(en)20 b(that)i(the)g(corner)e (has)i(been)f(detected)f(at)i FI(\()p FH(x;)14 b(y)s FI(\))p FQ(,)23 b(subpix)o(el)d(accurac)o(y)g(may)h(be)h(gained)e(by) 515 808 y(\002tting)g(a)h(quadratic)d(approximation)f(to)k FI(\010\()p FH(x;)14 b(y)s FI(\))1214 979 y FH(ax)1305 944 y FG(2)1361 979 y FI(+)k FH(by)1524 944 y FG(2)1579 979 y FI(+)g FH(cxy)k FI(+)c FH(dx)h FI(+)f FH(ey)j FI(+)d FH(f)31 b FI(=)23 b(\010\()p FH(x;)14 b(y)s FI(\))541 b FQ(\(2.5\))515 1149 y(using)20 b(the)h(nine)f(pix)o(els)g(around)f FI(\()p FH(x;)14 b(y)s FI(\))22 b FQ(this)g(leads)f(to)f(9)h(equations) f(in)h(6)g(unkno)n(wns)d(that)j(can)g(be)515 1249 y(solv)o(ed)e(by)h (least)h(squares.)515 1527 y FD(2.2)119 b(Detecting)30 b(Cor)n(ners,)h Fu(torr)p 2070 1527 36 4 v 42 w(charris)515 1712 y FO(function)48 b([c_coord])g(=)515 1812 y(torr_charris\(im,)f (ncorners,)h(width,)g(sigma,)h(subpixel\))515 2031 y FM(Input:)639 2187 y FA(\017)41 b FO(im)p FQ(,)30 b(is)f(the)f(image)g (which)f(is)i(passed)f(as)h(an)e(array)h(of)f(doubles)g(\(at)h(present) g(only)f(gre)o(y)722 2287 y(le)n(v)o(el)20 b(images)g(are)g(used,)g (one)g(day)f(maybe)g(I')o(ll)i(implement)e(colour)f(corners\))722 2474 y FO([i1,map1])48 b(=)i(imread\([pathname1)d(filename1]\);)722 2573 y(iii)j(=)99 b(size\(size\(i1\)\);)722 2673 y(if)50 b(iii\(2\))e(==)i(3)922 2773 y(g1)f(=)h(rgb2gray\(i1\);)922 2872 y(disp\('converting)c(to)k(rgb'\);)722 2972 y(else)922 3072 y(g1)f(=)h(i1;)722 3171 y(end)722 3271 y(im)g(=)f(double\(g1\);) 639 3458 y FA(\017)41 b FO(ncorners)p FQ(,)25 b(is)h(the)e(number)f(of) h(corners)g(requested,)g(note)h(that)f(due)g(to)h(non-maximal)722 3557 y(suppression,)19 b(sometimes)h(fe)n(wer)f(corners)g(than)h FO(ncorners)f FQ(may)h(be)g(returned.)639 3719 y FA(\017)41 b FO(width)p FQ(,)20 b(is)h(the)f(width)g(of)g(the)g(Gaussian)g(used)g (to)h(smooth)e(the)h(image.)639 3880 y FA(\017)41 b FO(sigma)p FQ(,)20 b(is)h(the)f(standard)f(de)n(viation)g(of)h(the)g(Gaussian)g (used)g(to)g(smooth)g(the)g(image.)639 4041 y FA(\017)41 b FO(subpixel)p FQ(,)31 b(is)f(a)g(boolean)e(v)n(ariable)g(which)h(is)h (set)g(to)g(one)f(if)g(subpix)o(el)f(detection)h(is)722 4141 y(requested,)19 b(at)i(the)f(moment)f(this)h(is)i(not)d (implemented.)515 4359 y FM(Output:)82 b FO(c)p 939 4359 25 4 v 29 w(coord)20 b FQ(a)h FH(n)d FA(\002)g FI(2)i FQ(array)g(of)g FI(\()p FH(x;)14 b(y)s FI(\))21 b FQ(positions.)515 4637 y FD(2.3)119 b(Display:)33 b Fu(display)p 1753 4637 36 4 v 41 w(corners)p 2298 4637 V 42 w(in)p 2484 4637 V 42 w(figure\(handles\))515 4823 y FQ(A)20 b(simple)h(function)d(to)i (display)g(corners)f(on)h(an)g(image)g(in)g(torr)p 2384 4823 25 4 v 29 w(tool,)f(the)i(meat)f(of)g(it)h(is:)515 4993 y FO(plot\(ccr1\(:,1\),)47 b(ccr1\(:,2\),)h('g+','Parent',)f (ax_handle2\);)p eop %%Page: 9 11 9 10 bop 515 232 a FJ(CHAPTER)21 b(2.)45 b(FEA)-9 b(TURE)21 b(DETECTION)1462 b FQ(9)515 523 y FD(2.4)119 b(F)m(eatur)n(e)30 b(Generation)g(Example,)g Fu(torr)p 2621 523 36 4 v 42 w(cor)p 2879 523 V 42 w(script)515 709 y FQ(T)-7 b(o)17 b(get)g(an)g(easy)h(e)o(xample)d(of)i(the)g(corner)f(detector)g(in)i (action)e(run)h(script)g FO(torr)p 2856 709 25 4 v 29 w(cor)p 3035 709 V 29 w(script)p FQ(,)515 808 y(this)g(will)h(read)f (in)g(the)h(image)e(j1.bmp)g(\(the)h(def)o(ault)f(image\))h(and)f (display)h(a)g(little)h(\002gure)f(together)515 908 y(with)j(crosses)h (sho)n(wing)e(Harris)h(corners.)k(The)c(e)o(xample)e(is)j(as)g(simple)g (as:)515 1074 y FO(figure)515 1174 y(i1)49 b(=)h (imread\('j1.bmp','bmp'\);)515 1273 y(g1)f(=)h(rgb2gray\(i1\);)515 1373 y(d1)f(=)h(double\(g1\);)515 1572 y(ncorners)e(=)i(500)515 1672 y(width)f(=)g(4)515 1771 y(sigma)g(=)g(1)515 1871 y(subpixel)f(=)i(0)515 2070 y([ccr1])98 b(=)50 b(torr_charris\(d1,)d (ncorners,)h(width,)h(sigma,)f(subpixel\);)515 2369 y(imshow\(g1\);)515 2469 y(\045display)g(corners)515 2568 y(hold)h(on)515 2668 y(plot\(ccr1\(:,1\),)e(ccr1\(:,2\),)h('g+'\);)515 2768 y(hold)h(off)515 3048 y FD(2.5)119 b(T)-11 b(o)29 b(do)h(List)619 3233 y FQ(1.)40 b(Subpix)o(el)19 b(accurac)o(y)-5 b(.)619 3399 y(2.)40 b(Colour)20 b(corners)619 3565 y(3.)40 b(Cann)o(y)20 b(Edges.)619 3731 y(4.)40 b(Other)20 b(features)p eop %%Page: 10 12 10 11 bop 515 1146 a FE(Chapter)44 b(3)515 1561 y FN(F)-5 b(eatur)l(e)51 b(Matching)515 1993 y FQ(Feature)21 b(matching)g(is)i (perhaps)e(one)h(of)g(the)g(weak)o(est)h(parts)f(of)g(the)i(S)t(A)t(M)h (edi\002ce.)31 b(The)22 b(geom-)515 2092 y(etry)g(and)g(algorithms)g (in)h(the)g(later)g(chapters)f(are)g(well)i(understood,)c(b)n(ut)j(the) g(whole)f(algorithm)515 2192 y(is)j(b)n(uilt)f(on)f(shak)o(y)g (foundations.)34 b(Correlation)22 b(matching)h(can)g(w)o(ork)h(v)o(ery) e(well)j(for)e(the)h(small)515 2291 y(baseline)f(case,)h(when)f(there)g (is)h(only)f(a)h(small)g(change)e(in)h(illumination.)34 b(Ho)n(we)n(v)o(er)21 b(for)i(signif-)515 2391 y(icant)k(changes)g(in)g (perspecti)n(v)o(e)f(and)h(lighting)f(the)i(intensities)f(of)h (corresponding)c(features)i(in)515 2491 y(tw)o(o)19 b(images)g(can)g (under)o(go)d(lar)o(ge)i(changes,)g(resulting)g(in)i(the)f(f)o(ailure)f (of)h(correlation)e(matching,)515 2590 y(and)i(hence)h(the)i(S)t(A)t(M) h(algorithm.)h(It)c(is)h(hoped)e(in)h(future)f(v)o(ersion)g(of)h(the)g (code)g(to)g(address)g(this)515 2690 y(fundamental)g(problem.)29 b(An)23 b(important)d(thing)i(to)h(note)e(is)j(that)e(image)g (coordinates)f(are)h(repre-)515 2790 y(sented)f(as)i(homogeneous)18 b(v)o(ectors)j FI(\()p FH(x;)14 b(y)s(;)22 b FO(m3)p FI(\))p FQ(,)h(where)e FO(m3)h FQ(is)g(the)g(v)n(ariable)f(used)g(in)h (the)i(S)t(A)t(M)515 2889 y(for)19 b(the)i(third)e(homogeneous)e (coordinate)h(of)i(image)g(points.)639 2989 y(All)26 b(corners)d(within)i(a)g(certain)f(disparity)g(limit)h(are)g(compared)d (o)o(v)o(er)i(the)h(tw)o(o)g(images.)38 b(In)515 3088 y(the)27 b(absence)f(of)h Ft(a)g(priori)g FQ(information,)f(this)i (limit)g(is)f(set)h(to)f FO(max)p 2545 3088 25 4 v 30 w(disparity)f FQ(pix)o(els.)45 b(In)515 3188 y(the)23 b(course)f(of)h(the)g(matching)f(process)h(there)g(are)g(often)f(se)n (v)o(eral)h(candidate)f(matches)g(for)h(each)515 3288 y(feature.)k(Initially)20 b(the)i(one)e(that)i(is)g(most)f(correlated)f (in)h(image)g(intensities)g(at)h(the)f(corner)f(posi-)515 3387 y(tions)j(is)i(selected,)f(in)g(a)g(similar)g(manner)e(to)i(the)f (Droid)g(system)h([18)n(].)36 b(As)24 b(auto-correlation)d(is)515 3487 y(used)d(to)g(de\002ne)f(a)i(feature,)e(the)h(strength)f(of)h (match)g(is)h(obtained)d(by)i(cross-correlation)d(of)j(image)515 3587 y(intensity)h(o)o(v)o(er)g(tw)o(o)i FO(half)p 1337 3587 V 29 w(size)f FA(\002)g FO(half)p 1871 3587 V 29 w(size)g FQ(pix)o(el)g(patches)g(centred)f(on)h(each)f(feature,)1368 3783 y FH(C)29 b FI(=)1624 3704 y FC(X)1544 3899 y FB(ij)s FF(2)p FQ(patch)1837 3783 y FI(\()p FH(I)1905 3795 y FG(2)1943 3783 y FI(\()p FH(i;)14 b(j)5 b FI(\))18 b FA(\000)g FH(I)2249 3795 y FG(1)2287 3783 y FI(\()p FH(i;)c(j)5 b FI(\)\))2488 3741 y FG(2)3220 3783 y FQ(\(3.1\))515 4079 y(where)23 b FH(I)778 4091 y FB(n)823 4079 y FI(\()p FH(i;)14 b(j)5 b FI(\))25 b FQ(is)f(the)g(image)f(intensity)g(at)h (coordinate)d FI(\()p FH(i;)14 b(j)5 b FI(\))25 b FQ(in)e(the)h FH(n)p FQ(th)f(image.)35 b(The)23 b(match)515 4179 y(with)18 b(the)g(maximum)e(strength)i(is)h(stored)e(for)h(each)f(corner)g(from)g (the)h(\002rst)h(to)f(the)g(second)f(image.)515 4278 y(The)g(same)h(process)f(is)h(then)f(applied)g(in)g(re)n(v)o(erse)g (from)f(the)i(second)f(to)g(the)h(\002rst)g(image.)24 b(Matches)515 4378 y(are)k(accepted)g(into)g(the)g(initial)h(set)g(if)g (the)o(y)f(are)g(e)o(xhibit)g(a)h(maximum)d(in)j(both)f(comparisons.) 515 4478 y(This)18 b(has)g(the)g(ef)n(fect)f(of)g(remo)o(ving)f (corners)g(which)i(are)f(ambiguous)f(in)i(that)g(the)o(y)f(ha)n(v)o(e)g (multiple)515 4577 y(candidate)i(matches.)1905 5255 y(10)p eop %%Page: 11 13 11 12 bop 515 232 a FJ(CHAPTER)21 b(3.)45 b(FEA)-9 b(TURE)21 b(MA)-9 b(TCHING)1449 b FQ(11)515 523 y FD(3.1)119 b(Corr)n(elation)30 b(Matching:)37 b Fu(torr)p 2272 523 36 4 v 42 w(corn)p 2602 523 V 42 w(matcher)515 709 y FQ(At)21 b(the)g(heart)f(of)h(this)g (code)f(is)i(the)f(function)e FO(patch)p 2092 709 25 4 v 29 w(match)p FQ(,)h(this)h(tak)o(es)g(as)h(input)e(tw)o(o)h(corner) 515 808 y(positions)g(and)h(outputs)f(the)h(correlation)e(between)h (them,)h(because)f(this)i(is)f(v)o(ery)f(computation-)515 908 y(ally)f(intensi)n(v)o(e)f(and)h(in)m(v)n(olv)o(es)f(loops)g(it)i (is)g(implemented)e(as)i(a)f(MEX)h(\002le.)515 1090 y FO(function)48 b([matches12,minc,mat12])e(=)565 1190 y(torr_corn_matcher\(im1,)f(im2,)k(clist1,)g(clist2,)f (max_disparity,half_size\))515 1411 y FM(Input:)639 1577 y FA(\017)41 b FO(im1,)49 b(im2)21 b FQ(the)f(tw)o(o)g(input)g(images,) f(arrays)h(of)g(doubles)f(as)i(described)e(in)h(Section)g(2.2.)639 1743 y FA(\017)41 b FO(clist1,)49 b(clist2)21 b FQ(tw)o(o)h FH(nc)d FA(\002)g FI(2)j FQ(arrays)f(of)g(corner)f(positions)h(as)h (described)f(in)g(Sec-)722 1843 y(tion)f(2.2,)g FH(nc)g FQ(is)h(the)f(number)f(of)h(corners.)639 2009 y FA(\017)41 b FO(max)p 877 2009 V 30 w(disparity)19 b FQ(the)h(size)h(of)f(the)g (search)g(windo)n(w)f(\(square\))g(in)h(the)g(ne)o(xt)g(image.)639 2175 y FA(\017)41 b FO(half)p 927 2175 V 29 w(size)20 b FQ(the)h(half)f(size)g(of)g(the)g(correlation)f(windo)n(w)-5 b(.)515 2396 y FM(Output:)639 2562 y FA(\017)41 b FO(matches12)19 b FQ(matches)h(in)g(an)g FH(n)e FA(\002)g FI(4)i FQ(array)f(of)h (matches)g FI(\()p FH(x;)14 b(y)s(;)g(x)2699 2532 y FF(0)2723 2562 y FH(y)2767 2532 y FF(0)2790 2562 y FI(\))p FQ(,)21 b(in)f(this)h(case)f FH(n)h FQ(is)722 2662 y(the)f(number)f(of)h (matches.)639 2828 y FA(\017)41 b FO(minc)20 b FQ(is)h(the)g(minimum)d (v)n(alue)i(of)g FH(C)27 b FQ(for)19 b(each)h(corner)-5 b(.)639 2994 y FA(\017)41 b FO(mat12)20 b FQ(is)h(de\002ned)e(such)h (that)h FO(mat\(i\))48 b(=)i(j)21 b FQ(means)e(corner)g FH(i)i FQ(matches)e(to)i(corner)d FH(j)5 b FQ(.)515 3231 y Fs(3.1.1)99 b(Corr)n(elation)25 b(of)g(tw)o(o)f(patches,)i Fr(patch)p 2298 3231 30 4 v 35 w(match)1063 3387 y FO(Correlation)47 b(=)j(patch_match\(im1,im2,x,y,x',y',half_s)o(ize,m)o(inC\)\))o(;)515 3608 y FM(Input:)639 3774 y FA(\017)41 b FO(im1,)49 b(im2)21 b FQ(the)f(tw)o(o)h(input)e(images,)h(arrays)g(of)g(doubles)f(as)i (described)e(in)i(Section)e(2.2.)722 3874 y(These)h(are)g(needed)f(to)i (access)f(the)h(intensity)e(v)n(alues)h(for)g(correlation.)639 4040 y FA(\017)41 b FH(x;)14 b(y)s(;)g(x)934 4010 y FF(0)958 4040 y FH(;)g(y)1039 4010 y FF(0)1083 4040 y FQ(the)20 b(coordinates)e(of)i(tw)o(o)h(prospecti)n(v)o(e)d(matches.)639 4206 y FA(\017)41 b FO(half)p 927 4206 25 4 v 29 w(size)20 b FQ(the)h(half)f(size)g(of)g(the)g(correlation)f(windo)n(w)-5 b(.)639 4372 y FA(\017)41 b FO(minC)30 b FQ(is)h(the)f(minimum)f(v)n (alue)h(of)f FH(C)37 b FQ(so)31 b(f)o(ar)f(for)f(that)i(corner)d (\(this)j(allo)n(ws)f(an)g(early)722 4472 y(jump)20 b(out)g(if)h(the)g (computed)d(correlation)h(goes)h(greater)f(than)h FH(C)28 b FQ(as)21 b(we)g(sum)f(o)o(v)o(er)f(all)j(the)722 4571 y(pix)o(els\).)515 4792 y FM(Output:)82 b FO(Correlation)18 b FQ(The)i(correlation)f(between)g(the)h(tw)o(o)h(corners.)p eop %%Page: 12 14 12 13 bop 515 232 a FJ(CHAPTER)21 b(3.)45 b(FEA)-9 b(TURE)21 b(MA)-9 b(TCHING)1449 b FQ(12)515 523 y FD(3.2)119 b(Bir)n(ch\002eld)31 b(and)g(T)-11 b(omasi)28 b(Corr)n(elation)515 709 y FQ(W)m(ithin)g (this)h(section)f(an)g(error)g(measure)f(is)i(described)e(based)h(on)g (a)h(modi\002cation)e(to)h(that)h(of)515 808 y(Birch\002eld)23 b(and)h(T)-7 b(omasi)24 b([5)o(].)36 b(Comparati)n(v)o(e)22 b(tests)k([42)n(])e(ha)n(v)o(e)g(sho)n(wn)f(that)h(this)g(method)f(gi)n (v)o(es)515 908 y(better)e(results)i(than)e(standard)g(correlation.)29 b(Referring)21 b(to)h(the)g(notation)f(of)h(their)f(paper)m(,)g(rather) 515 1007 y(than)c(compute)f FH(I)1016 1035 y FQ(min)1169 1007 y(and)h FH(I)1343 1019 y FQ(max)1509 1007 y(o)o(v)o(er)f(the)i (left)g(and)g(right)f(pix)o(el,)g(which)h(w)o(ould)f(only)g(be)h (correct)515 1107 y(if)25 b(the)g(epipolar)e(lines)j(lay)f(e)o(xactly)f (along)g(the)g(scanlines,)i(we)f(compute)f(these)h(tw)o(o)g(quantities) 515 1207 y(o)o(v)o(er)20 b(the)i(8-connected)d(neighbourhood)f(of)j (each)h(corresponding)c(pix)o(el)k(in)g(both)f(images.)30 b(De-)515 1306 y(\002ne)20 b(the)g(follo)n(wing)f(quantities:)621 1461 y FH(e)660 1473 y FG(12)730 1461 y FI(\()p FH(I)798 1473 y FG(1)836 1461 y FI(\()p FH(x;)14 b(y)s FI(\))p FH(;)g(I)1101 1473 y 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v 1851 2753 a(2)1903 2696 y FA(f)p FI(\()p FH(I)2013 2708 y FG(2)2050 2696 y FI(\()p FH(x;)14 b(y)s FI(\))19 b(+)f FH(I)2380 2708 y FG(2)2418 2696 y FI(\()p FH(x)h FI(+)f FH(\016)s(x;)c(y)22 b FI(+)c FH(\016)s(y)s FI(\)\))p FA(g)848 2912 y FH(I)891 2878 y FG(min)884 2933 y(2)1005 2912 y FI(\()p FH(x;)c(y)s FI(\))83 b(=)214 b(min)1428 2967 y FB(\016)r(x;\016)r(y)r FG(=)p FF(\000)p FG(1)p FB(;)p FG(0)p FB(;)p FG(1)1851 2856 y FI(1)p 1851 2893 V 1851 2969 a(2)1903 2912 y FA(f)p FI(\()p FH(I)2013 2924 y FG(2)2050 2912 y FI(\()p FH(x;)14 b(y)s FI(\))19 b(+)f FH(I)2380 2924 y FG(2)2418 2912 y FI(\()p FH(x)h FI(+)f FH(\016)s(x;)c(y)22 b FI(+)c FH(\016)s(y)s FI(\)\))p FA(g)515 3106 y FQ(W)m(ith)i(these)h(quantities)e(de\002ned:)596 3261 y FH(e)635 3273 y FG(12)705 3261 y FI(\()p FH(I)773 3273 y FG(1)811 3261 y FI(\()p FH(x;)14 b(y)s FI(\))p FH(;)g(I)1076 3273 y FG(2)1115 3261 y FI(\()p FH(x)1194 3226 y FF(0)1218 3261 y FH(;)g(y)1299 3226 y FF(0)1322 3261 y FI(\)\))23 b(=)g(max)o FA(f)p FI(0)p 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y(Fundamental)g(Matrix)515 2242 y FQ(This)22 b(chapter)f(is)h(tak)o(en)g(lar)o(gely)e(from)h(my)h (thesis,)h(b)n(ut)e(pro)o(vides)g(some)g(useful)h(background.)k(It)515 2341 y(w)o(as)16 b(published)e(in)i([52)n(,)g(51)o(,)g(53)o(].)24 b(Other)15 b(e)o(xcellent)f(sources)h(for)g(estimation)g(of)g FL(F)h FQ(are)g(the)f(re)n(vie)n(w)515 2441 y(article)23 b(by)g(Zhang)f([64)o(],)i(and)f(the)g(book)f(of)h(Hartle)o(y)g(and)g (Zisserman)f([25)o(])i(or)f([63)n(].)35 b(The)23 b(main)515 2540 y(function)17 b(that)i(you)f(need)h(to)g(estimate)g(the)g (fundamental)e(matrix,)i FL(F)g FQ(is)h FO(torr)p 2829 2540 25 4 v 29 w(estimateF)p FQ(,)e(a)515 2640 y(wrapper)e(for)i(all)h (the)f FL(F)h FQ(estimation)e(methods,)h(which)f(is)i(described)e(in)i (the)f(ne)o(xt)f(section.)24 b(There)515 2740 y(is)h(a)f(confusing)e (plethora)h(of)h(w)o(ays)g(to)g(compute)f FL(F)p FQ(,)i(I)f(suggest)g (using)g(a)g(rob)n(ust)g(estimator)f(lik)o(e)517 2839 y(M)t(A)t(P)t(S)t(A)q(C)f(to)d(get)f(a)i(\002rst)f(pass)g(at)g 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973 V 29 w(estf)p 1512 973 V 29 w(bookstein)p FQ(.)801 1098 y(4.)41 b(`BooSam',)22 b(\002tting)g(using)h(the)f (method)f(of)i(Section)f(4.5.7,)g(follo)n(wed)f(by)h(iterati)n(v)o(e) 905 1197 y(least)30 b(squares,)h(using)d(Sampson')-5 b(s)29 b(weighting)e(described)h(in)i(Section)e(4.7,)j(calls)905 1297 y FO(torr)p 1110 1297 V 29 w(estf)p 1339 1297 V 29 w(bookstein)p 1818 1297 V 28 w(sampson)p FQ(.)801 1422 y(5.)41 b(`non)p 1064 1422 V 28 w(linear')20 b(a)i(constrained)e (estimate)h(of)g FL(F)h FQ(enforcing)d FA(j)p FL(F)p FA(j)25 b FI(=)f(0)e FQ(as)g(described)d(in)905 1522 y(Section)h(4.9,)f(call)i FO(torr)p 1666 1522 V 29 w(nonlinf)p 2045 1522 V 28 w(mincon2x2)p FQ(.)801 1647 y(6.)41 b(`lin+non)p 1199 1647 V 27 w(lin':)26 b(`linear')19 b(follo)n(wed)g(by)g(`non)p 2227 1647 V 28 w(linear'.)639 1805 y FA(\017)41 b FO(set)p 877 1805 V 30 w(rank2)p FQ(:)24 b(boolean)17 b(v)n(ariable,)g(if)i(it)g (is)g(1)g(then)f(the)g(SVD)h(is)h(used)e(to)h(enforce)e FA(j)p FL(F)p FA(j)23 b FI(=)f(0)722 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4322 y FM(Input:)639 4472 y FA(\017)41 b FO(x1,y1,x2,y2)26 b FQ(are)i(arrays)f(of)g(the)h FH(n)g FQ(corresponding)c FI(\()p FH(x;)14 b(y)s(;)g(x)2711 4442 y FF(0)2735 4472 y FH(;)g(y)2816 4442 y FF(0)2839 4472 y FI(\))29 b FQ(coordinates.)45 b(I)722 4572 y(could)19 b(ha)n(v)o(e)g(passed)g(them)g(in)h(one)f FH(n)c FA(\002)g FI(4)20 b FQ(matrix)f(b)n(ut)g(in)h(the)f(subsequent)f(calculations)h (it)722 4671 y(often)h(mak)o(es)g(things)f(clearer)h(to)g(identity)g(e) o(xplicitly)f(the)h(coordinates.)639 4830 y FA(\017)41 b FO(no)p 827 4830 V 30 w(matches)19 b FQ(is)i(the)g(number)d(of)i (matches.)639 4988 y FA(\017)41 b FO(m3)21 b FQ(is)g(the)f(third)g (homogeneous)d(coordinate.)p eop %%Page: 16 18 16 17 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(16)515 523 y FM(Output:)82 b FO(f)20 b FQ(the)h(solution)e(v)o(ector)g FL(f)32 b FI(=)23 b(\()p FH(f)1814 535 y FG(1)1865 523 y FH(:)14 b(:)g(:)f(f)2016 535 y 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y([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2,tr)o(ue_C,t)o(rue_R)o (,true_)o(t,)c(true_E])560 2560 y(=)44 b(torr_gen_2view_matches;)515 2652 y(no_matches)f(=)i(length\(nx1\);)515 2743 y(matches)e(=)i ([nx1,ny1,nx2,ny2];)515 3017 y(\045first)f(estimate)f(F)515 3108 y([f,)h(e1,)g(n_inliers,inlier_index,estimateF])515 3199 y(=)g(torr_estimateF\()f(matches,)g(m3,)i([],)f(method,)g (set_rank2\);)515 3382 y(\045check)g(errors)515 3473 y(e)g(=)h(torr_errf2\(f,)e(nx1,ny1,nx2,ny2,)f(no_matches,)h(m3\);)515 3656 y(\045display)g(the)i(result)515 3747 y(if)f(compare)694 3839 y(torr_compare_epipoles\(estimateF,true_F,m)o(atche)o(s,)39 b(m3\))515 3930 y(else)694 4021 y(torr_display_epipoles\(nF,matches,)g (m3\))515 4113 y(end)639 4387 y FQ(There)26 b(are)g(tw)o(o)h(display)f (functions)f(for)h(matches)g(and)g(epipolar)g(geometry)-5 b(,)25 b(described)g(in)515 4486 y(the)20 b(ne)o(xt)f(section,)h FO(torr)p 1281 4486 V 29 w(display)p 1660 4486 V 29 w(epipoles)f FQ(or)h FO(torr)p 2398 4486 V 29 w(compare)p 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1648 L 5135 1612 mt 5135 1684 L 5990 3446 mt 6062 3446 L 6026 3410 mt 6026 3482 L 2659 3315 mt 2731 3315 L 2695 3279 mt 2695 3351 L 5635 603 mt 5707 603 L 5671 567 mt 5671 639 L 4889 1123 mt 4961 1123 L 4925 1087 mt 4925 1159 L 4201 1336 mt 4273 1336 L 4237 1300 mt 4237 1372 L 2871 2307 mt 2943 2307 L 2907 2271 mt 2907 2343 L 5294 3034 mt 5366 3034 L 5330 2998 mt 5330 3070 L 5094 2963 mt 5166 2963 L 5130 2927 mt 5130 2999 L 4032 3616 mt 4104 3616 L 4068 3580 mt 4068 3652 L 2865 2978 mt 2937 2978 L 2901 2942 mt 2901 3014 L 3508 1656 mt 3580 1656 L 3544 1620 mt 3544 1692 L 1684 1631 mt 1756 1631 L 1720 1595 mt 1720 1667 L 2974 2259 mt 3046 2259 L 3010 2223 mt 3010 2295 L 5067 2648 mt 5139 2648 L 5103 2612 mt 5103 2684 L 1547 4159 mt 1619 4159 L 1583 4123 mt 1583 4195 L 2295 3081 mt 2367 3081 L 2331 3045 mt 2331 3117 L 2941 3885 mt 3013 3885 L 2977 3849 mt 2977 3921 L 4722 2690 mt 4794 2690 L 4758 2654 mt 4758 2726 L 4527 965 mt 4599 965 L 4563 929 mt 4563 1001 L 4089 2666 mt 4161 2666 L 4125 2630 mt 4125 2702 L 1952 1503 mt 2024 1503 L 1988 1467 mt 1988 1539 L 5109 754 mt 5181 754 L 5145 718 mt 5145 790 L 1040 3756 mt 1112 3756 L 1076 3720 mt 1076 3792 L 4595 2550 mt 4667 2550 L 4631 2514 mt 4631 2586 L 4110 3980 mt 4182 3980 L 4146 3944 mt 4146 4016 L 1693 1806 mt 1765 1806 L 1729 1770 mt 1729 1842 L 1827 1753 mt 1899 1753 L 1863 1717 mt 1863 1789 L 4885 1215 mt 4957 1215 L 4921 1179 mt 4921 1251 L 1561 961 mt 1633 961 L 1597 925 mt 1597 997 L 5473 4170 mt 5545 4170 L 5509 4134 mt 5509 4206 L 3632 1066 mt 3704 1066 L 3668 1030 mt 3668 1102 L 4723 2996 mt 4795 2996 L 4759 2960 mt 4759 3032 L 3140 705 mt 3212 705 L 3176 669 mt 3176 741 L 1173 782 mt 1245 782 L 1209 746 mt 1209 818 L 2338 3284 mt 2410 3284 L 2374 3248 mt 2374 3320 L 4142 2237 mt 4214 2237 L 4178 2201 mt 4178 2273 L 2750 840 mt 2822 840 L 2786 804 mt 2786 876 L 4083 4267 mt 4155 4267 L 4119 4231 mt 4119 4303 L 3350 3706 mt 3422 3706 L 3386 3670 mt 3386 3742 L 1636 2482 mt 1708 2482 L 1672 2446 mt 1672 2518 L 4746 3459 mt 4818 3459 L 4782 3423 mt 4782 3495 L 4445 3067 mt 4517 3067 L 4481 3031 mt 4481 3103 L 2451 3349 mt 2523 3349 L 2487 3313 mt 2487 3385 L 3694 3332 mt 3766 3332 L 3730 3296 mt 3730 3368 L 3704 831 mt 3776 831 L 3740 795 mt 3740 867 L 1586 1124 mt 1658 1124 L 1622 1088 mt 1622 1160 L 2457 3138 mt 2529 3138 L 2493 3102 mt 2493 3174 L 3445 4087 mt 3517 4087 L 3481 4051 mt 3481 4123 L 3020 3387 mt 3092 3387 L 3056 3351 mt 3056 3423 L gs 899 389 5357 4228 MR c np gr 4278 1028 mt 4350 1028 L 4314 992 mt 4314 1064 L 2138 3866 mt 2210 3866 L 2174 3830 mt 2174 3902 L 5483 1308 mt 5555 1308 L 5519 1272 mt 5519 1344 L 4289 3497 mt 4361 3497 L 4325 3461 mt 4325 3533 L 6194 3915 mt 6266 3915 L 6230 3879 mt 6230 3951 L 1576 4616 mt 1648 4616 L 1612 4580 mt 1612 4652 L 5067 3013 mt 5139 3013 L 5103 2977 mt 5103 3049 L 2598 1278 mt 2670 1278 L 2634 1242 mt 2634 1314 L 3163 2634 mt 3235 2634 L 3199 2598 mt 3199 2670 L 1948 4034 mt 2020 4034 L 1984 3998 mt 1984 4070 L 4268 2616 mt 4340 2616 L 4304 2580 mt 4304 2652 L 5391 2790 mt 5463 2790 L 5427 2754 mt 5427 2826 L 4569 3599 mt 4641 3599 L 4605 3563 mt 4605 3635 L 3178 3478 mt 3250 3478 L 3214 3442 mt 3214 3514 L 5409 2980 mt 5481 2980 L 5445 2944 mt 5445 3016 L 2698 2515 mt 2770 2515 L 2734 2479 mt 2734 2551 L 3075 3458 mt 3147 3458 L 3111 3422 mt 3111 3494 L 1906 1631 mt 1978 1631 L 1942 1595 mt 1942 1667 L 5511 3191 mt 5583 3191 L 5547 3155 mt 5547 3227 L 3492 1482 mt 3564 1482 L 3528 1446 mt 3528 1518 L 3951 3384 mt 4023 3384 L 3987 3348 mt 3987 3420 L 2610 3271 mt 2682 3271 L 2646 3235 mt 2646 3307 L 2937 3378 mt 3009 3378 L 2973 3342 mt 2973 3414 L 5870 3397 mt 5942 3397 L 5906 3361 mt 5906 3433 L 3546 2728 mt 3618 2728 L 3582 2692 mt 3582 2764 L 4029 949 mt 4101 949 L 4065 913 mt 4065 985 L 2948 1994 mt 3020 1994 L 2984 1958 mt 2984 2030 L 1638 1897 mt 1710 1897 L 1674 1861 mt 1674 1933 L 1479 1689 mt 1551 1689 L 1515 1653 mt 1515 1725 L 3544 1041 mt 3616 1041 L 3580 1005 mt 3580 1077 L 1869 3704 mt 1941 3704 L 1905 3668 mt 1905 3740 L 5380 1267 mt 5452 1267 L 5416 1231 mt 5416 1303 L 3341 2146 mt 3413 2146 L 3377 2110 mt 3377 2182 L 5239 2548 mt 5311 2548 L 5275 2512 mt 5275 2584 L 3266 4023 mt 3338 4023 L 3302 3987 mt 3302 4059 L 3426 3089 mt 3498 3089 L 3462 3053 mt 3462 3125 L 2755 1760 mt 2827 1760 L 2791 1724 mt 2791 1796 L 4707 3062 mt 4779 3062 L 4743 3026 mt 4743 3098 L 2016 3166 mt 2088 3166 L 2052 3130 mt 2052 3202 L 3716 3587 mt 3788 3587 L 3752 3551 mt 3752 3623 L 4331 1397 mt 4403 1397 L 4367 1361 mt 4367 1433 L 4022 1852 mt 4094 1852 L 4058 1816 mt 4058 1888 L 2496 4198 mt 2568 4198 L 2532 4162 mt 2532 4234 L 2248 1660 mt 2320 1660 L 2284 1624 mt 2284 1696 L 1997 2317 mt 2069 2317 L 2033 2281 mt 2033 2353 L 2605 3997 mt 2677 3997 L 2641 3961 mt 2641 4033 L 3593 1972 mt 3665 1972 L 3629 1936 mt 3629 2008 L 2398 1517 mt 2470 1517 L 2434 1481 mt 2434 1553 L 4914 2787 mt 4986 2787 L 4950 2751 mt 4950 2823 L 4355 3368 mt 4427 3368 L 4391 3332 mt 4391 3404 L 4891 1714 mt 4963 1714 L 4927 1678 mt 4927 1750 L 5919 3097 mt 5991 3097 L 5955 3061 mt 5955 3133 L 2982 3389 mt 3054 3389 L 3018 3353 mt 3018 3425 L 5206 909 mt 5278 909 L 5242 873 mt 5242 945 L 4674 1367 mt 4746 1367 L 4710 1331 mt 4710 1403 L 4122 1533 mt 4194 1533 L 4158 1497 mt 4158 1569 L 3166 2392 mt 3238 2392 L 3202 2356 mt 3202 2428 L 5361 2958 mt 5433 2958 L 5397 2922 mt 5397 2994 L 5144 2889 mt 5216 2889 L 5180 2853 mt 5180 2925 L 4280 3567 mt 4352 3567 L 4316 3531 mt 4316 3603 L 3504 2666 mt 3576 2666 L 3540 2630 mt 3540 2702 L 3679 1805 mt 3751 1805 L 3715 1769 mt 3715 1841 L 2063 1942 mt 2135 1942 L 2099 1906 mt 2099 1978 L 3260 2344 mt 3332 2344 L 3296 2308 mt 3296 2380 L 4971 2356 mt 5043 2356 L 5007 2320 mt 5007 2392 L 2007 4283 mt 2079 4283 L 2043 4247 mt 2043 4319 L 3097 2780 mt 3169 2780 L 3133 2744 mt 3133 2816 L 3312 3902 mt 3384 3902 L 3348 3866 mt 3348 3938 L 4744 2534 mt 4816 2534 L 4780 2498 mt 4780 2570 L 4363 1263 mt 4435 1263 L 4399 1227 mt 4399 1299 L 4200 2677 mt 4272 2677 L 4236 2641 mt 4236 2713 L 2334 1824 mt 2406 1824 L 2370 1788 mt 2370 1860 L 4792 1060 mt 4864 1060 L 4828 1024 mt 4828 1096 L 1489 3939 mt 1561 3939 L 1525 3903 mt 1525 3975 L 4633 2494 mt 4705 2494 L 4669 2458 mt 4669 2530 L 4423 3891 mt 4495 3891 L 4459 3855 mt 4459 3927 L 2243 2050 mt 2315 2050 L 2279 2014 mt 2279 2086 L 2298 1998 mt 2370 1998 L 2334 1962 mt 2334 2034 L 4682 1425 mt 4754 1425 L 4718 1389 mt 4718 1461 L 1956 1436 mt 2028 1436 L 1992 1400 mt 1992 1472 L 5752 4029 mt 5824 4029 L 5788 3993 mt 5788 4065 L 3692 1376 mt 3764 1376 L 3728 1340 mt 3728 1412 L 4806 2786 mt 4878 2786 L 4842 2750 mt 4842 2822 L 3196 1152 mt 3268 1152 L 3232 1116 mt 3232 1188 L 1731 1317 mt 1803 1317 L 1767 1281 mt 1767 1353 L 3174 2899 mt 3246 2899 L 3210 2863 mt 3210 2935 L 4176 2312 mt 4248 2312 L 4212 2276 mt 4212 2348 L 2893 1264 mt 2965 1264 L 2929 1228 mt 2929 1300 L 4440 4204 mt 4512 4204 L 4476 4168 mt 4476 4240 L 3827 3327 mt 3899 3327 L 3863 3291 mt 3863 3363 L 2245 2574 mt 2317 2574 L 2281 2538 mt 2281 2610 L 4900 3286 mt 4972 3286 L 4936 3250 mt 4936 3322 L 4598 3009 mt 4670 3009 L 4634 2973 mt 4634 3045 L 2818 3374 mt 2890 3374 L 2854 3338 mt 2854 3410 L 4003 3167 mt 4075 3167 L 4039 3131 mt 4039 3203 L 3703 1193 mt 3775 1193 L 3739 1157 mt 3739 1229 L 2376 1528 mt 2448 1528 L 2412 1492 mt 2412 1564 L 3182 2841 mt 3254 2841 L 3218 2805 mt 3218 2877 L 3865 3909 mt 3937 3909 L 3901 3873 mt 3901 3945 L 3541 3116 mt 3613 3116 L 3577 3080 mt 3577 3152 L gs 899 389 5357 4228 MR c np /c9 { 0.000000 0.000000 1.000000 sr} bdef c9 -153 421 4467 607 2 MP stroke 1028 -400 1146 4266 2 MP stroke -398 175 5917 1133 2 MP stroke 298 -225 4027 3722 2 MP stroke 115 -484 6115 4399 2 MP stroke 442 206 1170 4410 2 MP stroke 65 -445 5038 3458 2 MP stroke 215 432 2419 846 2 MP stroke 225 99 2974 2535 2 MP stroke 873 -227 1111 4261 2 MP stroke 99 18 4205 2598 2 MP stroke 0 -151 5427 2941 2 MP stroke 212 -26 4393 3625 2 MP stroke 843 -597 2371 4075 2 MP stroke 4 -267 5441 3247 2 MP stroke 538 24 2196 2491 2 MP stroke 349 28 2762 3430 2 MP stroke 839 399 1103 1232 2 MP stroke -9 -428 5556 3619 2 MP stroke 247 316 3281 1166 2 MP stroke 347 -183 3640 3567 2 MP stroke 463 14 2183 3257 2 MP stroke 580 -193 2393 3571 2 MP stroke 101 -165 5805 3562 2 MP stroke 413 -181 3169 2909 2 MP stroke -133 422 4198 527 2 MP stroke 592 123 2392 1871 2 MP stroke 517 337 1157 1560 2 MP stroke 457 434 1058 1255 2 MP stroke 33 479 3547 562 2 MP stroke 586 7 1319 3697 2 MP stroke -409 207 5825 1060 2 MP stroke 504 21 2873 2125 2 MP stroke -64 -171 5339 2719 2 MP stroke 374 4 2928 4019 2 MP stroke 374 -114 3088 3203 2 MP stroke 425 277 2366 1483 2 MP stroke 145 -112 4598 3174 2 MP stroke 969 -176 1083 3342 2 MP stroke 343 -75 3409 3662 2 MP stroke -68 263 4435 1134 2 MP stroke 103 115 3955 1737 2 MP stroke 407 89 2125 4109 2 MP stroke 626 352 1658 1308 2 MP stroke 456 240 1577 2077 2 MP stroke 410 50 2231 3947 2 MP stroke 213 124 3416 1848 2 MP stroke 703 409 1731 1108 2 MP stroke 88 -57 4862 2844 2 MP stroke 203 1 4188 3367 2 MP stroke -208 66 5135 1648 2 MP stroke -71 -349 6026 3446 2 MP stroke 323 74 2695 3315 2 MP stroke -429 306 5671 603 2 MP stroke -215 244 4925 1123 2 MP stroke -79 197 4237 1336 2 MP stroke 295 85 2907 2307 2 MP stroke 67 -76 5330 3034 2 MP stroke 50 -74 5130 2963 2 MP stroke 248 -49 4068 3616 2 MP stroke 639 -312 2901 2978 2 MP stroke 171 149 3544 1656 2 MP stroke 379 311 1720 1631 2 MP stroke 286 85 3010 2259 2 MP stroke -96 -292 5103 2648 2 MP stroke 460 124 1583 4159 2 MP stroke 802 -301 2331 3081 2 MP stroke 371 17 2977 3885 2 MP stroke 22 -156 4758 2690 2 MP stroke -164 298 4563 965 2 MP stroke 111 11 4125 2666 2 MP stroke 382 321 1988 1503 2 MP stroke -317 306 5145 754 2 MP stroke 449 183 1076 3756 2 MP stroke 38 -56 4631 2550 2 MP stroke 313 -89 4146 3980 2 MP stroke 550 244 1729 1806 2 MP stroke 471 245 1863 1753 2 MP stroke -203 210 4921 1215 2 MP stroke 395 475 1597 961 2 MP stroke 279 -141 5509 4170 2 MP stroke 60 310 3668 1066 2 MP stroke 83 -210 4759 2996 2 MP stroke 56 447 3176 705 2 MP stroke 558 535 1209 782 2 MP stroke 836 -385 2374 3284 2 MP stroke 34 75 4178 2237 2 MP stroke 143 424 2786 840 2 MP stroke 357 -63 4119 4267 2 MP stroke 477 -379 3386 3706 2 MP stroke 609 92 1672 2482 2 MP stroke 154 -173 4782 3459 2 MP stroke 153 -58 4481 3067 2 MP stroke 367 25 2487 3349 2 MP stroke 309 -165 3730 3332 2 MP stroke -1 362 3740 831 2 MP stroke 790 404 1622 1124 2 MP stroke 725 -297 2493 3138 2 MP stroke 420 -178 3481 4087 2 MP stroke 521 -271 3056 3387 2 MP stroke -8 -152 -108 126 4422 902 3 MP stroke -284 220 394 -43 1780 3909 3 MP stroke 107 -92 -155 24 5674 1284 3 MP stroke -68 100 130 -48 4195 3545 3 MP stroke 26 153 105 -150 6125 4065 3 MP stroke -174 -30 118 106 1494 4510 3 MP stroke 40 153 83 -141 5020 3154 3 MP stroke -131 -124 11 161 2623 1117 3 MP stroke -88 -13 60 52 3139 2582 3 MP stroke -257 150 319 0 1665 4034 3 MP stroke -35 3 30 15 4274 2601 3 MP stroke 21 50 21 -50 5406 2840 3 MP stroke -66 27 73 10 4532 3589 3 MP stroke -196 269 361 -125 2853 3603 3 MP stroke 36 89 38 -88 5407 3068 3 MP stroke -181 38 174 54 2560 2461 3 MP stroke -119 21 112 39 2999 3419 3 MP stroke -332 -59 222 203 1720 1428 3 MP stroke 62 141 56 -142 5491 3333 3 MP stroke -125 -83 38 125 3490 1357 3 MP stroke -89 90 139 -31 3848 3415 3 MP stroke -155 35 151 44 2495 3227 3 MP stroke -165 113 218 -14 2755 3392 3 MP stroke -10 63 56 -46 5850 3443 3 MP stroke -112 96 161 -25 3421 2753 3 MP stroke -15 -151 -102 128 4167 821 3 MP stroke -212 10 178 92 2806 1902 3 MP stroke -217 -67 124 155 1550 1742 3 MP stroke -211 -104 91 182 1424 1507 3 MP stroke -77 -156 -55 161 3635 880 3 MP stroke -194 48 192 53 1713 3651 3 MP stroke 107 -103 -164 33 5580 1234 3 MP stroke -169 36 164 50 3213 2096 3 MP stroke 45 51 2 -62 5273 2610 3 MP stroke -124 31 123 33 3179 3990 3 MP stroke -108 70 139 -6 3323 3095 3 MP stroke -178 -55 102 128 2689 1632 3 MP stroke -33 50 63 -25 4680 3087 3 MP stroke -296 141 344 25 1708 3141 3 MP stroke -102 54 123 5 3629 3582 3 MP stroke -14 -93 -59 81 4426 1316 3 MP stroke -50 -29 18 47 4040 1805 3 MP stroke -147 6 122 64 2410 4134 3 MP stroke -255 -62 158 170 2126 1490 3 MP stroke -184 -40 118 119 1915 2198 3 MP stroke -142 19 128 52 2513 3945 3 MP stroke -88 -22 53 59 3576 1913 3 MP stroke -288 -75 176 196 2258 1321 3 MP stroke -22 27 37 -11 4913 2798 3 MP stroke -67 17 67 18 4324 3350 3 MP stroke 59 -40 -78 4 5005 1710 3 MP stroke 72 109 24 -121 5931 3218 3 MP stroke -117 3 96 52 2922 3337 3 MP stroke 99 -138 -184 64 5426 845 3 MP stroke 38 -99 -105 62 4815 1305 3 MP stroke -1 -72 -53 59 4211 1474 3 MP stroke -109 -3 86 53 3116 2339 3 MP stroke -12 31 33 -20 5364 2978 3 MP stroke -6 29 27 -20 5153 2909 3 MP stroke -75 37 89 5 4227 3562 3 MP stroke -168 158 254 -48 3286 2714 3 MP stroke -77 -34 36 64 3679 1741 3 MP stroke -168 -70 82 135 2017 1807 3 MP stroke -106 -3 83 52 3213 2292 3 MP stroke 72 88 8 -105 4999 2461 3 MP stroke -169 -1 134 80 1909 4203 3 MP stroke -223 168 306 -30 2827 2810 3 MP stroke -125 26 120 37 3228 3865 3 MP stroke 14 53 29 -49 4751 2583 3 MP stroke 13 -112 -95 84 4494 1179 3 MP stroke -38 6 35 13 4201 2664 3 MP stroke -170 -73 82 139 2288 1685 3 MP stroke 62 -129 -147 74 4975 986 3 MP stroke -173 -21 123 99 1402 3840 3 MP stroke -5 22 20 -15 4649 2509 3 MP stroke -91 57 115 -3 4344 3894 3 MP stroke -215 -33 147 128 2132 1922 3 MP stroke -189 -41 122 121 2212 1877 3 MP stroke 38 -87 -96 52 4814 1373 3 MP stroke -196 -123 65 191 1927 1245 3 MP stroke -72 71 111 -23 5677 4052 3 MP stroke -63 -97 -23 108 3751 1268 3 MP stroke 2 76 56 -62 4786 2848 3 MP stroke -80 -142 -43 152 3275 1000 3 MP stroke -258 -129 110 225 1657 1092 3 MP stroke -223 199 329 -55 2881 2954 3 MP stroke -21 -22 1 27 4211 2285 3 MP stroke -106 -128 -11 152 2940 1112 3 MP stroke -109 51 126 10 4350 4194 3 MP stroke -105 166 210 -84 3653 3411 3 MP stroke -214 22 188 82 2093 2492 3 MP stroke -27 71 75 -44 4861 3330 3 MP stroke -42 32 59 -6 4575 3015 3 MP stroke -125 24 118 39 2736 3335 3 MP stroke -79 81 124 -28 3915 3195 3 MP stroke -50 -119 -51 119 3790 1074 3 MP stroke -316 -66 205 202 2207 1326 3 MP stroke -198 160 280 -36 2938 2877 3 MP stroke -114 94 163 -22 3738 3931 3 MP stroke -135 134 209 -45 3368 3161 3 MP stroke gr c9 /c10 { 0.000000 1.000000 0.000000 sr} bdef c10 2276 3438 mt 2348 3438 L 2312 3402 mt 2312 3474 L gs 899 389 5357 4228 MR c np gr 0 j 0 -58 -58 0 0 58 58 0 2283 3409 5 MP DP gs 899 389 5357 4228 MR c np gr 1927 2389 mt 1999 2389 L 1963 2353 mt 1963 2425 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 1934 2360 5 MP DP gs 899 389 5357 4228 MR c np gr 3191 1484 mt 3263 1484 L 3227 1448 mt 3227 1520 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3198 1455 5 MP DP gs 899 389 5357 4228 MR c np gr 4908 3664 mt 4980 3664 L 4944 3628 mt 4944 3700 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4915 3635 5 MP DP gs 899 389 5357 4228 MR c np gr 3818 3746 mt 3890 3746 L 3854 3710 mt 3854 3782 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3825 3717 5 MP DP gs 899 389 5357 4228 MR c np gr 3911 2934 mt 3983 2934 L 3947 2898 mt 3947 2970 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3918 2905 5 MP DP gs 899 389 5357 4228 MR c np gr 4404 1731 mt 4476 1731 L 4440 1695 mt 4440 1767 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4411 1702 5 MP DP gs 899 389 5357 4228 MR c np gr 4897 1072 mt 4969 1072 L 4933 1036 mt 4933 1108 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4904 1043 5 MP DP gs 899 389 5357 4228 MR c np gr 5535 2553 mt 5607 2553 L 5571 2517 mt 5571 2589 L gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 5542 2524 5 MP DP gs 899 389 5357 4228 MR c np gr end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 547 1486 a currentpoint currentpoint translate 1 0.33821 div 1 0.33821 div scale neg exch neg exch translate 547 1486 a 1997 1486 a currentpoint currentpoint translate 0.33821 0.33821 scale neg exch neg exch translate 1997 1486 a @beginspecial 67 @llx 212 @lly 546 @urx 583 @ury 4790 @rwi @setspecial %%BeginDocument: mat2.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: MATLAB, The Mathworks, Inc. %%Title: D:\Matlab\AAAFaceGUI\Manual\mat2.eps %%CreationDate: 05/16/2002 12:02:19 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 67 212 546 583 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 67 212 546 583 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 594 341 5747 4448 rc 85 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5187 rf 6 w 0 4227 5356 0 0 -4227 899 4616 4 MP PP -5356 0 0 4227 5356 0 0 -4227 899 4616 5 MP stroke 4 w DO SO 6 w 0 sg 899 4616 mt 6255 4616 L 899 4616 mt 899 389 L 1086 4616 mt 1086 4562 L %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 916 4761 mt (-250) s 1584 4616 mt 1584 4562 L 1414 4761 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3704 1066 L 3668 1030 mt 3668 1102 L 4723 2996 mt 4795 2996 L 4759 2960 mt 4759 3032 L 3140 705 mt 3212 705 L 3176 669 mt 3176 741 L 1173 782 mt 1245 782 L 1209 746 mt 1209 818 L 2338 3284 mt 2410 3284 L 2374 3248 mt 2374 3320 L 4142 2237 mt 4214 2237 L 4178 2201 mt 4178 2273 L 2750 840 mt 2822 840 L 2786 804 mt 2786 876 L 4083 4267 mt 4155 4267 L 4119 4231 mt 4119 4303 L 3350 3706 mt 3422 3706 L 3386 3670 mt 3386 3742 L 1636 2482 mt 1708 2482 L 1672 2446 mt 1672 2518 L 4746 3459 mt 4818 3459 L 4782 3423 mt 4782 3495 L 4445 3067 mt 4517 3067 L 4481 3031 mt 4481 3103 L 2451 3349 mt 2523 3349 L 2487 3313 mt 2487 3385 L 3694 3332 mt 3766 3332 L 3730 3296 mt 3730 3368 L 3704 831 mt 3776 831 L 3740 795 mt 3740 867 L 1586 1124 mt 1658 1124 L 1622 1088 mt 1622 1160 L 2457 3138 mt 2529 3138 L 2493 3102 mt 2493 3174 L 3445 4087 mt 3517 4087 L 3481 4051 mt 3481 4123 L 3020 3387 mt 3092 3387 L 3056 3351 mt 3056 3423 L gs 899 389 5357 4228 rc gr 4278 1028 mt 4350 1028 L 4314 992 mt 4314 1064 L 2138 3866 mt 2210 3866 L 2174 3830 mt 2174 3902 L 5483 1308 mt 5555 1308 L 5519 1272 mt 5519 1344 L 4289 3497 mt 4361 3497 L 4325 3461 mt 4325 3533 L 6194 3915 mt 6266 3915 L 6230 3879 mt 6230 3951 L 1576 4616 mt 1648 4616 L 1612 4580 mt 1612 4652 L 5067 3013 mt 5139 3013 L 5103 2977 mt 5103 3049 L 2598 1278 mt 2670 1278 L 2634 1242 mt 2634 1314 L 3163 2634 mt 3235 2634 L 3199 2598 mt 3199 2670 L 1948 4034 mt 2020 4034 L 1984 3998 mt 1984 4070 L 4268 2616 mt 4340 2616 L 4304 2580 mt 4304 2652 L 5391 2790 mt 5463 2790 L 5427 2754 mt 5427 2826 L 4569 3599 mt 4641 3599 L 4605 3563 mt 4605 3635 L 3178 3478 mt 3250 3478 L 3214 3442 mt 3214 3514 L 5409 2980 mt 5481 2980 L 5445 2944 mt 5445 3016 L 2698 2515 mt 2770 2515 L 2734 2479 mt 2734 2551 L 3075 3458 mt 3147 3458 L 3111 3422 mt 3111 3494 L 1906 1631 mt 1978 1631 L 1942 1595 mt 1942 1667 L 5511 3191 mt 5583 3191 L 5547 3155 mt 5547 3227 L 3492 1482 mt 3564 1482 L 3528 1446 mt 3528 1518 L 3951 3384 mt 4023 3384 L 3987 3348 mt 3987 3420 L 2610 3271 mt 2682 3271 L 2646 3235 mt 2646 3307 L 2937 3378 mt 3009 3378 L 2973 3342 mt 2973 3414 L 5870 3397 mt 5942 3397 L 5906 3361 mt 5906 3433 L 3546 2728 mt 3618 2728 L 3582 2692 mt 3582 2764 L 4029 949 mt 4101 949 L 4065 913 mt 4065 985 L 2948 1994 mt 3020 1994 L 2984 1958 mt 2984 2030 L 1638 1897 mt 1710 1897 L 1674 1861 mt 1674 1933 L 1479 1689 mt 1551 1689 L 1515 1653 mt 1515 1725 L 3544 1041 mt 3616 1041 L 3580 1005 mt 3580 1077 L 1869 3704 mt 1941 3704 L 1905 3668 mt 1905 3740 L 5380 1267 mt 5452 1267 L 5416 1231 mt 5416 1303 L 3341 2146 mt 3413 2146 L 3377 2110 mt 3377 2182 L 5239 2548 mt 5311 2548 L 5275 2512 mt 5275 2584 L 3266 4023 mt 3338 4023 L 3302 3987 mt 3302 4059 L 3426 3089 mt 3498 3089 L 3462 3053 mt 3462 3125 L 2755 1760 mt 2827 1760 L 2791 1724 mt 2791 1796 L 4707 3062 mt 4779 3062 L 4743 3026 mt 4743 3098 L 2016 3166 mt 2088 3166 L 2052 3130 mt 2052 3202 L 3716 3587 mt 3788 3587 L 3752 3551 mt 3752 3623 L 4331 1397 mt 4403 1397 L 4367 1361 mt 4367 1433 L 4022 1852 mt 4094 1852 L 4058 1816 mt 4058 1888 L 2496 4198 mt 2568 4198 L 2532 4162 mt 2532 4234 L 2248 1660 mt 2320 1660 L 2284 1624 mt 2284 1696 L 1997 2317 mt 2069 2317 L 2033 2281 mt 2033 2353 L 2605 3997 mt 2677 3997 L 2641 3961 mt 2641 4033 L 3593 1972 mt 3665 1972 L 3629 1936 mt 3629 2008 L 2398 1517 mt 2470 1517 L 2434 1481 mt 2434 1553 L 4914 2787 mt 4986 2787 L 4950 2751 mt 4950 2823 L 4355 3368 mt 4427 3368 L 4391 3332 mt 4391 3404 L 4891 1714 mt 4963 1714 L 4927 1678 mt 4927 1750 L 5919 3097 mt 5991 3097 L 5955 3061 mt 5955 3133 L 2982 3389 mt 3054 3389 L 3018 3353 mt 3018 3425 L 5206 909 mt 5278 909 L 5242 873 mt 5242 945 L 4674 1367 mt 4746 1367 L 4710 1331 mt 4710 1403 L 4122 1533 mt 4194 1533 L 4158 1497 mt 4158 1569 L 3166 2392 mt 3238 2392 L 3202 2356 mt 3202 2428 L 5361 2958 mt 5433 2958 L 5397 2922 mt 5397 2994 L 5144 2889 mt 5216 2889 L 5180 2853 mt 5180 2925 L 4280 3567 mt 4352 3567 L 4316 3531 mt 4316 3603 L 3504 2666 mt 3576 2666 L 3540 2630 mt 3540 2702 L 3679 1805 mt 3751 1805 L 3715 1769 mt 3715 1841 L 2063 1942 mt 2135 1942 L 2099 1906 mt 2099 1978 L 3260 2344 mt 3332 2344 L 3296 2308 mt 3296 2380 L 4971 2356 mt 5043 2356 L 5007 2320 mt 5007 2392 L 2007 4283 mt 2079 4283 L 2043 4247 mt 2043 4319 L 3097 2780 mt 3169 2780 L 3133 2744 mt 3133 2816 L 3312 3902 mt 3384 3902 L 3348 3866 mt 3348 3938 L 4744 2534 mt 4816 2534 L 4780 2498 mt 4780 2570 L 4363 1263 mt 4435 1263 L 4399 1227 mt 4399 1299 L 4200 2677 mt 4272 2677 L 4236 2641 mt 4236 2713 L 2334 1824 mt 2406 1824 L 2370 1788 mt 2370 1860 L 4792 1060 mt 4864 1060 L 4828 1024 mt 4828 1096 L 1489 3939 mt 1561 3939 L 1525 3903 mt 1525 3975 L 4633 2494 mt 4705 2494 L 4669 2458 mt 4669 2530 L 4423 3891 mt 4495 3891 L 4459 3855 mt 4459 3927 L 2243 2050 mt 2315 2050 L 2279 2014 mt 2279 2086 L 2298 1998 mt 2370 1998 L 2334 1962 mt 2334 2034 L 4682 1425 mt 4754 1425 L 4718 1389 mt 4718 1461 L 1956 1436 mt 2028 1436 L 1992 1400 mt 1992 1472 L 5752 4029 mt 5824 4029 L 5788 3993 mt 5788 4065 L 3692 1376 mt 3764 1376 L 3728 1340 mt 3728 1412 L 4806 2786 mt 4878 2786 L 4842 2750 mt 4842 2822 L 3196 1152 mt 3268 1152 L 3232 1116 mt 3232 1188 L 1731 1317 mt 1803 1317 L 1767 1281 mt 1767 1353 L 3174 2899 mt 3246 2899 L 3210 2863 mt 3210 2935 L 4176 2312 mt 4248 2312 L 4212 2276 mt 4212 2348 L 2893 1264 mt 2965 1264 L 2929 1228 mt 2929 1300 L 4440 4204 mt 4512 4204 L 4476 4168 mt 4476 4240 L 3827 3327 mt 3899 3327 L 3863 3291 mt 3863 3363 L 2245 2574 mt 2317 2574 L 2281 2538 mt 2281 2610 L 4900 3286 mt 4972 3286 L 4936 3250 mt 4936 3322 L 4598 3009 mt 4670 3009 L 4634 2973 mt 4634 3045 L 2818 3374 mt 2890 3374 L 2854 3338 mt 2854 3410 L 4003 3167 mt 4075 3167 L 4039 3131 mt 4039 3203 L 3703 1193 mt 3775 1193 L 3739 1157 mt 3739 1229 L 2376 1528 mt 2448 1528 L 2412 1492 mt 2412 1564 L 3182 2841 mt 3254 2841 L 3218 2805 mt 3218 2877 L 3865 3909 mt 3937 3909 L 3901 3873 mt 3901 3945 L 3541 3116 mt 3613 3116 L 3577 3080 mt 3577 3152 L gs 899 389 5357 4228 rc /c9 { 0.000000 0.000000 1.000000 sr} bdef c9 -153 421 4467 607 2 MP stroke 1028 -400 1146 4266 2 MP stroke -398 175 5917 1133 2 MP stroke 298 -225 4027 3722 2 MP stroke 115 -484 6115 4399 2 MP stroke 442 206 1170 4410 2 MP stroke 65 -445 5038 3458 2 MP stroke 215 432 2419 846 2 MP stroke 225 99 2974 2535 2 MP stroke 873 -227 1111 4261 2 MP stroke 99 18 4205 2598 2 MP stroke 0 -151 5427 2941 2 MP stroke 212 -26 4393 3625 2 MP stroke 843 -597 2371 4075 2 MP stroke 4 -267 5441 3247 2 MP stroke 538 24 2196 2491 2 MP stroke 349 28 2762 3430 2 MP stroke 839 399 1103 1232 2 MP stroke -9 -428 5556 3619 2 MP stroke 247 316 3281 1166 2 MP stroke 347 -183 3640 3567 2 MP stroke 463 14 2183 3257 2 MP stroke 580 -193 2393 3571 2 MP stroke 101 -165 5805 3562 2 MP stroke 413 -181 3169 2909 2 MP stroke -133 422 4198 527 2 MP stroke 592 123 2392 1871 2 MP stroke 517 337 1157 1560 2 MP stroke 457 434 1058 1255 2 MP stroke 33 479 3547 562 2 MP stroke 586 7 1319 3697 2 MP stroke -409 207 5825 1060 2 MP stroke 504 21 2873 2125 2 MP stroke -64 -171 5339 2719 2 MP stroke 374 4 2928 4019 2 MP stroke 374 -114 3088 3203 2 MP stroke 425 277 2366 1483 2 MP stroke 145 -112 4598 3174 2 MP stroke 969 -176 1083 3342 2 MP stroke 343 -75 3409 3662 2 MP stroke -68 263 4435 1134 2 MP stroke 103 115 3955 1737 2 MP stroke 407 89 2125 4109 2 MP stroke 626 352 1658 1308 2 MP stroke 456 240 1577 2077 2 MP stroke 410 50 2231 3947 2 MP stroke 213 124 3416 1848 2 MP stroke 703 409 1731 1108 2 MP stroke 88 -57 4862 2844 2 MP stroke 203 1 4188 3367 2 MP stroke -208 66 5135 1648 2 MP stroke -71 -349 6026 3446 2 MP stroke 323 74 2695 3315 2 MP stroke -429 306 5671 603 2 MP stroke -215 244 4925 1123 2 MP stroke -79 197 4237 1336 2 MP stroke 295 85 2907 2307 2 MP stroke 67 -76 5330 3034 2 MP stroke 50 -74 5130 2963 2 MP stroke 248 -49 4068 3616 2 MP stroke 639 -312 2901 2978 2 MP stroke 171 149 3544 1656 2 MP stroke 379 311 1720 1631 2 MP stroke 286 85 3010 2259 2 MP stroke -96 -292 5103 2648 2 MP stroke 460 124 1583 4159 2 MP stroke 802 -301 2331 3081 2 MP stroke 371 17 2977 3885 2 MP stroke 22 -156 4758 2690 2 MP stroke -164 298 4563 965 2 MP stroke 111 11 4125 2666 2 MP stroke 382 321 1988 1503 2 MP stroke -317 306 5145 754 2 MP stroke 449 183 1076 3756 2 MP stroke 38 -56 4631 2550 2 MP stroke 313 -89 4146 3980 2 MP stroke 550 244 1729 1806 2 MP stroke 471 245 1863 1753 2 MP stroke -203 210 4921 1215 2 MP stroke 395 475 1597 961 2 MP stroke 279 -141 5509 4170 2 MP stroke 60 310 3668 1066 2 MP stroke 83 -210 4759 2996 2 MP stroke 56 447 3176 705 2 MP stroke 558 535 1209 782 2 MP stroke 836 -385 2374 3284 2 MP stroke 34 75 4178 2237 2 MP stroke 143 424 2786 840 2 MP stroke 357 -63 4119 4267 2 MP stroke 477 -379 3386 3706 2 MP stroke 609 92 1672 2482 2 MP stroke 154 -173 4782 3459 2 MP stroke 153 -58 4481 3067 2 MP stroke 367 25 2487 3349 2 MP stroke 309 -165 3730 3332 2 MP stroke -1 362 3740 831 2 MP stroke 790 404 1622 1124 2 MP stroke 725 -297 2493 3138 2 MP stroke 420 -178 3481 4087 2 MP stroke 521 -271 3056 3387 2 MP stroke -8 -152 -108 126 4422 902 3 MP stroke -284 220 394 -43 1780 3909 3 MP stroke 107 -92 -155 24 5674 1284 3 MP stroke -68 100 130 -48 4195 3545 3 MP stroke 26 153 105 -150 6125 4065 3 MP stroke -174 -30 118 106 1494 4510 3 MP stroke 40 153 83 -141 5020 3154 3 MP stroke -131 -124 11 161 2623 1117 3 MP stroke -88 -13 60 52 3139 2582 3 MP stroke -257 150 319 0 1665 4034 3 MP stroke -35 3 30 15 4274 2601 3 MP stroke 21 50 21 -50 5406 2840 3 MP stroke -66 27 73 10 4532 3589 3 MP stroke -196 269 361 -125 2853 3603 3 MP stroke 36 89 38 -88 5407 3068 3 MP stroke -181 38 174 54 2560 2461 3 MP stroke -119 21 112 39 2999 3419 3 MP stroke -332 -59 222 203 1720 1428 3 MP stroke 62 141 56 -142 5491 3333 3 MP stroke -125 -83 38 125 3490 1357 3 MP stroke -89 90 139 -31 3848 3415 3 MP stroke -155 35 151 44 2495 3227 3 MP stroke -165 113 218 -14 2755 3392 3 MP stroke -10 63 56 -46 5850 3443 3 MP stroke -112 96 161 -25 3421 2753 3 MP stroke -15 -151 -102 128 4167 821 3 MP stroke -212 10 178 92 2806 1902 3 MP stroke -217 -67 124 155 1550 1742 3 MP stroke -211 -104 91 182 1424 1507 3 MP stroke -77 -156 -55 161 3635 880 3 MP stroke -194 48 192 53 1713 3651 3 MP stroke 107 -103 -164 33 5580 1234 3 MP stroke -169 36 164 50 3213 2096 3 MP stroke 45 51 2 -62 5273 2610 3 MP stroke -124 31 123 33 3179 3990 3 MP stroke -108 70 139 -6 3323 3095 3 MP stroke -178 -55 102 128 2689 1632 3 MP stroke -33 50 63 -25 4680 3087 3 MP stroke -296 141 344 25 1708 3141 3 MP stroke -102 54 123 5 3629 3582 3 MP stroke -14 -93 -59 81 4426 1316 3 MP stroke -50 -29 18 47 4040 1805 3 MP stroke -147 6 122 64 2410 4134 3 MP stroke -255 -62 158 170 2126 1490 3 MP stroke -184 -40 118 119 1915 2198 3 MP stroke -142 19 128 52 2513 3945 3 MP stroke -88 -22 53 59 3576 1913 3 MP stroke -288 -75 176 196 2258 1321 3 MP stroke -22 27 37 -11 4913 2798 3 MP stroke -67 17 67 18 4324 3350 3 MP stroke 59 -40 -78 4 5005 1710 3 MP stroke 72 109 24 -121 5931 3218 3 MP stroke -117 3 96 52 2922 3337 3 MP stroke 99 -138 -184 64 5426 845 3 MP stroke 38 -99 -105 62 4815 1305 3 MP stroke -1 -72 -53 59 4211 1474 3 MP stroke -109 -3 86 53 3116 2339 3 MP stroke -12 31 33 -20 5364 2978 3 MP stroke -6 29 27 -20 5153 2909 3 MP stroke -75 37 89 5 4227 3562 3 MP stroke -168 158 254 -48 3286 2714 3 MP stroke -77 -34 36 64 3679 1741 3 MP stroke -168 -70 82 135 2017 1807 3 MP stroke -106 -3 83 52 3213 2292 3 MP stroke 72 88 8 -105 4999 2461 3 MP stroke -169 -1 134 80 1909 4203 3 MP stroke -223 168 306 -30 2827 2810 3 MP stroke -125 26 120 37 3228 3865 3 MP stroke 14 53 29 -49 4751 2583 3 MP stroke 13 -112 -95 84 4494 1179 3 MP stroke -38 6 35 13 4201 2664 3 MP stroke -170 -73 82 139 2288 1685 3 MP stroke 62 -129 -147 74 4975 986 3 MP stroke -173 -21 123 99 1402 3840 3 MP stroke -5 22 20 -15 4649 2509 3 MP stroke -91 57 115 -3 4344 3894 3 MP stroke -215 -33 147 128 2132 1922 3 MP stroke -189 -41 122 121 2212 1877 3 MP stroke 38 -87 -96 52 4814 1373 3 MP stroke -196 -123 65 191 1927 1245 3 MP stroke -72 71 111 -23 5677 4052 3 MP stroke -63 -97 -23 108 3751 1268 3 MP stroke 2 76 56 -62 4786 2848 3 MP stroke -80 -142 -43 152 3275 1000 3 MP stroke -258 -129 110 225 1657 1092 3 MP stroke -223 199 329 -55 2881 2954 3 MP stroke -21 -22 1 27 4211 2285 3 MP stroke -106 -128 -11 152 2940 1112 3 MP stroke -109 51 126 10 4350 4194 3 MP stroke -105 166 210 -84 3653 3411 3 MP stroke -214 22 188 82 2093 2492 3 MP stroke -27 71 75 -44 4861 3330 3 MP stroke -42 32 59 -6 4575 3015 3 MP stroke -125 24 118 39 2736 3335 3 MP stroke -79 81 124 -28 3915 3195 3 MP stroke -50 -119 -51 119 3790 1074 3 MP stroke -316 -66 205 202 2207 1326 3 MP stroke -198 160 280 -36 2938 2877 3 MP stroke -114 94 163 -22 3738 3931 3 MP stroke -135 134 209 -45 3368 3161 3 MP stroke c8 3992 -4228 1639 4617 2 MP stroke 5357 -2620 899 3264 2 MP stroke 5357 -1257 899 2310 2 MP stroke 992 4228 4402 389 2 MP stroke 935 -4228 3942 4617 2 MP stroke 1485 -4227 3527 4616 2 MP stroke 2505 -4229 2756 4617 2 MP stroke 2307 -4228 2924 4617 2 MP stroke 2454 3032 3802 389 2 MP stroke 5357 -384 899 1700 2 MP stroke gr end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 1997 1486 a currentpoint currentpoint translate 1 0.33821 div 1 0.33821 div scale neg exch neg exch translate 1997 1486 a 1176 1585 a FQ(\(a\))1354 b(\(b\))515 1763 y(Figure)69 b(4.1:)125 b Ft(Output)70 b(of)h FO(torr)p 1698 1763 25 4 v 29 w(display)p 2077 1763 V 28 w(epipoles)p Ft(:)125 b(\(a\))69 b(A)i(set)g(of)g(syn-)515 1863 y(thetic)88 b(matc)o(hes)f(g)o(ener)o(ated)f(by)i FO(torr)p 1937 1863 V 29 w(gen)p 2116 1863 V 29 w(2view)p 2395 1863 V 29 w(matches)p Ft(,)104 b(displayed)87 b(by)515 1963 y FO(torr)p 720 1963 V 29 w(display)p 1099 1963 V 28 w(matches)p Ft(.)96 b(The)44 b(squar)m(es)g(show)g(using)g(input)f (points,)49 b(with)c(corr)m(e-)515 2062 y(sponding)18 b(epipolar)h(lines)i(shown)f(in)g(\(b\))515 2321 y FQ(epipolar)31 b(line)h(will)h(appear)e(in)h(the)g(second)g(image)g(consistent)f(with) i FO(F)f FQ(and)g(corresponding)515 2421 y(to)c(the)f(point)g(click)o (ed)g(on.)47 b(The)27 b(function)f(uses)i FO(ginput)f FQ(and)g(so)h(input)f(is)i(terminated)d(by)h(a)515 2521 y(carriage)19 b(return.)24 b(An)c(e)o(xample)f(of)g(the)i(output)e(is)i (sho)n(wn)e(in)i(\002gure)e(4.1.)515 2694 y FO(function)48 b(torr_display_epipoles\(F,matches,)c(m3\))515 2914 y FM(Input:)639 3073 y FA(\017)d FO(F)p FQ(,)22 b(the)g FI(3)e FA(\002)f FI(3)j FQ(fundamental)e(matrix)h FL(F)p FQ(,)i(for)e(which)h(epipolar)e(lines)j(are)f(to)g(be)g(displayed)722 3172 y(\(if)e(you)g(are)g(only)f(interested)h(in)g(displaying)f(the)h (matches,)g(this)g(can)g(be)g(arbitrary\).)639 3335 y FA(\017)41 b FO(matches)p FQ(,)19 b(the)i FH(n)d FA(\002)g FI(4)i FQ(match)g(array)-5 b(.)639 3497 y FA(\017)41 b FO(m3)21 b FQ(the)f(third)f(homogeneous)e(coordinate.)515 3671 y FO(function)48 b(torr_compare_epipoles\(Fmat1,Fmat2,ma)o(tches)o (,)c(m3\))515 3965 y FM(Input:)639 4124 y FA(\017)d FO(Fmat1,)49 b(Fmat2)p FQ(,)20 b(tw)o(o)g FI(3)e FA(\002)g FI(3)i FQ(fundamental)e(matrices)i(to)g(be)h(compared.)639 4286 y FA(\017)41 b FO(matches)p FQ(,)19 b(the)i FH(n)d FA(\002)g FI(4)i FQ(match)g(array)-5 b(.)639 4449 y FA(\017)41 b FO(m3)21 b FQ(the)f(third)f(homogeneous)e(coordinate.)639 4608 y(When)26 b(this)h(function)d(is)j(called,)g(tw)o(o)f(\002gures)g (appear)f(sho)n(wing)g(the)h(disparity)f(v)o(ectors)g(of)515 4707 y(the)e(matches.)33 b(Click)23 b(in)h(\002gure)e(1)h(and)g(tw)o(o) g(epipolar)f(lines)h(appear)f(in)h(the)g(second)f(\002gure,)h(one)515 4807 y(in)e(green)g(\(for)f FO(Fmat1)p FQ(\))h(and)g(one)g(in)h(red)f (\()p FO(Fmat2)p FQ(\).)28 b(This)21 b(function)f(is)i(particularly)e (useful)h(for)515 4907 y(comparing)e(the)i(estimated)g FL(F)h FQ(to)g(the)f(ground)e(truth)i(\(as)h(in)g FO(torr)p 2494 4907 V 29 w(test)p 2723 4907 V 29 w(F)p FQ(\))f(or)g(in)h (comparing)515 5006 y(the)e(result)g(of)g(tw)o(o)h(dif)n(ferent)d (algorithms)h(for)h(estimating)f FL(F)p FQ(.)p eop %%Page: 18 20 18 19 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(18)515 523 y FD(4.2)119 b(An)30 b(Ov)o(er)o(view)g(of)g (methods)f(to)h(estimate)f Fo(F)515 709 y FQ(This)c(chapter)f (introduces)f(the)i(elementary)e(concepts)h(necessary)g(for)g(accurate) g(estimation)h(of)515 808 y(the)j(fundamental)d(matrix)j FL(F)g FQ(which)g(encapsulates)f(all)h(the)g(information)e(on)h(camera) h(motion)515 908 y(and)17 b(camera)g(parameters)g(a)n(v)n(ailable)h (from)f(a)h(gi)n(v)o(en)f(set)i(of)e(point)h(correspondences)c([19)o(,) k(12)o(].)25 b(In)515 1007 y(subsequent)e(chapters)g(it)i(is)g(sho)n (wn)f(that)g(the)h(fundamental)d(matrix)h(can)h(be)h(used)f(to)g(guide) g(the)515 1107 y(feature)c(matching)h(process)g(and)g(to)h(initialize)f (motion)g(se)o(gmentation.)27 b(The)22 b(fundamental)d(ma-)515 1207 y(trix)g(has)g(also)h(been)f(used)g(to)g(initialize)h(structure.)j (This)d(no)n(w)e(near)h(ubiquitous)f(emplo)o(yment)f(of)515 1306 y(the)i(fundamental)e(matrix)h(in)h(vision)g(algorithms)f (prompted)f(the)i(content)f(of)h(this)h(chapter)-5 b(.)24 b(Spe-)515 1406 y(ci\002c)k(issues)g(that)g(ha)n(v)o(e)f(not)h(been)e (seriously)h(analysed)g(pre)n(viously)f(are)h(highlighted\227issues)515 1506 y(such)21 b(as)h(rob)n(ustness)e(to)i(gross)f(outliers)f(and)h (the)g(detection)f(of)h(de)o(generac)o(y)d(in)k(the)f(data\226and)e(in) 515 1605 y(subsequent)24 b(chapters)h(the)o(y)g(are)h(addressed.)41 b(It)26 b(will)g(be)g(seen)g(that)g(without)f(address)g(to)h(these)515 1705 y(issues,)21 b(the)f(estimate)g(of)g FL(F)h FQ(can)f(be)g(v)o(ery) f(poor)-5 b(.)639 1804 y(In)28 b(Section)f(4.3)g(the)h(basic)g (terminology)d(that)j(is)g(used)g(throughout)d(this)j(report)e(is)j(gi) n(v)o(en.)515 1904 y(Section)21 b(4.4)h(describes)f(the)h(fundamental)d (matrix)j(and)f(the)h(distinction)f(between)g(the)h(intrinsic)515 2004 y(and)d(e)o(xtrinsic)g(parameters)f(of)i(a)g(pair)f(of)h(cameras.) k(Methods)19 b(for)g(estimating)g(the)h(fundamental)515 2103 y(matrix)j(are)g(then)g(analysed.)35 b(There)22 b(are)i(se)n(v)o(eral)f(k)o(e)o(y)g(points)g(to)h(consider:)31 b(\002rst)24 b(the)g(choice)f(of)515 2203 y(error)c(function)g(to)h(be) h(minimized,)e(secondly)g(the)h(choice)g(of)g(parameterization)e(for)i (the)g(funda-)515 2303 y(mental)h(matrix)h(to)g(enforce)e(the)i (constraint)f(that)h(the)g(determinant)e(is)j(zero,)f(and)f(thirdly)g (which)515 2402 y(estimator)26 b(should)g(be)h(used.)44 b(The)26 b(latter)h(is)h(determined)d(to)i(a)g(lar)o(ge)f(e)o(xtent)g (by)g(the)h(\002rst)h(tw)o(o.)515 2502 y(Three)22 b(classes)i(of)f (estimators)g(are)g(described.)32 b(An)23 b(unnormalized)d(least)k (squares)f(approach)e(is)515 2601 y(described)27 b(in)i(Section)f(4.5,) i(an)f(iterati)n(v)o(e)f(re)n(weighted)f(least)j(squares)e(method)f (adapted)h(from)515 2701 y(Sampson)22 b([41)o(])i(and)f(used)g(by)g(W) -7 b(eng)24 b Ft(et)g(al.)48 b FQ([60)o(])23 b(is)i(described)d(in)i (Section)f(4.7,)h(and)f(gradient)515 2801 y(descent)h(estimators)i(are) f(set)h(out)f(in)g(Section)g(4.8.)39 b(T)-6 b(esting)25 b(methodology)d(and)j(functions)f(are)515 2900 y(described)19 b(in)h(Section)g(4.10)515 3179 y FD(4.3)119 b(Homogeneous)29 b(and)i(Pr)n(ojecti)o(v)o(e)e(coordinates)515 3364 y FQ(In)f(this)h(section)f(basic)g(concepts)g(and)g(terminology)d(which)j (will)h(be)g(used)f(in)g(the)h(rest)f(of)h(the)515 3464 y(w)o(ork)e(are)g(de\002ned.)46 b(The)28 b(language)e(of)h(projecti)n (v)o(e)f(geometry)g(is)i(adopted,)g(about)e(which)i(the)515 3564 y(interested)20 b(reader)f(may)h(consult)g(Semple)g(and)g (Kneebone)e([43)o(])i(for)g(a)h(te)o(xt)f(book)f(on)h(projecti)n(v)o(e) 515 3663 y(geometry)-5 b(,)15 b(Mundy)g(and)i(Zisserman)f([35)o(])h (for)f(an)h(ele)o(gant)f(joining)g(of)h(projecti)n(v)o(e)e(geometry)g (and)515 3763 y(computer)g(vision,)i(and)f(Kanatani)g([27)o(])h(for)g (a)g(discussion)g(of)f(some)h(of)g(the)g(basic)g(computational)515 3863 y(theory)i(associated)h(with)g(the)g(application)f(of)h(projecti)n (v)o(e)e(geometry)-5 b(.)639 3962 y(V)c(ectors)18 b(will)h(be)g (denoted)e(by)h(boldf)o(ace)f(type:)24 b FL(a)p FQ(,)19 b(and)f(matrices)g(by)g(boldf)o(ace)f(and)h(capitals:)515 4062 y FL(A)p FQ(.)25 b(The)20 b(determinant)f(of)h(a)g(matrix)g(as)h FA(j)p FL(A)p FA(j)p FQ(,)f(the)h(Fr)7 b(\250)-35 b(obenius)18 b(norm)h(of)h(a)h(matrix)f(as)g FA(jj)p FL(A)q FA(jj)p FQ(.)639 4161 y(The)h(image)g(plane)f(is)i(re)o(garded)d(as)i(a)h(2-D)f (projecti)n(v)o(e)e(space:)27 b FA(P)2554 4125 y FG(2)2591 4161 y FQ(,)22 b(a)f(point)g(of)g FA(P)3040 4125 y FG(2)3099 4161 y FQ(is)h(desig-)515 4261 y(nated)f(as)h(a)g(triplet)g FL(x)k FI(=)f(\()p FH(x)1333 4273 y FG(1)1371 4261 y FH(;)14 b(x)1455 4273 y FG(2)1493 4261 y FH(;)g(x)1577 4273 y FG(3)1614 4261 y FI(\))23 b FQ(of)e(real)h(numbers,)e(not)h(all) h(of)g(them)f(equal)g(to)h(zero.)28 b(This)515 4361 y(triplet)18 b(is)i(termed)e(a)h(homogeneous)d(coordinate.)23 b(It)c(is)g (bene\002cial)g(to)g(w)o(ork)f(with)h(homogeneous)515 4460 y(coordinates)e(as)j(this)g(often)f(simpli\002es)h(the)f (mathematical)f(deri)n(v)n(ations.)23 b(T)-7 b(w)o(o)20 b(points)f(of)g FA(P)3220 4424 y FG(2)3278 4460 y FQ(are)515 4560 y(equal)g(if)i(there)f(e)o(xists)g(a)h(non-zero)d(scalar)i FH(\025)h FQ(such)f(that)g FH(x)2204 4572 y FB(i)2255 4560 y FI(=)j FH(\025x)2438 4572 y FB(i)2487 4560 y FQ(for)d FH(i)j FI(=)f(1)p FH(;)14 b(:)g(:)g(:)f FI(3)p FQ(,)20 b(thus)1475 4733 y FI(\()p FH(x)1554 4745 y FG(1)1592 4733 y FH(;)14 b(x)1676 4745 y FG(2)1713 4733 y FH(;)g(x)1797 4745 y FG(3)1835 4733 y FI(\))23 b(=)g FH(\025)p FI(\()p FH(x)2105 4745 y FG(1)2143 4733 y FH(;)14 b(x)2227 4745 y FG(2)2265 4733 y FH(;)g(x)2349 4745 y FG(3)2387 4733 y FI(\))801 b FQ(\(4.1\))515 4907 y(W)-7 b(e)33 b(introduce)c(the)j (notation)f FL(a)44 b FA(\030)g FL(b)33 b FQ(to)e(indicate)g(equality)g (up)h(to)f(a)i(scale)f(f)o(actor)f(when)g(it)515 5006 y(is)g(not)f(ob)o(vious)e(that)j(we)f(are)g(dealing)f(with)i (homogeneous)c(coordinates.)53 b(The)30 b(embedding)p eop %%Page: 19 21 19 20 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(19)515 523 y FA(R)585 487 y FG(2)652 523 y FA(\032)30 b(P)811 487 y FG(2)873 523 y FQ(is)25 b FI(\()p FH(x;)14 b(y)s FI(\))31 b FA(7!)e FI(\()p FH(x;)14 b(y)s(;)g(\020)6 b FI(\))26 b FQ(where)d FL(x)31 b FI(=)e(\()p FH(x;)14 b(y)s FI(\))25 b FQ(is)g(the)f(inhomogeneous)c(image)k(point.) 515 623 y(The)i(constant)g FH(\020)34 b FQ(is)28 b(chosen)e(to)h (elicit)h(the)f(best)g(numerical)e(conditioning)f(of)j(our)f (algorithms,)515 722 y(as)e(e)o(xplained)e(belo)n(w)-5 b(.)34 b(In)23 b(the)j(S)t(A)t(M)g(algorithm)c(its)j(v)n(ariable)d(is)j FO(m3)f FQ(\(the)f(third)g(homogeneous)515 822 y(coordinate.)639 922 y(A)32 b(line)g(in)f(the)g(2-D)g(projecti)n(v)o(e)f(space)h(is)i (also)e(de\002ned)g(by)g(a)g(triplet)h(of)f(numbers)f FL(n)43 b FI(=)515 1021 y(\()p FH(n)597 1033 y FG(1)634 1021 y FH(;)14 b(n)721 1033 y FG(2)758 1021 y FH(;)g(n)845 1033 y FG(3)882 1021 y FI(\))p FQ(,)32 b(not)c(all)i(of)f(them)f(equal) h(to)g(zero.)51 b(The)28 b(line)h(appears)f(on)h(the)g(image)f(plane)h (as)515 1121 y FH(n)565 1133 y FG(1)602 1121 y FH(x)23 b FI(+)f FH(n)809 1133 y FG(2)846 1121 y FH(y)j FI(+)e FH(n)1050 1133 y FG(3)1120 1121 y FI(=)33 b(0)p FQ(.)42 b(If)26 b FH(n)1455 1133 y FG(1)1526 1121 y FI(=)33 b FH(n)1674 1133 y FG(2)1744 1121 y FI(=)g(0)26 b FQ(the)g(line)g(is)h (interpreted)d(to)i(be)f(the)h(ideal)g(line)g(at)515 1220 y(in\002nity)-5 b(,)18 b FH(l)815 1232 y FF(1)885 1220 y FQ(,)h(de\002ned)f(by)h(the)g(set)h FA(P)1589 1184 y FG(2)1627 1220 y FA(nR)1739 1184 y FG(2)1776 1220 y FQ(.)25 b(The)19 b(points)f FL(x)24 b FI(=)e(\()p FH(x;)14 b(y)s(;)g FI(0\))20 b FQ(that)f(lie)g(on)g(this)h(line)f(are)515 1320 y(the)h(points)g(at)h(in\002nity)-5 b(.)24 b(It)d(is)g(important)e 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900 y FG(2)2038 888 y FH(t)2068 900 y FG(1)2266 888 y FI(0)2394 622 y FC(3)2394 772 y(5)3220 789 y FQ(\(4.7\))515 1052 y(which)19 b(allo)n(ws)i(v)o(ector)e (products)g(to)h(be)g(written)g(as:)1657 1222 y FI([)p FL(t)q FI(])1741 1234 y FF(\002)1797 1222 y FL(v)k FI(=)f FL(t)c FA(\002)f FL(v)1072 b FQ(\(4.8\))1578 1346 y FL(v)1630 1312 y FF(>)1686 1346 y FI([)p FL(t)p FI(])1769 1358 y FF(\002)1848 1346 y FI(=)23 b FL(v)d FA(\002)e FL(t)p FH(:)1071 b FQ(\(4.9\))515 1516 y(F)o(or)21 b(an)o(y)h(non-zero)d(v)o (ector)m(,)i FI([)p FL(t)p FI(])1440 1528 y FF(\002)1519 1516 y FQ(has)h(rank)g FI(2)p FQ(.)30 b(Furthermore)20 b(the)i(null)g(space)g(of)g FI([)p FL(t)p FI(])3012 1528 y FF(\002)3091 1516 y FQ(is)h(gener)n(-)515 1615 y(ated)i(by)h(the)f(v) o(ector)g FL(t)h FQ(i.e.)g FL(t)1377 1583 y FF(>)1433 1615 y FI([)p FL(t)p FI(])1516 1627 y FF(\002)1606 1615 y FI(=)33 b([)p FL(t)p FI(])1787 1627 y FF(\002)1843 1615 y FL(t)g FI(=)g FL(0)p FQ(.)42 b(If)25 b FL(a)i 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b(the)f(epipolar)g(geometry)-5 b(.)33 b(The)23 b(fundamental)f(matrix)h(is)h(an)g(e)o(xtension)e(of)i(the)f(the)h (essential)515 3289 y(matrix)d(formulated)e(by)j(Longuet-Higgins)c([31) o(])k(to)f(the)h(case)g(where)f(calibration)g(is)h(unkno)n(wn.)515 3389 y(The)e(case)h(for)e(algorithms)h(that)g(do)g(not)g(require)f (calibration)g(has)i(been)e(strongly)h(made)f(in)i([12)o(].)515 3488 y(Camera)26 b(calibration)g(is)i(at)f(best)h(dif)n(\002cult,)g (possibly)e(introducing)e(correlated)i(errors)g(into)h(the)515 3588 y(system,)20 b(and)g(at)g(w)o(orst)h(it)g(is)g(impossible.)j(The)c (fundamental)e(matrix)h(is)i(no)n(w)f(de\002ned.)515 3757 y FM(Theor)o(em)g(1)41 b Ft([12)o(,)22 b(19)o(])g(Given)e(an)h (uncalibr)o(ated)e(camer)o(a)h(let)i(the)f(set)h(of)f(homo)o(g)o (eneous)e(ima)o(g)o(e)515 3857 y(points)28 b FA(f)p FL(x)839 3869 y FB(i)866 3857 y FA(g)p FH(;)14 b(i)38 b FI(=)g(1)p FH(;)14 b(:)g(:)g(:)f(n;)29 b Ft(be)g(tr)o(ansformed)f(to)g(the)h(set)g FA(f)p FL(x)2385 3827 y FF(0)2385 3878 y FB(i)2413 3857 y FA(g)f Ft(on)g(the)h(ima)o(g)o(e)f(plane)g(by)h(the)515 3956 y(motion)23 b(par)o(ameter)o(s)i FA(f)p FL(R)o FH(;)14 b FL(t)p FA(g)24 b Ft(suc)o(h)g(that)h FL(t)31 b FA(6)p FI(=)g FL(0)o Ft(.)39 b(Then)24 b(ther)m(e)h(e)n(xists)g(a)g FI(3)c FA(\002)g FI(3)k Ft(matrix)g FL(F)g Ft(suc)o(h)515 4056 y(that)1737 4156 y FL(x)1788 4096 y Fn(0)1810 4121 y FF(>)1788 4176 y FB(i)1866 4156 y FL(Fx)1977 4168 y FB(i)2027 4156 y FI(=)e(0)1021 b FQ(\(4.10\))515 4297 y Ft(for)20 b(all)h FH(i)p Ft(.)515 4466 y FM(Pr)o(oof)o(.)i FQ(Appendix)18 b(A.)639 4566 y(The)30 b(fundamental)d(matrix)i(may)g (be)g(written)h(in)g(terms)f(of)h(the)f(intrinsic)h(matrix)e(and)i(the) 515 4666 y(essential)20 b(matrix)1663 4765 y FL(F)j FI(=)f FL(C)1902 4729 y FF(\000>)2010 4765 y FL(EC)2142 4729 y FF(\000)p FG(1)3178 4765 y FQ(\(4.11\))515 4907 y(pro)o(v)o(ed)17 b(in)k(Appendix)d(A.)25 b(Because)20 b FL(E)h FQ(is)g(rank)e(tw)o(o)h ([31)o(],)g(the)g(fundamental)e(matrix)h(at)i(most)f(of)515 5006 y(rank)f(tw)o(o.)25 b(This)c(leads)f(to)g(the)g(follo)n(wing)f (lemma.)p eop %%Page: 22 24 22 23 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(22)515 523 y FM(Lemma)21 b(1)41 b Ft(If)25 b FL(F)h Ft(is)g(a)f(fundamental)e(matrix)i(corr)m(esponding)e(to)j(a)f (pair)g(of)g(ima)o(g)o(e)g(and)f FL(x)i Ft(is)g(a)515 623 y(point)17 b(in)i(the)f(\002r)o(st)h(ima)o(g)o(e)o(,)g(then)e FL(Fx)j Ft(is)f(the)f(epipolar)f(line)i(in)f(the)g(second)g(ima)o(g)o (e)g(corr)m(esponding)515 722 y(to)i FL(x)p Ft(.)639 905 y FQ(The)k(epipoles)f(correspond)e(to)j(projections)e(of)i(the)f (directions)g(of)h(translations)f(in)h(the)f(tw)o(o)515 1005 y(images,)37 b(and)d(can)h(be)f(reco)o(v)o(ered)e(by)i(the)h(left) g(and)f(right)f(nullspace)h(of)g FL(F)q FQ(.)68 b(The)34 b(function)515 1104 y FO(torr)p 720 1104 25 4 v 29 w(get)p 899 1104 V 29 w(right)p 1178 1104 V 29 w(epipole)20 b FQ(is)h(pro)o(vided)c(to)k(calculate)f(the)g(epipole:)714 1270 y FO(\045)50 b(returns)e(epipole)h(such)g(that)g(Fmat1)g(*)g (epipole)g(=)g(0)515 1569 y(function)98 b(epipole)48 b(=)i(torr_get_right_epipole\(Fmat1,m3\))714 1768 y([v,d])f(=)g (eig\(Fmat1\);)714 1968 y(dd)g(=)h([d\(1,1\)\2102,)e(d\(2,2\)\2102,)g (d\(3,3\)\2102];)714 2067 y([Y)h(Index])g(=)h(min\(dd\);)714 2366 y(epipole)e(=)i(v\(:,Index\);)714 2466 y(epipole)e(=)i(epipole)f (*)g(\(m3/epipole\(3\)\);)714 2565 y(\045Fmat1)g(*)g(epipole)639 2831 y FQ(From)24 b FL(F)g FQ(and)g(the)g(image)f(correspondences)e(it) k(is)g(straightforw)o(ard)d(to)i(reco)o(v)o(er)e Ft(pr)l(ojective)515 2931 y FQ(structure)d(as)i(has)f(been)g(pointed)f(out)g(in)i([12)o(,)f (19)o(]:)515 3113 y FM(Theor)o(em)g(2)g(\([12)n(,)h(19)o(]\))41 b Ft(Given)30 b(a)g(set)i(of)e(ima)o(g)o(e)h(corr)m(espondences)d(suf)o (\002cient)h(to)i(determine)515 3213 y(the)18 b(fundamental)f(matrix,)i (the)f(corr)m(esponding)e(world)j(space)g(coor)m(dinates)e(ar)m(e)h (determined)g(up)515 3313 y(to)i(a)g(collineation)f(of)h(pr)l(ojective) g FI(3)p Ft(-space)f FA(P)1858 3276 y FG(3)1895 3313 y Ft(.)515 3495 y FQ(In)j(the)g(rest)h(of)g(the)f(chapter)g(non-rob)n (ust)e(estimators)i(for)g(reco)o(v)o(ering)e(the)i(fundamental)e (matrix)515 3595 y(from)f(corrupted)f(data)i(are)g(re)n(vie)n(wed.)515 3875 y FD(4.5)119 b(Linear)30 b(Methods:)37 b Fo(O)1782 3893 y FL(1)1834 3875 y FD(,)30 b Fo(O)1995 3893 y FL(2)515 4061 y FQ(Implicit)h(in)h(Equation)d(\(4.10\))h(is)j(the)e(assumption)g (that)g(image)g(coordinates)f(are)i(noise)f(free)515 4160 y(quantities.)49 b(Ho)n(we)n(v)o(er)m(,)29 b(in)g(real)g (situations)f(reco)o(v)o(ery)e(of)j(the)f(fundamental)f(matrix)h (becomes)515 4260 y(non-tri)n(vial)17 b(as)j(the)f(image)g(coordinates) e(are)i(perturbed)e(by)i(noise,)g(leading)f(to)h(inconsistenc)o(y)e(in) 515 4359 y(the)j(the)g(set)h(of)f(constraints)g(pro)o(vided)d(by)j (\(4.10\).)639 4459 y(In)25 b(the)f(ensuing)g(sections)h(pre)n(vious)e (w)o(ork)h(on)g(estimation)g(of)h(the)f(fundamental)f(matrix)h(is)515 4559 y(summarised,)19 b(describing)g(a)i(linear)f(method,)f(an)h (iterati)n(v)o(e)g(method)f(and)h(a)h(descent)f(method)f(for)515 4658 y(estimating)27 b FL(F)p FQ(.)48 b(The)27 b(choice)g(of)g(the)h (algorithm)e(is)j(critical,)g(as)f(some)g(perform)d(substantially)515 4758 y(better)d(than)h(others.)33 b(There)23 b(are)g(tw)o(o)g(k)o(e)o (y)g(points)f(in)h(the)h(estimation)e(of)h(the)g(fundamental)e(ma-)515 4858 y(trix.)34 b(First)25 b(is)f(the)f(choice)g(of)g(error)f(function) g(to)i(be)f(minimized:)31 b(this)24 b(in)m(v)n(olv)o(es)e (consideration)515 4957 y(of)k(the)h(assumptions)f(to)g(be)h(made)f (about)g(the)g(noise)h(distrib)n(utions)f(of)g(the)h(data)f(\(and,)h (as)h(will)p eop %%Page: 23 25 23 24 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(23)515 523 y(be)21 b(seen)h(later)g(outlier)f(distrib)n (utions\).)29 b(W)m(ithin)21 b(this)h(section)g(linear)f(methods)g(are) h(used)f(to)h(esti-)515 623 y(mate)c FL(F)p FQ(,)i(which)e(although)f (computational)f(ef)n(\002cient,)i(do)g(not)h(pro)o(vide)d(a)j(maximum) e(lik)o(elihood)515 722 y(solution.)45 b(In)27 b(Sections)h(4.7)e(and)h (4.8)g(tw)o(o)h(dif)n(ferent)d(estimators)i(that)h(purport)d(to)j(gi)n (v)o(e)e(maxi-)515 822 y(mum)c(lik)o(elihood)g(solutions)g(are)i(e)o (xplained.)32 b(Second)22 b(is)i(the)f(choice)g(of)g(parameterization)d (for)515 922 y(the)h(fundamental)e(matrix)h(to)i(enforce)d(the)j (constraint)e(that)h(its)h(determinant)e(is)i(zero.)27 b(It)22 b(will)g(be)515 1021 y(seen,)k(in)g(Section)f(4.8,)g(that)h (there)f(is)h(no)f(ideal)g(parameterization,)f(b)n(ut)h(that)h(an)f (adequate)f(one)515 1121 y(can)c(be)g(de\002ned.)639 1220 y(W)-7 b(ell)21 b(kno)n(wn)d(results)i(in)f(linear)g(re)o (gression)f(are)i(summarised)e(ne)o(xt.)24 b(Re)o(gression)19 b(is)h(used)f(to)515 1320 y(study)f(the)g(relationship)g(between)f (parameters,)h(and)g(linear)g(re)o(gression)f(deals)i(with)g(the)f (class)i(of)515 1420 y(relationships)i(that)h(are)h(linear)f(in)g(the)g (parameters.)34 b(Although)21 b(bilinear)i(in)g(the)g(image)g(coordi-) 515 1519 y(nates)d(estimation)g(of)g FL(F)h FQ(is)g(linear)e(in)i(the)f (parameters.)k(This)c(can)g(be)g(seen)g(by)g(e)o(xpanding)1737 1698 y FL(x)p 1737 1711 51 4 v 1788 1639 a Fn(0)1810 1664 y FF(>)1788 1719 y FB(i)1866 1698 y FL(Fx)p 1926 1711 V 1977 1719 a FB(i)2027 1698 y FI(=)j(0)1021 b FQ(\(4.12\))515 1868 y(to)20 b(gi)n(v)o(e)556 2037 y FH(f)597 2049 y FG(1)634 2037 y FH(x)p 634 2050 48 4 v 682 2003 a FF(0)682 2058 y FB(i)709 2037 y FH(x)p 709 2050 V 757 2058 a FB(i)788 2037 y FI(+)t FH(f)898 2049 y FG(2)934 2037 y FH(x)p 934 2050 V 982 2003 a FF(0)982 2058 y FB(i)1009 2037 y FH(y)p 1009 2067 44 4 v 1053 2074 a FB(i)1084 2037 y FI(+)t FH(f)1194 2049 y FG(3)1231 2037 y FH(x)p 1231 2050 48 4 v -34 x FF(0)1278 2058 y FB(i)1306 2037 y FH(\020)10 b FI(+)t FH(f)1462 2049 y FG(4)1498 2037 y FH(y)p 1498 2067 44 4 v 1542 2003 a FF(0)1542 2074 y FB(i)1569 2037 y FH(x)p 1569 2050 48 4 v 1617 2058 a FB(i)1648 2037 y FI(+)t FH(f)1758 2049 y FG(5)1794 2037 y FH(y)p 1794 2067 44 4 v 1838 2003 a FF(0)1838 2074 y FB(i)1866 2037 y FH(y)p 1866 2067 V 1909 2074 a FB(i)1941 2037 y FI(+)t FH(f)2051 2049 y FG(6)2087 2037 y FH(y)p 2087 2067 V 2131 2003 a FF(0)2131 2074 y FB(i)2158 2037 y FH(\020)g FI(+)t FH(f)2314 2049 y FG(7)2351 2037 y FH(x)p 2351 2050 48 4 v 21 x FB(i)2426 2037 y FH(\020)g FI(+)t FH(f)2582 2049 y FG(8)2618 2037 y FH(y)p 2618 2067 44 4 v 2662 2074 a FB(i)2690 2037 y FH(\020)g FI(+)t FH(f)2846 2049 y FG(9)2882 2037 y FH(\020)2924 2003 y FG(2)2985 2037 y FI(=)22 b(0)p FH(;)41 b FQ(\(4.13\))515 2207 y(where)19 b(the)i(terms)f(of)g(the)g(fundamental)e(matrix)h(are)1546 2457 y FL(F)k FI(=)1717 2290 y FC(2)1717 2439 y(4)1814 2356 y FH(f)1855 2368 y FG(1)1975 2356 y FH(f)2016 2368 y FG(2)2136 2356 y FH(f)2177 2368 y FG(3)1814 2456 y FH(f)1855 2468 y FG(4)1975 2456 y FH(f)2016 2468 y FG(5)2136 2456 y FH(f)2177 2468 y FG(6)1814 2555 y FH(f)1855 2567 y FG(7)1975 2555 y FH(f)2016 2567 y FG(8)2136 2555 y FH(f)2177 2567 y FG(9)2255 2290 y FC(3)2255 2439 y(5)2324 2457 y FH(:)831 b FQ(\(4.14\))639 2720 y(There)15 b(are)g(tw)o(o)g (main)g(approaches)e(to)j(re)o(gression:)21 b(ordinary)13 b(\(or)i(classical\))g(least)h(squares)f([37)o(])515 2820 y(\(common)i(in)i(statistics\))i(in)e(which)g(the)g(error)f(is)i (subsumed)e(into)h(one)g(of)g(the)g(v)n(ariables,)f(and)h(or)n(-)515 2920 y(thogonal)h(re)o(gression)h([17)o(])h(when)g(there)g(is)h(error)f (in)g(all)h(the)g(v)n(ariables.)30 b(F)o(or)22 b(ease)h(of)f(reference) 515 3019 y(classical)i(least)f(squares)g(is)h(referred)d(to)i(as)h FL(O)1886 3031 y FK(1)1951 3019 y FQ(and)f(orthogonal)d(re)o(gression)h (as)j FL(O)3006 3031 y FK(2)3048 3019 y FQ(.)34 b(The)22 b(tw)o(o)515 3119 y(methods)g(are)i(no)n(w)f(outlined)g(and)g(then)g (it)i(is)f(e)o(xplained)e(ho)n(w)h(the)o(y)g(might)h(be)f(used)h(to)g (reco)o(v)o(er)515 3218 y FL(F)d FQ(from)e(image)g(correspondences.)515 3454 y Fs(4.5.1)99 b(Ordinary)25 b(Least)g(Squar)n(es)h(Regr)n(ession:) 31 b Fp(O)2477 3469 y Fm(1)515 3609 y FQ(Consider)19 b(the)h(set)h(of)f FH(n)h FQ(measurement)d(equations)1259 3779 y FH(a)1303 3791 y FB(i)1353 3779 y FI(=)23 b FL(d)1494 3742 y FF(>)1494 3800 y FB(i)1550 3779 y FL(b)p 1550 3792 54 4 v 18 w FI(+)18 b FH(\017)519 b(i)23 b FI(=)f(1)14 b FH(:)g(:)g(:)f(n;)543 b FQ(\(4.15\))515 3949 y(where)29 b FH(a)792 3961 y FB(i)851 3949 y FQ(is)i(a)f(measured)f(scalar)h(with) h(error)m(,)g FH(\017)p FQ(,)h(which)e(is)h(assumed)f(to)g(be)g (Gaussian)g(with)515 4048 y(standard)h(de)n(viation)f FH(\033)s FQ(;)40 b FL(d)1334 4060 y FB(i)1394 4048 y FQ(is)33 b(a)g(kno)n(wn)d FH(p)j FQ(dimensional)d(v)o(ector)h(and)h FL(b)p 2713 4061 V 32 w FQ(is)i(an)e(unkno)n(wn)d FH(p)515 4148 y FQ(dimensional)16 b(v)o(ector)h(of)h(parameters)e(which)i(is)h (to)f(be)g(reco)o(v)o(ered.)j(If)d FH(n)g FQ(is)h(greater)e(then)h 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b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(24)515 523 y Fs(4.5.2)99 b(Orthogonal)24 b(Least)i(Squar)n(es)g(Regr)n(ession:)31 b Fp(O)2566 538 y Fm(2)515 679 y FQ(In)c(ordinary)e(least)j(squares)e(the)i(error)e (is)i(assumed)e(to)i(be)f(in)g(only)g(one)f(coordinate,)h(whereas)515 778 y(in)d(orthogonal)d(least)j(squares)g(it)g(is)h(assumed)e(that)h (all)g(coordinates)e(are)i(measured)f(with)g(error)-5 b(.)515 878 y(Consider)29 b(\002tting)i(a)g(hyperplane)c FL(f)51 b FI(=)41 b(\()p FH(f)1801 890 y FG(1)1839 878 y FH(;)14 b(f)1917 890 y FG(2)1953 878 y FH(;)g(:)g(:)g(:)g(;)g(f)2179 890 y FB(p)2217 878 y FI(\))31 b FQ(through)d(a)j(set)g(of)f FH(n)h FQ(points)f(in)g FA(R)3341 842 y FB(p)515 978 y FQ(with)21 b(coordinates)e FL(z)1130 990 y FB(i)1182 978 y FI(=)24 b(\()p FH(z)1342 990 y FB(i)1365 998 y Fq(1)1402 978 y FH(;)14 b(z)1478 990 y FB(i)1501 998 y Fq(2)1538 978 y FH(;)g(:)g(:)g(:)f(;)h(z)1761 990 y FB(i)1784 998 y Fl(p)1823 978 y FI(\))p FQ(,)22 b(with)f(the)g 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FB(n)1264 2541 y FH(x)1311 2510 y FF(0)1311 2561 y FB(n)1357 2541 y FH(y)1398 2553 y FB(n)1525 2541 y FH(x)1572 2510 y FF(0)1572 2561 y FB(n)1618 2541 y FH(\020)90 b(y)1788 2510 y FF(0)1785 2561 y FB(n)1829 2541 y FH(x)1876 2553 y FB(n)2005 2541 y FH(y)2049 2510 y FF(0)2046 2561 y FB(n)2091 2541 y FH(y)2132 2553 y FB(n)2260 2541 y FH(y)2304 2510 y FF(0)2301 2561 y FB(n)2346 2541 y FH(\020)f(x)2518 2553 y FB(n)2564 2541 y FH(\020)g(y)2730 2553 y FB(n)2775 2541 y FH(\020)h(\020)2943 2510 y FG(2)3022 1871 y FC(3)3022 2017 y(7)3022 2067 y(7)3022 2117 y(7)3022 2167 y(7)3022 2217 y(7)3022 2266 y(7)3022 2316 y(7)3022 2366 y(7)3022 2419 y(5)3178 2237 y FQ(\(4.21\))515 2719 y(where)27 b FL(W)i FQ(is)g(a)f(diagonal)e (matrix)h(of)h(the)f(weights)h(gi)n(v)o(en)e(to)i(each)g(feature)e (correspondence,)515 2818 y(corresponding)i(to)33 b(the)f(in)m(v)o (erse)e(standard)h(de)n(viation)g(of)h(each)f(error)-5 b(.)60 b(\(This)32 b(is)h(assumed)e(to)515 2918 y(be)20 b(homogeneous)e(at)j(present,)e(in)i(the)g(ne)o(xt)f(section)g(its)h (estimation)g(by)f(iterati)n(v)o(ely)f(re)n(weighted)515 3018 y(least)f(squares)f(is)i(e)o(xplained.\))j(If)17 b(the)h(v)n(ariances)f(of)g(the)h(image)f(coordinates)f(are)i(dif)n (ferent)e(along)515 3117 y(the)22 b(tw)o(o)g(ax)o(es)g(e.g.)g(x-axis)f (is)i FH(\033)1450 3087 y FG(2)1447 3138 y FB(x)1512 3117 y FQ(and)f(y-axis)f(is)i FH(\033)2008 3087 y FG(2)2005 3138 y FB(y)2046 3117 y FQ(,)g(the)f(image)g(coordinates)e(are)i (weighted)f(by)515 3217 y(di)n(viding)d(them)i(by)g(their)g(respecti)n (v)o(e)f(v)n(ariances.)639 3316 y(If)24 b FL(M)k FI(=)h FL(Z)991 3280 y FF(>)1047 3316 y FL(Z)24 b FQ(is)g(the)g(moment)e (matrix)h(then)g(the)h(estimate)f FL(f)34 b FQ(minimizes)23 b FL(f)2935 3279 y FF(>)2991 3316 y FL(Mf)33 b FQ(subject)515 3416 y(to)20 b FL(f)638 3379 y FF(>)694 3416 y FL(J)q(f)32 b FI(=)23 b FQ(constant)n(,)e(where)1490 3599 y FL(J)i FI(=)g FQ(diag)o FI(\(1)p FH(;)14 b FI(1)p FH(;)g FI(1)p FH(;)g(:)g(:)g(:)e(;)i FI(1)p FH(;)g FI(0\))20 b FH(;)774 b FQ(\(4.22\))515 3781 y(is)26 b(a)g(normalization)e(chosen)h(to)h (realise)f(a)i(solution)d(from)h(the)h(equi)n(v)n(alence)d(class)k(of)e (solution)515 3881 y(with)31 b(dif)n(ferent)f(scalings.)58 b(This)31 b(estimate)h(is)g(gi)n(v)o(en)e(by)h(the)g(eigen)m(v)o(ector) e(corresponding)e(to)515 3981 y(the)g(minimum)e(eigen)m(v)n(alue)g(of)i (the)g(centred)f(moment)f(matrix.)45 b(Centring)26 b(\()p Ft(sic)p FQ(\))h(is)h(a)g(standard)515 4080 y(statistical)e(technique)d (that)i(in)m(v)n(olv)o(es)f(shifting)g(the)h(coordinate)e(system)i(of)g (the)g(data)g(points)f(so)515 4180 y(that)f(the)g(centroid)f(lies)i(at) f(the)h(origin.)32 b(This)23 b(can)g(be)g(ef)n(fected)f(by)h (subtracting)2879 4162 y FH(~)2886 4180 y FI(1)p 2941 4134 43 4 v 13 w FH(z)2983 4192 y FB(j)3042 4180 y FQ(from)f(each)515 4289 y(column)g(of)i FL(Z)p FQ(,)h(where)1207 4271 y FH(~)1214 4289 y FI(1)f FQ(is)h(an)f FH(n)g FQ(dimensional)e(v)o(ector) h(such)h(that)2517 4271 y FH(~)2524 4289 y FI(1)29 b(=)h(\(1)p FH(;)14 b FI(1)p FH(;)g FI(1)p FH(;)g(:)g(:)g(:)e(;)i FI(1\))3179 4259 y FF(>)3259 4289 y FQ(and)p 515 4343 V 515 4389 a FH(z)557 4401 y FB(j)613 4389 y FQ(is)21 b(the)f(mean)g(of)g(that)g(column.)j(The)d(proof)f(is)i(gi)n(v)o(en)e (in)h(Appendix)e(C.)639 4488 y(The)23 b(third)f(homogeneous)d (coordinate)i FH(\020)30 b FQ(is)24 b(set)f(to)g(be)g(equal)f(to)h(the) g(estimate)g(of)f(the)h(focal)515 4588 y(length)e(measured)f(in)i(pix)o (els,)f(in)h(order)f(to)h(impro)o(v)o(e)d(the)j(conditioning)d(of)i (the)h(solution,)f(and)g(to)515 4688 y(maintain)h(consistenc)o(y)f(of)h (units.)33 b(If)23 b(no)f(estimate)h(is)g(a)n(v)n(ailable)g(then)f FH(\020)34 b FI(=)27 b(256)p FQ(,)c(to)g(ensure)e(that)515 4787 y(it)g(is)g(of)f(the)g(same)h(order)e(of)h(magnitude)e(as)j(the)g (image)f(coordinates.)j(Because)e FL(F)g FQ(is)g(bilinear)e(in)515 4887 y(the)i(image)g(coordinates)f(the)i(orthogonal)d(re)o(gression)h (result)i(is)g(not)g(in)m(v)n(ariant)e(to)h(the)h(choice)f(of)515 4986 y FH(\020)6 b FQ(.)36 b(This)24 b(is)g(a)g(phenomenon)c(well)k (kno)n(wn)e(in)i(conic)f(\002tting)h([6)o(,)g(41)o(])f(and)g(stems)i (from)d(the)i(poor)p eop %%Page: 26 28 26 27 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(26)515 523 y(normalization)18 b(gi)n(v)o(en)h(in)i (\(4.22\).)j(This)c(normalization)f(is)i(inappropriate)d(for)h(the)i (fundamental)515 623 y(matrix)f(as)h(it)h(is)f(not)g(in)m(v)n(ariant)e (to)h(Euclidean)g(transformations)e(of)i(the)h(coordinate)e(system.)26 b(T)-7 b(a-)515 722 y(ble)17 b(4.1)g(gi)n(v)o(es)g(the)g(v)n(ariance)f (of)h(the)h(distance)f(of)g Ft(noise)g(fr)m(ee)h(points)f(to)h(their)f (estimated)h(epipolar)515 822 y(lines)k FQ(v)n(arying)f(the)h(v)n(alue) g(of)g FH(\020)29 b FQ(for)22 b(10)g(sets)i(of)e FH(n)g FQ(synthetic)g(correspondences.)28 b(It)23 b(can)f(be)g(seen)p 1273 943 1368 4 v 1273 959 V 1271 1059 4 100 v 1288 1059 V 1447 1029 a(n)p 1643 1059 V 237 w(25)161 b(50)140 b(100)119 b(200)p 2623 1059 V 2640 1059 V 1273 1062 1368 4 v 1271 1162 4 100 v 1288 1162 V 1381 1132 a FH(\020)27 b FQ(=)21 b(1)p 1643 1162 V 141 w(8.94)98 b(7.60)g(7.45)f(4.33)p 2623 1162 V 2640 1162 V 1271 1262 V 1288 1262 V 1340 1232 a FH(\020)27 b FQ(=)21 b(256)p 1643 1262 V 98 w(5.16)98 b(3.19)g(1.37)f(0.48)p 2623 1262 V 2640 1262 V 1271 1361 V 1288 1361 V 1340 1331 a FH(\020)27 b FQ(=)21 b(600)p 1643 1361 V 98 w(6.02)98 b(3.36)g(1.55)f(0.89)p 2623 1361 V 2640 1361 V 1273 1365 1368 4 v 1273 1381 V 515 1529 a(T)-7 b(able)24 b(4.1:)33 b Ft(Showing)23 b(the)i(variance)e(of)i (the)f(distance)g(of)h FQ(noise)f(free)g(points)g Ft(to)h(epipolar)e (lines)515 1629 y(varying)c FH(\020)27 b Ft(for)21 b(10)f(sets)h(of)f FH(n)h Ft(synthetic)f(corr)m(espondences.)515 1827 y FQ(that)g(minimization)g(of)g(the)h(algebraic)e(distance)h(is)i(not)e (in)m(v)n(ariant)f(to)i FH(\020)28 b FQ(with)21 b(respect)f(to)h(the)f (dis-)515 1927 y(tances)k(of)g(features)f(on)h(the)g(image)g(from)f (their)h(epipolar)f(lines.)38 b(Setting)24 b FH(\020)31 b FQ(to)24 b(the)g(same)h(order)515 2027 y(of)19 b(magnitude)e(as)j (the)f(image)g(coordinates)e(produces)h(the)h(best)g(result:)25 b(in)19 b(this)h(case)g(the)f(images)515 2126 y(were)k FI(512)c FA(\002)h FI(512)i FQ(thus)h(we)h(set)g FH(\020)35 b FI(=)28 b(256)p FQ(.)33 b(These)23 b(problems)f(do)g(not)h(arise)h (when)e(minimizing)515 2226 y(the)17 b(geometric)e(distances)i(on)g (the)g(image)f(plane,)h(which)f(will)i(be)f(described)f(in)h(the)g(ne)o (xt)f(section.)515 2464 y Fs(4.5.3)99 b(Linear)25 b(Estimation:)31 b Fr(torr)p 1901 2464 30 4 v 35 w(ls)515 2619 y FQ(In)20 b(Matlab)g(linear)f(estimation)h(by)i(S)t(V)t(D)h(is)e(easy)g(as)f (pie,)g(and)g(is)h(pro)o(vided)d(by:)515 2802 y FO(function)48 b([vec,)h(error])g(=)g(torr_ls\(Z\))515 3023 y FM(Input:)83 b FO(Z)20 b FQ(is)i(the)e FH(n)e FA(\002)g FH(p)j FL(Z)g FQ(matrix.)515 3244 y FM(Output:)639 3410 y FA(\017)41 b FO(vec)20 b FQ(is)i(the)e(solution)f(v)o(ector)-5 b(.)639 3576 y FA(\017)41 b FO(error)20 b FQ(is)h FH(\025)1116 3588 y FG(1)1177 3576 y FI(=)1265 3514 y FC(P)1366 3576 y FH(r)1405 3546 y FG(2)1403 3598 y FB(i)1443 3576 y FQ(.)515 3814 y Fs(4.5.4)99 b(Linear)25 b(Estimation)g(of)g Fp(F)q Fs(,)f Fr(torr)p 2090 3814 V 35 w(estf)515 3969 y FQ(Most)c(of)g(the)g(functions)f(for)h(estimating)f FL(F)i FQ(ha)n(v)o(e)f(the)g(same)g(parameters)515 4135 y FO(function)48 b(f)i(=)f(torr_estf\(x1,y1,x2,y2,)d(no_matches,m3\)) 515 4301 y FQ(with)20 b(parameters)f(de\002ned)g(as)i(in)f(Section)g (4.1.1.)515 4539 y Fs(4.5.5)99 b(The)26 b(Shortcomings)f(of)g(the)h (Linear)g(Methods)515 4695 y FQ(Unfortunately)d(both)i(produce)f (inaccurate)h(results.)42 b(Classical)28 b(least)e(squares)g(to)g (estimate)g FL(F)p FQ(,)515 4794 y(as)e(suggested)f(by)g(Tsai)h Ft(et)g(al.)48 b FQ([58)n(])24 b(and)f(Olsen)h([36)o(],)g(where)f(a)h (chosen)f(v)n(ariable)f(\(the)i(`obser)n(-)515 4894 y(v)n(ation'\))e (is)j(re)o(gressed)d(against,)i(is)h(unsuitable)e(as)i(there)e(is)i(a)f (tacit)h(assumption)d(that)i(there)g(are)515 4994 y(only)f(errors)g(in) h(this)g(v)n(ariable.)34 b(The)24 b(problem)e(is)i(sho)n(wn)f(in)h (Figure)f(4.3)h(where)f(it)h(can)g(be)f(seen)p eop %%Page: 27 29 27 28 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(27)515 523 y(that)25 b(the)g(ordinary)d(least)k (squares)e(distance,)i(measured)d(in)i(the)g(direction)f(of)g(one)h(of) f(the)h(ax)o(es,)515 623 y(becomes)19 b(less)i(stable)g(as)g(the)f (line)g(to)g(be)h(estimated)e(tends)h(to)h(parallelism)f(with)g(this)h (axis.)k(That)515 722 y(is,)g(the)e(v)n(ariable)f(chosen)h(to)g(re)o (gress)g(against)g(may)g(ha)n(v)o(e)g(zero)f(coef)n(\002cient,)h(in)h (which)f(case)h(the)515 822 y(estimate,)c FL(f)9 b FQ(,)21 b(w)o(ould)e(be)h(rendered)e(meaningless.)639 922 y(Orthogonal)24 b(least)j(squares)f(also)g(produces)f(poor)g(results.)43 b(This)26 b(is)h(due)f(to)g(three)g(f)o(actors.)515 1021 y(First)21 b(the)f(residuals)g FH(r)1163 1033 y FB(i)1212 1021 y FQ(that)g(we)h(minimize:)556 1204 y FH(r)593 1216 y FB(i)644 1204 y FI(=)i FH(f)773 1216 y FG(1)810 1204 y FH(x)857 1170 y FF(0)857 1224 y FB(i)885 1204 y FH(x)932 1216 y FB(i)965 1204 y FI(+)5 b FH(f)1076 1216 y FG(2)1112 1204 y FH(x)1159 1170 y FF(0)1159 1224 y FB(i)1187 1204 y FH(y)1228 1216 y FB(i)1260 1204 y FI(+)g FH(f)1371 1216 y FG(3)1408 1204 y FH(x)1455 1170 y FF(0)1455 1224 y FB(i)1483 1204 y FH(\020)11 b FI(+)5 b FH(f)1641 1216 y FG(4)1677 1204 y FH(y)1721 1170 y FF(0)1718 1224 y FB(i)1746 1204 y FH(x)1793 1216 y FB(i)1826 1204 y FI(+)g FH(f)1937 1216 y FG(5)1973 1204 y FH(y)2017 1170 y FF(0)2014 1224 y FB(i)2041 1204 y FH(y)2082 1216 y FB(i)2114 1204 y FI(+)g FH(f)2225 1216 y FG(6)2262 1204 y FH(y)2306 1170 y FF(0)2303 1224 y FB(i)2330 1204 y FH(\020)11 b FI(+)5 b FH(f)2488 1216 y FG(7)2525 1204 y FH(x)2572 1216 y FB(i)2600 1204 y FH(\020)11 b FI(+)5 b FH(f)2758 1216 y FG(8)2794 1204 y FH(y)2835 1216 y FB(i)2863 1204 y FH(\020)11 b FI(+)5 b FH(f)3021 1216 y FG(9)3057 1204 y FH(\020)3099 1170 y FG(2)3178 1204 y FQ(\(4.23\))515 1386 y(are)16 b(not)f(Gaussian;)j(secondly)c(the)i(constraint)f(that)h (the)g(determinant)e(of)i FL(F)h FQ(should)e(be)h(zero)f(is)i(not)515 1486 y(enforced;)j(thirdly)g(the)h(choice)g(of)f(normalization)f(in)j (orthogonal)c(re)o(gression)i(is)i(inappropriate)515 1586 y(to)30 b FL(F)p FQ(.)54 b(In)29 b(the)h(ne)o(xt)f(section)h Ft(iter)o(atively)f(r)m(e)o(weighted)g(least)i(squar)m(es)e FQ(is)i(described)d(in)i(which)515 1685 y(some)22 b(compensation)e(is)j (made)f(for)g(the)g(\002rst)h(and)f(third)g(shortcoming)e([6)o(,)j(41)o (].)31 b(In)22 b(Section)g(4.8)515 1785 y(we)e(summarise)g (parameterisations)e(that)i(may)g(be)g(used)g(to)h(o)o(v)o(ercome)c (the)j(second.)515 2023 y Fs(4.5.6)99 b(Imposing)24 b(the)i(cubic)g (constraint)f Fk(det)q(\()p Fp(F)p Fk(\))j(=)f(0)515 2178 y FQ(The)d(second)g(constraint)g(which)g FL(F)i FQ(should)d(satisfy)j(is)f(a)h(cubic)e(polynomial)e(in)j(the)g(matrix)f (el-)515 2278 y(ements)i(imposing)f FI(det\()p FL(F)q FI(\))34 b(=)g(0)p FQ(.)44 b(If)26 b(it)h(is)h(not)e(imposed)f(then)h (the)g(epipolar)f(lines)i(do)f(not)g(all)515 2378 y(intersect)20 b(in)h(a)h(single)e(epipole.)26 b(Assume)21 b(that)g(we)g(ha)n(v)o(e)f (an)h(estimate)g(of)g(the)f(fundamental)f(ma-)515 2477 y(trix,)685 2456 y FI(^)676 2477 y FL(F)p FQ(.)36 b(T)-7 b(ypically)i(,)23 b(when)g(performing)e(only)i(linear)g(estimation)g (to)h(obtain)2832 2456 y FI(^)2823 2477 y FL(F)g FQ(this)h(constraint) 515 2577 y(is)j(imposed)e(by)h(projecting)e(the)i(linear)g(solution)f (onto)g(the)i(space)f(of)f(Fundamental)g(matrices)515 2676 y(such)c(that)h FI(det\()p FL(F)q FI(\))28 b(=)f(0)p FQ(,)c(such)g(that)g(the)g(Frobenius)e(norm)h FA(jj)2342 2655 y FI(^)2333 2676 y FL(F)e FA(\000)g FL(F)p FA(jj)k FQ(is)f(a)g(minimum.)32 b(Let)23 b(the)515 2776 y(singular)c(v)n(alue)g (decomposition)f([17)o(])i(of)g(the)g(reco)o(v)o(ered)e FL(F)j FQ(be)1709 2959 y FL(F)i FI(=)g FL(V)r FI(\003)o FL(U)2085 2922 y FF(>)2161 2959 y FH(:)994 b FQ(\(4.24\))515 3141 y(Due)15 b(to)g(noise)g FL(F)h FQ(will)g(ha)n(v)o(e)f(full)g(rank) f(with)h(non)g(zero)f(singular)h(v)n(alues:)22 b FI(\003)h(=)f FQ(diag)o FI(\()p FH(\025)3028 3089 y Fq(1)p 3029 3098 29 4 v 3029 3131 a(2)3071 3153 y FG(1)3109 3141 y FH(;)14 b(\025)3204 3089 y Fq(1)p 3204 3098 V 3204 3131 a(2)3246 3153 y FG(2)3284 3141 y FH(;)g(\025)3379 3089 y Fq(1)p 3379 3098 V 3379 3131 a(2)3421 3153 y FG(3)3458 3141 y FI(\))p FQ(.)515 3250 y(T)-7 b(o)21 b(approximate)e FL(F)i FQ(by)g(a)g(rank)f(tw)o(o)i(matrix,)e(let)i FI(\003)2039 3213 y FG(+)2119 3250 y FI(=)i FQ(diag)o FI(\()p FH(\025)2441 3197 y Fq(1)p 2442 3206 V 2442 3239 a(2)2484 3262 y FG(1)2521 3250 y FH(;)14 b(\025)2616 3197 y Fq(1)p 2617 3206 V 2617 3239 a(2)2659 3262 y FG(2)2696 3250 y FH(;)g FI(0\))22 b FQ(then)e(the)h(reduced)515 3349 y(rank)e(approximation)e(of)j(the)g (fundamental)e(matrix)h(is)1682 3532 y FL(F)k FI(=)f FL(V)r FI(\003)1984 3496 y FG(+)2039 3532 y FL(U)2112 3495 y FF(>)2189 3532 y FH(:)966 b FQ(\(4.25\))515 3715 y(This)20 b(is)h(easily)g(coded)e(in)h(MA)-9 b(TLAB)20 b(via)g(SVD)h(as)714 3881 y FO([U,S,V])48 b(=)i(svd\(F\);)714 3980 y(S\(3,3\))f(=)g(0;)714 4080 y(F)h(=)f(U*S*V';)714 4179 y(f)h(=)f(reshape\(F,9,1\);)639 4345 y FQ(A)30 b(problem)d(is)i (that)g(smaller)g(elements)g(will)g(ha)n(v)o(e)f(a)i(relati)n(v)o(ely)d (greater)h(perturbation)e(in)515 4445 y(relation)g(to)g(their)h(size.) 44 b(Thus)27 b(the)f(method)f(of)i([22)n(])g(is)h(adopted.)42 b(This)27 b(is)h(not)e(entirely)g(satis-)515 4545 y(f)o(actory)d(as)h (ideally)g(we)g(w)o(ould)g(lik)o(e)g(to)g(choose)f(the)2088 4523 y FI(^)2089 4545 y FL(f)34 b FQ(which)24 b(minimizes)f(some)h (Mahalanobis)515 4644 y(distance)d FI(\()839 4622 y(^)841 4644 y FL(f)30 b FA(\000)19 b FL(f)9 b FI(\))1054 4614 y FF(>)1111 4644 y FL(M)1201 4608 y FF(\000)p FG(1)1201 4668 y FB(f)1290 4644 y FI(\()1320 4622 y(^)1322 4644 y FL(f)30 b FA(\000)19 b FL(f)9 b FI(\))p FQ(,)23 b(taking)e(into)h (account)f(the)h(co)o(v)n(ariance)e(of)h FL(F)q FQ(.)30 b(W)-7 b(ork)22 b(on)g(this)515 4744 y(is)f(in)f(progress.)p eop %%Page: 28 30 28 29 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(28)515 523 y Fs(4.5.7)99 b(In)l(v)o(ariant)25 b(linear)g(\002tting)515 679 y FQ(Ne)o(xt)19 b(an)g(in)m(v)n(ariant)f (linear)h(method)f(based)h(on)h(Bookstein)e(is)j(described)d(noting)g (that)i(the)f(funda-)515 778 y(mental)i(matrix)h(is)g(lik)o(e)h(a)f(4D) g(conic.)30 b(W)-7 b(e)23 b(seek)f(an)g(estimation)f(rule)h(which)f(is) i Ft(g)o(ener)o(al,)e(simple)515 878 y(to)31 b(compute)e FQ(and)i Ft(in)m(variant)p FQ(.)55 b(Simplicity)30 b(suggests)h(we)g (seek)g(a)h(quadratic)d(norm,)j FL(f)3128 841 y FF(>)3184 878 y FL(Jf)52 b FI(=)515 978 y FQ(constant)n(,)27 b(on)e(the)h (parameters)e(of)h FL(F)h FQ(to)f(enforce)f(the)i(scaling)f(constraint) f(as)i(this)g(will)g(lead)f(to)515 1077 y(a)h(eigen)m(v)o(ector)e (solution.)42 b(In)m(v)n(ariance)24 b(is)j(to)g(be)f(with)h(respect)f (to)g(Euclidean)f(transformations)515 1177 y(of)d(both)f(image)h (planes)f(\(possibly)h(dif)n(ferent)e(transformations)g(to)i(dif)n (ferent)f(planes\))g(i.e.)31 b(if)23 b(the)515 1276 y(coordinate)16 b(system)i(is)i(changed)c(in)j(one)e(or)h(both)g(of)g(the)g(images,)g (then)g(the)g(best)h(\002tting)3124 1255 y FI(~)3115 1276 y FL(F)g FQ(to)g(the)515 1376 y(transformed)g(points)j(must)g(be)f (e)o(xactly)g(the)h(result)g(of)g(the)g(same)g(transformation\(s\))d (applied)h(to)515 1476 y(the)g(best)g(\002tting)h FL(F)f FQ(of)g(the)h(original)e(points.)639 1575 y(Bookstein[6)n(])24 b(suggested)f(an)g(in)m(v)n(ariant)f(norm)h(for)g(conics)g(under)f (Euclidean)h(transforma-)515 1675 y(tions.)g(It)17 b(has)f(been)f (observ)o(ed)f(that)i(the)g(fundamental)e(matrix)h(is)i(lik)o(e)f(a)g (conic)g(in)g(the)g(four)f(dimen-)515 1775 y(sions)j(of)g(the)g(joint)g (image)g(space)g FA(<)1568 1744 y FG(4)1624 1775 y FQ([49)o(].)24 b(F)o(ollo)n(wing)17 b(Bookstein)h(we)g(seek)g(a)h(parametrization)515 1874 y(of)k FL(F)h FQ(in)m(v)n(ariant)e(to)i(Euclidean)e (transformations)g(in)h(the)h(image)f(planes)h(\(which)e(is)j(a)f (subgroup)515 1974 y(of)j(the)h(Euclidean)e(transformations)f(in)j(the) g(joint)f(image)h(space)f FA(<)2558 1944 y FG(4)2595 1974 y FQ(\).)48 b(F)o(ortunately)26 b(the)h(con-)515 2073 y(struction)19 b(of)h(these)g(in)m(v)n(ariants)f(is)i(a)g(well)g (studied)e(problem)g([35)n(].)639 2173 y(Consider)26 b(the)g(transformations)e(of)i(the)g(image)g(coordinates)f FL(G)i FQ(in)f(image)g(one)f(such)h(that)515 2273 y FL(G)594 2272 y FI(~)590 2273 y FL(x)h FI(=)f FL(x)p FQ(,)d(and)e(image)h(tw)o (o,)g FL(G)1462 2236 y FF(0)1490 2272 y FI(~)1486 2273 y FL(x)1536 2237 y FF(0)1586 2273 y FI(=)k FL(x)1727 2243 y FF(0)1750 2273 y FQ(,)d(which)f(leads)g(to)g(a)g(transformation) e(on)h FL(F)i FQ(such)f(that,)524 2351 y FI(~)515 2372 y FL(F)h FI(=)g FL(G)761 2336 y FF(0>)836 2372 y FL(FG)e FQ(with)567 2577 y FL(G)i FI(=)753 2460 y FC(\024)854 2524 y FL(R)101 b(t)838 2628 y(0)886 2595 y FF(>)1025 2628 y FI(1)1108 2460 y FC(\025)1165 2577 y FH(;)83 b FL(G)1346 2540 y FF(0)1393 2577 y FI(=)1480 2460 y FC(\024)1570 2524 y FL(R)1642 2488 y FF(0)1753 2524 y FL(t)1790 2492 y FF(0)1566 2628 y FL(0)1614 2595 y FF(>)1762 2628 y FI(1)1855 2460 y FC(\025)1912 2577 y FH(;)g FL(F)23 b FI(=)2189 2460 y FC(\024)2287 2526 y FL(A)97 b(b)2274 2626 y(c)2317 2596 y FF(>)2461 2626 y FH(d)2550 2460 y FC(\025)2608 2577 y FH(;)2723 2556 y FI(~)2714 2577 y FL(F)23 b FI(=)2885 2460 y FC(\024)2999 2505 y FI(~)2983 2526 y FL(A)3157 2504 y FI(~)3152 2526 y FL(b)2971 2632 y FI(~)2970 2633 y FL(c)3013 2597 y FF(>)3171 2611 y FI(~)3157 2633 y FH(d)3246 2460 y FC(\025)3304 2577 y FH(:)3178 2728 y FQ(\(4.26\))515 2828 y(Thus)d(it)g(can)g(be)h(seen)f (that)571 3011 y FI(~)561 3032 y FL(F)j FI(=)g FL(G)807 2996 y FF(0>)882 3032 y FL(FG)h FI(=)1128 2915 y FC(\024)1242 2960 y FI(~)1227 2981 y FL(A)1401 2959 y FI(~)1395 2981 y FL(b)1214 3088 y FI(~)1213 3089 y FL(c)1256 3052 y FF(>)1415 3067 y FI(~)1400 3089 y FH(d)1490 2915 y FC(\025)1556 3032 y FI(=)1644 2915 y FC(\024)1848 2983 y FL(R)1919 2947 y FF(0>)1995 2983 y FL(AR)357 b(R)2568 2947 y FF(0>)2643 2983 y FL(At)19 b FI(+)f FL(R)2926 2947 y FF(0>)3001 2983 y FL(b)1729 3086 y(t)1766 3054 y FF(0>)1842 3086 y FL(AR)g FI(+)g FL(c)2129 3056 y FF(>)2185 3086 y FL(R)83 b(t)2377 3054 y FF(0>)2452 3086 y FL(A)q(t)18 b FI(+)g FL(t)2700 3054 y FF(0>)2775 3086 y FL(b)h FI(+)f FL(c)2972 3056 y FF(>)3028 3086 y FL(t)h FI(+)f FH(d)3252 2915 y FC(\025)3309 3032 y FH(:)3178 3183 y FQ(\(4.27\))515 3283 y(From)e(this)i(it)g(can)f(be)g(seen)h(that)f(the)g(norm)f(cannot) h(be)g(an)o(y)f(combination)f(of)i FH(f)2796 3295 y FG(3)2833 3283 y FH(;)d(f)2911 3295 y FG(6)2948 3283 y FH(;)g(f)3026 3295 y FG(7)3062 3283 y FH(;)g(f)3140 3295 y FG(8)3177 3283 y FH(;)g(f)3255 3295 y FG(9)3310 3283 y FQ(as)515 3383 y(these)24 b(can)g(be)g(transformed)e(to)i(arbitrary)e(v)n(alues)i (by)g(translations)f(of)h(the)g(image)g(coordinates.)515 3482 y(Unless)g(of)f(course)f FL(t)p FH(;)14 b FL(t)1209 3450 y FF(0)1262 3482 y FI(=)28 b(0)p FQ(,)c(in)f(which)g(case)h(the)f (norm)2249 3420 y FC(P)2336 3440 y FB(i)p FG(=9)2336 3507 y FB(i)p FG(=1)2462 3482 y FH(f)2512 3452 y FG(2)2503 3504 y FB(i)2577 3482 y FI(=)29 b(1)23 b FQ(\(amongst)f(others\))h(is) 515 3582 y(in)m(v)n(ariant)c(to)h(rotations)g(of)g(the)h(image)f (plane.)26 b(Discounting)19 b(the)h(non)g(generic)f(case,)i(this)g(lea) n(v)o(es)515 3681 y(the)27 b(elements)h(of)f(the)h(upper)e(left)i FI(2)23 b FA(\002)h FI(2)k FQ(submatrix)e(of)h FL(F)i FQ(to)e(de\002ne)g(the)h(norm.)46 b(Due)28 b(to)g(the)515 3781 y(special)20 b(nature)f(of)h(rotation)f(matrices)h(it)h(can)f(be)g (immediately)f(seen)h(that)691 3948 y FI(det\()854 3926 y(~)838 3948 y FL(A)q FI(\))j(=)g(det\()p FL(R)1273 3911 y FF(0>)1348 3948 y FL(AR)p FI(\))g(=)g(det\()p FL(A)q FI(\))p FH(;)166 b FA(jj)p FI(\()2170 3926 y(~)2154 3948 y FL(A)q FI(\))p FA(jj)23 b FI(=)g FA(jj)p FI(\()p FL(R)2566 3911 y FF(0>)2641 3948 y FL(AR)p FI(\))p FA(jj)g FI(=)g FA(jj)p FI(\()p FL(A)p FI(\))p FA(jj)515 4114 y FQ(where)c FA(jj)p FI(\(\))p FA(jj)i FQ(corresponds)c(to)j(the)f(Frobenius)f(norm) h(of)g(the)h(matrix.)k(Thus)19 b(we)h(ha)n(v)o(e)f(the)g(choice)515 4214 y(of)g(the)h(follo)n(wing)d(norms,)i(the)h(determinant)d(norm)i FI(det\()p FL(A)p FI(\))24 b(=)e(\()p FH(f)2484 4226 y FG(1)2522 4214 y FH(f)2563 4226 y FG(5)2616 4214 y FA(\000)15 b FH(f)2737 4226 y FG(2)2774 4214 y FH(f)2815 4226 y FG(4)2852 4214 y FI(\))p FQ(,)20 b(the)g(Frobenius)515 4322 y(norm)f FA(jj)p FI(\()p FL(A)q FI(\))p FA(jj)k FI(=)h(\()p FH(f)1134 4292 y FG(2)1125 4343 y(1)1189 4322 y FI(+)18 b FH(f)1322 4292 y FG(2)1313 4343 y(2)1378 4322 y FI(+)g FH(f)1511 4292 y FG(2)1502 4343 y(4)1567 4322 y FI(+)g FH(f)1700 4292 y FG(2)1691 4343 y(5)1737 4322 y FI(\))1779 4270 y Fq(1)p 1779 4279 29 4 v 1779 4312 a(2)1821 4322 y FQ(.)26 b(Ho)n(w)21 b(man)o(y)e(in)m(v)n(ariants)g (can)h(there)g(be?)26 b(Referring)515 4422 y(to)54 b([35)o(])27 b(the)g(counting)f(ar)o(gument)e(states:)40 b(\223suppose)26 b(there)h(is)h(a)f(con\002guration)d(space)j FA(S)6 b FQ(,)30 b(on)515 4521 y(which)d(a)i(group)d FH(G)j FQ(acts,)i(then)d (the)g(number)e(of)i(functionally)e(independent)g(primiti)n(v)o(e)g (scalar)515 4621 y(in)m(v)n(ariants)e(is)j(greater)e(than)g(or)h(equal) f(to)h FI(dim)15 b FA(S)29 b(\000)22 b FI(dim)14 b FH(G)2327 4591 y Fz(2)2360 4621 y FQ(.)43 b(In)25 b(this)i(case)f FI(dim)14 b FL(A)34 b FI(=)f(4)26 b FQ(and)515 4721 y FI(dim\()p FL(R)p FH(;)14 b FL(R)866 4684 y FF(0)889 4721 y FI(\))23 b(=)g(2)p FQ(,)d(thus)g(we)h(w)o(ould)e(e)o(xpect)g(at) i(least)g(tw)o(o)f(in)m(v)n(ariants)2596 4690 y Fz(3)2629 4721 y FQ(.)p 515 4767 1146 4 v 605 4823 a Fy(2)634 4846 y FP(In)d(general)i(equality)h(holds)d(e)o(xcept)i(in)f(the)f(special)i (case)f(of)f(isotopies)605 4904 y Fy(3)634 4927 y FP(If)f(only)g(one)h (rotation)h(w)o(as)e(applied)h(to)g(both)f(images)g(i.e.)21 b(we)16 b(kne)n(w)g(the)h(common)f(orientation)j(of)c(the)i(tw)o(o)f (images,)515 5006 y(then)i(we)f(could)h(e)o(xpect)h(another)g(in)m(v)n (ariant,)h(which)e(w)o(ould)g(correspond)g(to)g(trace)r Fm(A)p eop %%Page: 29 31 29 30 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(29)639 523 y(Which)19 b(of)g(these)g(norms)f(is)i(most) f(appropriate?)k(In)18 b(order)g(to)h(deduce)f(this)i(another)d (desider)n(-)515 623 y(atum)k(is)i(introduced;)e(that)g(the)h(norm)f (is)i(positi)n(v)o(e)e(de\002nite.)30 b(This)22 b(is)h(desirable)e (because)g(epipo-)515 722 y(lar)32 b(geometries)e(for)i(whose)f FL(F)i FQ(the)e(norm)g(is)i(zero)e(can)h(ne)n(v)o(er)e(be)i(\002tted)g (at)h(all,)i(e)n(v)o(en)30 b(if)j(the)515 822 y(data)24 b(lie)g(e)o(xactly)f(upon)g(them.)36 b(Therefore)21 b(we)k(must)f(say)g (goodbye)d(to)k(the)f(determinant)e(norm)515 922 y FI(det\()p FL(A)q FI(\))p FQ(,)29 b(which)e(e)o(xcludes)f(all)i FL(F)g FQ(for)f(which)g FI(det\()p FL(A)q FI(\))36 b(=)g(0)p FQ(.)47 b(The)27 b(square)f(of)i(the)f(Frobenius)515 1021 y(norm)17 b FA(jj)p FI(\()p FL(A)p FI(\))p FA(jj)937 991 y FG(2)998 1021 y FI(=)22 b(\()p FH(f)1167 991 y FG(2)1158 1042 y(1)1215 1021 y FI(+)10 b FH(f)1340 991 y FG(2)1331 1042 y(2)1387 1021 y FI(+)g FH(f)1512 991 y FG(2)1503 1042 y(4)1559 1021 y FI(+)g FH(f)1684 991 y FG(2)1675 1042 y(5)1721 1021 y FI(\))18 b FQ(does)g(not)g(e)o(xclude) e(general)h FL(F)p FQ(,)i(rather)e(it)i(will)g(\002t)f(all)h FL(F)515 1121 y FQ(e)o(xcept)f(for)h(data)g(for)g(which)g(a)h(linear)f (or)g(af)n(\002ne)g(fundamental)e(matrix)i FL(F)2659 1133 y FB(A)2733 1121 y FQ([35)o(])h(is)g(more)f(suited;)1122 1400 y FL(x)p 1122 1413 51 4 v 1173 1366 a FF(0>)1173 1421 y FB(i)1248 1400 y FL(F)1308 1412 y FB(A)1362 1400 y FL(x)1412 1412 y FB(i)p 1362 1426 79 4 v 1463 1400 a FI(=)k(0)61 b FQ(where)41 b FL(F)1959 1412 y FB(A)2036 1400 y FI(=)2124 1233 y FC(2)2124 1383 y(4)2238 1300 y FI(0)118 b(0)100 b FH(g)2580 1312 y FG(1)2238 1399 y FI(0)118 b(0)100 b FH(g)2580 1411 y FG(2)2220 1499 y FH(g)2260 1511 y FG(3)2380 1499 y FH(g)2420 1511 y FG(4)2540 1499 y FH(g)2580 1511 y FG(5)2659 1233 y FC(3)2659 1383 y(5)2748 1400 y FH(:)407 b FQ(\(4.28\))515 1677 y(Whether)15 b(or)g(not)g FL(F)1084 1689 y FB(A)1154 1677 y FQ(is)h(the)f(more)g(appropriate)e(model)h(can)i(be)f (determined)e(by)i(model)g(selection)515 1777 y(methods)23 b([57)n(].)37 b(If)23 b(it)i(is)g(then)e(an)h(e)o(xact)g(eigen)m(v)o (ector)d(solution)i([44)o(])h(e)o(xists)g(for)f FL(F)2971 1789 y FB(A)3050 1777 y FQ(that)h(mini-)515 1877 y(mizes)c (reprojection)d(error)i(which)g(should)g(al)o(w)o(ays)i(be)e(used)h (rather)f(than)g(a)i(more)e(general)f(algo-)515 1976 y(rithm)k(for)f(\002tting)h FL(F)q FQ(.)31 b(Thus)22 b(we)h(propose)e(to)h(minimize)g FL(f)2228 1939 y FF(>)2284 1976 y FL(Mf)32 b FQ(subject)22 b(to)g FL(f)2819 1939 y FF(>)2875 1976 y FL(J)q(f)36 b FI(=)27 b FQ(constant)n(,)515 2076 y(where)19 b FL(J)24 b FI(=)e FQ(diag)o FI(\(1)p FH(;)14 b FI(1)p FH(;)g FI(0)p FH(;)g FI(1)p FH(;)g FI(1)p FH(;)g FI(0)p FH(;)g FI(0)p FH(;)g FI(0)p FH(;)g FI(0\))p FQ(,)i(is)21 b(the)g(square)e(of)h(Frobenius)f(normalization.)639 2175 y(The)29 b(square)g(of)g(the)g(Frobenius)f(norm)g FA(jj)p FI(\()p FL(A)p FI(\))p FA(jj)2070 2145 y FG(2)2148 2175 y FI(=)39 b(\()p FH(f)2334 2145 y FG(2)2325 2196 y(1)2396 2175 y FI(+)25 b FH(f)2536 2145 y FG(2)2527 2196 y(2)2598 2175 y FI(+)g FH(f)2738 2145 y FG(2)2729 2196 y(4)2800 2175 y FI(+)f FH(f)2939 2145 y FG(2)2930 2196 y(5)2976 2175 y FI(\))30 b FQ(is)g(also)g(in-)515 2275 y(v)n(ariant)i(to)h(choice)f(of)g(scale.)64 b(W)m(ithout)32 b(loss)i(of)e(generality)g(consider)f(only)h(the)h(change)f(of)515 2375 y(scaling)j(in)g(one)f(of)h(the)g(images.)69 b(If)35 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4995 y FG(12)1119 4974 y FL(M)1210 4986 y FG(22)1322 4804 y FC(\025)1380 4921 y FH(;)83 b FQ(then)f FL(f)1750 4884 y FF(>)1806 4921 y FL(M)o(f)33 b FI(=)22 b FL(f)2084 4884 y FF(>)2084 4942 y FG(1)2140 4921 y FL(M)2230 4933 y FG(11)2301 4921 y FL(f)2339 4933 y FG(1)2395 4921 y FI(+)c(2)p FL(f)2557 4884 y FF(>)2557 4942 y FG(1)2613 4921 y FL(M)2704 4933 y FG(12)2774 4921 y FL(f)2813 4933 y FG(2)2868 4921 y FI(+)g FL(f)2989 4884 y FF(>)2989 4942 y FG(2)3046 4921 y FL(M)3136 4933 y FG(22)3206 4921 y FL(f)3245 4933 y FG(2)3282 4921 y FH(;)p eop %%Page: 30 32 30 31 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(30)515 523 y(as)18 b FL(M)f FQ(and)g(its)h(partitions)f (are)g(all)h(symmetric.)23 b(W)-7 b(e)18 b(must)f(minimize)g(this)h (subject)f(to)g FL(f)3040 486 y FF(>)3040 544 y FG(1)3096 523 y FL(J)3146 535 y FG(11)3216 523 y FL(f)3254 535 y FG(1)3314 523 y FI(=)515 623 y FQ(constant)n(,)k(where)f FL(J)1106 635 y FG(11)1199 623 y FI(=)j FQ(diag)o FI(\(1)p FH(;)14 b FI(1)p FH(;)g FI(1)p FH(;)g FI(1\))21 b(=)i FL(I)p FQ(.)j(F)o(or)19 b(an)o(y)h(\002x)o(ed)f FL(f)2463 635 y FG(1)2500 623 y FQ(,)h FL(f)2580 586 y FF(>)2636 623 y FL(M)o(f)30 b FQ(is)21 b(minimal)f(when)1351 808 y FH(@)5 b FL(f)1438 771 y FF(>)1494 808 y FL(M)o(f)p 1351 846 272 4 v 1425 922 a FH(@)g FL(f)1512 934 y FG(2)1656 865 y FI(=)22 b(2)p FL(M)1875 828 y FF(>)1875 885 y FG(12)1946 865 y FL(f)1984 877 y FG(1)2040 865 y FI(+)c(2)p FL(M)2255 877 y FG(22)2325 865 y FL(f)2363 877 y FG(2)2424 865 y FI(=)k(0)625 b FQ(\(4.31\))515 1078 y(which)19 b(implies)1613 1178 y FL(f)1652 1190 y FG(2)1712 1178 y FI(=)k FA(\000)p FL(M)1955 1141 y FF(\000)p FG(1)1955 1199 y(22)2044 1178 y FL(M)2135 1141 y FF(>)2135 1198 y FG(12)2205 1178 y FL(f)2243 1190 y FG(1)3178 1178 y FQ(\(4.32\))515 1325 y(Then)1093 1425 y FL(f)1131 1388 y FF(>)1187 1425 y FL(Mf)32 b FI(=)23 b FL(f)1465 1388 y FF(>)1465 1445 y FG(1)1521 1425 y FI(\()p FL(M)1644 1437 y FG(11)1733 1425 y FA(\000)18 b FL(M)1907 1437 y FG(12)1977 1425 y FL(M)2068 1388 y FF(\000)p FG(1)2068 1446 y(22)2157 1425 y FL(M)2247 1388 y FF(>)2247 1445 y FG(12)2318 1425 y FI(\))p FL(f)2388 1437 y FG(1)2448 1425 y FI(=)23 b FL(f)2574 1388 y FF(>)2574 1445 y FG(1)2630 1425 y FL(Qf)2740 1437 y FG(1)2778 1425 y FH(:)377 b FQ(\(4.33\))515 1572 y(T)-7 b(o)30 b(minimize)e(this)j(for)e FL(f)1290 1535 y FF(>)1290 1592 y FG(1)1346 1572 y FL(J)1395 1584 y FG(11)1466 1572 y FL(f)1504 1584 y FG(1)1582 1572 y FI(=)40 b FQ(constant)o(,)32 b(let)e FH(\025)h FQ(be)e(a)h(Lagrangian)e (multiplier)h(for)g(the)515 1671 y(constraint.)23 b(Then)18 b(we)h(must)f(set)h(the)g(deri)n(v)n(ati)n(v)o(e)e(with)h(respect)g(to) h FL(f)2492 1683 y FG(1)2549 1671 y FQ(of)f FL(f)2675 1634 y FF(>)2675 1692 y FG(1)2731 1671 y FL(Qf)2841 1683 y FG(1)2891 1671 y FA(\000)12 b FH(\025)p FL(f)3054 1634 y FF(>)3054 1692 y FG(1)3110 1671 y FL(f)3148 1683 y FG(1)3185 1671 y FQ(.)25 b(This)515 1771 y(yields)1744 1871 y FL(Qf)1854 1883 y FG(1)1915 1871 y FI(=)d FH(\025)p FL(f)2089 1883 y FG(1)2126 1871 y FH(;)1029 b FQ(\(4.34\))515 2018 y(thus)23 b FL(f)715 2030 y FG(1)775 2018 y FQ(may)g(be)g(reco)o (v)o(ered)d(from)i(as)h(the)g(eigen)m(v)o(ector)d(solution)i(of)h (\(4.34\).)31 b(Note)23 b(that)f(unless)515 2117 y(the)h(data)f(lie)i (on)e(lines)h(in)g(the)g(image)f FL(M)1724 2129 y FG(22)1817 2117 y FQ(al)o(w)o(ays)h(has)g(an)g(in)m(v)o(erse.)65 b(As)24 b(a)f(general)e(note,)i(fol-)515 2217 y(lo)n(wing)d(in)h(the)g (style)g(of)f(the)h(ellipse)g(speci\002c)g(\002tter)h([15)n(],)f(the)g (other)f(quadratic)g(constraints)g(can)515 2317 y(be)h(imposed)f(on)g (the)h(elements)g(of)g FL(F)g FQ(to)g(restrict)g(them)g(to)g(a)h (certain)e(subspace)g(of)h(fundamental)515 2416 y(matrices.)j(Indeed)17 b(it)i(is)h(a)f(simple)f(matter)g(to)h(also)g(add)f(an)g(arbitrary)f (linear)h(constraint)g(to)h FL(F)g FQ(into)515 2516 y(the)h (optimization)e(as)j(e)o(xplained)e(in)h([6)o(].)515 2753 y Fs(4.5.9)99 b(Imposing)24 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(sensible)g(to)h(impose)e(the)i(constraint)e(that)h(the)h(norm)515 4707 y(of)e(the)i(top)e FI(2)10 b FA(\002)g FI(2)18 b FQ(is)h(constant,)f(ho)n(we)n(v)o(er)e(once)h(e)o(xact)h(\(as)g (opposed)f(to)h(approximate\))d(geometric)515 4807 y(distance)20 b(is)h(minimized)e(then)h(this)h(should)f(not)g(mak)o(e)g(much)f(dif)n (ference.)24 b(Recently)c(Chojnacki)515 4907 y(et)c(al)g([9])g(ha)n(v)o (e)f(reported)f(e)o(xcellent)h(results)h(on)g(a)g(v)n(ariation)e(of)i (the)g(Sampson)f(distance)g(for)h(\002tting)515 5006 y FL(F)p FQ(,)k(the)o(y)g(do)g(not)f(consider)g(in)m(v)n(ariant)g (\002tting)h(ho)n(we)n(v)o(er)-5 b(.)p eop %%Page: 33 35 33 34 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(33)639 523 y(The)25 b(optimal)g(weights)g(con)m(v)o (ert)e(the)i(algebraic)f(distance)h(of)g(each)g(point)f(into)h(the)h (statisti-)515 623 y(cal)d(distance)f(in)h(noise)f(space,)h(which)f(is) i(equi)n(v)n(alent)d(to)h(the)h(\002rst)g(order)f(approximation)d(of)k (the)515 722 y(geometric)h(distance)i(as)h(sho)n(wn)e(in)h([41)o(,)g (38)o(].)43 b(The)25 b(weighting)g(breaks)g(do)n(wn)g(at)i(the)f (epipole,)515 822 y(the)21 b(numerator)d(and)i(the)h(denominator)d (both)i(approaching)e(zero,)i(indicating)g(that)g(there)h(is)g(less)515 922 y(information)k(about)i(correspondences)e(the)j(closer)g(the)o(y)g (are)g(to)g(the)g(epipole.)47 b(In)28 b(practice)f(to)515 1021 y(remo)o(v)o(e)c(unstable)h(constraints)g(all)i(points)f(within)g (a)g(pix)o(el)f(of)h(our)f(estimated)h(epipole)f(are)h(e)o(x-)515 1121 y(cluded)19 b(from)g(that)h(iteration)g(of)g(the)g(calculation.) 515 1341 y FM(Contention:)81 b FQ(In)32 b(the)h(book)e(of)h(Hartle)o(y) g(and)g(Zisserman)g([25)o(])h(there)f(is)h(a)g(claim)g(that)g(the)515 1441 y(Sampson)18 b(distance)g(breaks)g(do)n(wn)g(after)g(1)h(pix)o(el) f(of)h(noise)g(and)f(that)h(a)g(highly)f(non-linear)e(trian-)515 1541 y(gulation)22 b(method)h(is)i(necessary)-5 b(.)35 b(I)24 b(w)o(ould)f(dispute)g(this)i(and)e(in)m(vite)g(the)h (discerning)f(reader)f(to)515 1640 y(try)e(a)g(comparison.)515 1877 y Fs(4.7.1)99 b(Err)n(or)26 b(function:)32 b Fr(torr)p 1750 1877 30 4 v 35 w(errf2)515 2033 y FQ(T)-7 b(o)20 b(calculate)g(the)g(\002rst)h(order)e(error)g(of)h(a)h(set)g(of)e (matches)h(for)g(a)g(gi)n(v)o(en)f(fundamental)f(matrix)i FL(f)9 b FQ(.)515 2197 y FO(function)48 b(e)i(=)f(torr_errf2\(f,)f (nx1,ny1,nx2,ny2,)f(no_matches,)g(m3\))515 2417 y FM(Input:)83 b FQ(input)19 b(parameters)g(de\002ned)g(as)i(in)g(Section)e(4.1.1.)515 2659 y FM(Output:)82 b FO(e)20 b FQ(is)h(a)g FH(n)d FA(\002)g FI(1)j FQ(v)o(ector)e(of)h(squared)e(errors)i(with)g FH(i)p FQ(th)g(element)2696 2617 y FB(r)2729 2591 y Fq(2)2727 2633 y Fl(i)p 2688 2640 82 4 v 2688 2690 a FB(w)2738 2670 y Fq(2)2736 2711 y Fl(i)2780 2659 y FH(:)515 2909 y Fs(4.7.2)99 b(W)-6 b(eight)24 b(function:)32 b Fr(torr)p 1806 2909 30 4 v 35 w(grad)p 2081 2909 V 35 w(f)515 3064 y FO(torr)p 720 3064 25 4 v 29 w(grad)p 949 3064 V 29 w(f)21 b FQ(Calculates)f(Sampson')-5 b(s)20 b(weight)g(for)f(a)i(set)g (of)f(matches)f(and)h(a)h(gi)n(v)o(en)e FL(f)9 b FQ(.)515 3228 y FO(function)48 b(g)i(=)f(torr_grad_f\(f,)f(nx1,ny1,nx2,ny2,)e (no_matches,)i(m3\))515 3449 y FM(Input:)83 b FQ(input)19 b(parameters)g(de\002ned)g(as)i(in)g(Section)e(4.1.1.)515 3669 y FM(Output:)82 b FO(g)20 b FQ(is)h(a)g FH(n)d FA(\002)g FI(1)j FQ(v)o(ector)e(of)h(squared)e(errors)i(with)g FH(i)p FQ(th)g(element)g FH(w)2739 3639 y FG(2)2737 3691 y FB(i)2777 3669 y FQ(.)515 3906 y Fs(4.7.3)99 b(Sampson)25 b(function:)32 b Fr(torr)p 1890 3906 30 4 v 35 w(estf)p 2165 3906 V 35 w(bookstein)p 2740 3906 V 34 w(sampson)515 4062 y FO(function)48 b(f)i(=)f(torr_estf_bookstein_sampson\(x1,y1,x2,) o(y2,)44 b(no_matches,m3\))515 4242 y FQ(with)20 b(parameters)f (de\002ned)g(as)i(in)f(Section)g(4.1.1.)515 4522 y FD(4.8)119 b(P)o(arameterised)29 b(Descent)h(Methods:)37 b Fo(N)2611 4540 y FL(1)2663 4522 y FD(-)p Fo(N)2808 4540 y FL(3)515 4707 y FQ(This)22 b(section)f(describes)g(minimization)f(whilst)i (imposing)e(the)i(non-linear)d(constraint)i(that)h(the)515 4807 y(determinant)h(is)j(zero.)40 b(The)24 b(iterati)n(v)o(e)h(method) f(gi)n(v)o(en)g(abo)o(v)o(e)f(has)j(the)f(de\002cienc)o(y)f(that)h(it)h (does)515 4907 y(not)19 b(directly)g(enforce)e(the)j(constraint)e(that) i(the)f(determinant)f(of)h(the)h(fundamental)d(matrix)i(must)515 5006 y(be)26 b(zero.)42 b(Rather)25 b(a)i(procrustean)d(de)n(vice)h(is) i(used)f(to)g(con)m(v)o(ert)e(the)i(solution)f(from)g(one)h(that)g(is)p eop %%Page: 34 36 34 35 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(34)515 523 y(in)m(v)n(alid)23 b(to)i(one)g(that)g(is)g (v)n(alid.)39 b(Another)23 b(w)o(ay)i(of)g(o)o(v)o(ercoming)c(this)k (de\002cienc)o(y)f(is)i(to)f(perform)515 623 y(a)f(constrained)e (non-linear)g(minimization.)34 b(As)25 b(a)f(general)f(rule)g (constrained)g(non-linear)e(min-)515 722 y(imizations)26 b(are)i(trick)o(y)e(and)h(it)g(is)i(usually)d(better)h(to)g(use)h(a)g (parameterisation)d(that)i(implicitly)515 822 y(includes)19 b(the)h(constraint.)k(F)o(or)c(instance,)g FL(F)h FQ(might)e (parameterized)f(as)977 1099 y FL(F)24 b FI(=)1148 932 y FC(2)1148 1081 y(4)1429 998 y FH(!)1481 1010 y FG(1)1969 998 y FH(!)2021 1010 y FG(2)2510 998 y FH(!)2562 1010 y FG(3)1429 1098 y FH(!)1481 1110 y FG(4)1969 1098 y FH(!)2021 1110 y FG(5)2510 1098 y FH(!)2562 1110 y FG(6)1245 1198 y FH(!)1297 1210 y FG(7)1334 1198 y FH(!)1386 1210 y FG(1)1441 1198 y FI(+)18 b FH(!)1576 1210 y FG(8)1613 1198 y FH(!)1665 1210 y FG(4)1785 1198 y FH(!)1837 1210 y FG(7)1874 1198 y FH(!)1926 1210 y FG(2)1982 1198 y FI(+)g FH(!)2117 1210 y FG(8)2153 1198 y FH(!)2205 1210 y FG(5)2325 1198 y FH(!)2377 1210 y FG(7)2414 1198 y FH(!)2466 1210 y FG(3)2522 1198 y FI(+)g FH(!)2657 1210 y FG(8)2694 1198 y FH(!)2746 1210 y FG(6)2824 932 y FC(3)2824 1081 y(5)2893 1099 y FH(;)262 b FQ(\(4.38\))515 1373 y(pro)o(viding)25 b(that)j(the)g(third)g(ro)n(w)g(is)h(not)f(zero)f (\(if)h(it)h(is)g(then)f(it)h(is)g(a)f(tri)n(vial)g(matter)g(to)g (alter)h(the)515 1472 y(parameterization\).)f(Luong)21 b Ft(et)i(al)f FQ([32)o(])h(suggested)e(a)i(parameterization)d(of)i FL(F)h FQ(in)f(terms)g(of)h(the)515 1572 y(non-homogeneous)16 b(coordinates)k(of)h(the)g(epipoles)g FL(e)k FI(=)g(\()p FH(e)2298 1584 y FG(1)2335 1572 y FH(;)14 b(e)2411 1584 y FG(2)2448 1572 y FI(\))p FQ(,)22 b FL(e)2567 1542 y FF(0)2615 1572 y FI(=)j(\()p FH(e)2776 1542 y FF(0)2776 1593 y FG(1)2813 1572 y FH(;)14 b(e)2889 1542 y FF(0)2889 1593 y FG(2)2926 1572 y FI(\))22 b FQ(and)f(three)g(of)515 1672 y(the)f(four)f(coef)n(\002cients)g(of)h(the)h(homography)16 b(between)j(the)h(epipolar)f(lines:)714 1948 y FL(F)k FI(=)884 1782 y FC(2)884 1931 y(4)1129 1848 y FH(b)376 b(a)564 b FA(\000)p FH(ae)2297 1860 y FG(2)2352 1848 y FA(\000)18 b FH(be)2510 1860 y FG(1)1093 1948 y FA(\000)p FH(d)312 b FA(\000)p FH(c)567 b(ce)2256 1960 y FG(2)2312 1948 y FI(+)18 b FH(de)2477 1960 y FG(1)981 2047 y FH(de)1063 2017 y FF(0)1063 2068 y FG(2)1119 2047 y FA(\000)g FH(be)1277 2017 y FF(0)1277 2068 y FG(1)1396 2047 y FH(ce)1471 2017 y FF(0)1471 2068 y FG(2)1527 2047 y FA(\000)g FH(ae)1693 2017 y FF(0)1693 2068 y FG(1)1812 2047 y FH(ae)1895 2059 y FG(2)1932 2047 y FH(e)1971 2017 y FF(0)1971 2068 y FG(1)2027 2047 y FI(+)g FH(be)2185 2059 y FG(1)2221 2047 y FH(e)2260 2017 y FF(0)2260 2068 y FG(1)2315 2047 y FA(\000)h FH(ce)2474 2059 y FG(2)2510 2047 y FH(e)2549 2017 y FF(0)2549 2068 y FG(2)2605 2047 y FA(\000)f FH(de)2770 2017 y FF(0)2770 2068 y FG(2)2807 2047 y FH(e)2846 2059 y FG(1)2924 1782 y FC(3)2924 1931 y(5)3178 1948 y FQ(\(4.39\))515 2286 y(where)744 2169 y FC(\024)829 2236 y FH(a)87 b(b)833 2335 y(c)g(d)1041 2169 y FC(\025)1111 2286 y FQ(is)26 b(the)g(homography)21 b(between)k(the)h(epipolar)e(lines.)42 b(As)26 b(the)g(homography)515 2430 y(has)32 b(only)g(three)g(de)o (grees)f(of)h(freedom,)h(its)h(matrix')-5 b(s)32 b(determinant)e(is)j (constrained)e(to)h(be)h FI(1)p FQ(,)515 2530 y(gi)n(ving)25 b FH(ad)f FA(\000)f FH(bc)35 b FI(=)g(1)p FQ(,)29 b(from)d(which)g(it)i (can)f(be)g(seen)g(that)g(the)g(fundamental)d(matrix)j(de\002ned)515 2629 y(in)22 b(Equation)f(\(4.39\))g(has)h(determinant)f(zero.)31 b(Gi)n(v)o(en)21 b(an)i(estimate)g(of)f FL(F)h FQ(it)g(is)g(con)m(v)o (erted)d(to)j(the)515 2729 y(epipolar)c(parameterization)e(as)k(follo)n (ws:)1506 2909 y FH(a)83 b FI(=)f FH(F)1833 2921 y FG(12)1514 3033 y FH(b)h FI(=)f FH(F)1833 3045 y FG(11)1514 3158 y FH(c)h FI(=)f FA(\000)p FH(F)1898 3170 y FG(22)1507 3317 y FH(d)h FI(=)1790 3261 y(1)18 b(+)g FH(bc)p 1790 3298 215 4 v 1876 3374 a(a)1507 3464 y(g)86 b FI(=)c FH(F)1833 3476 y FG(22)1904 3464 y FH(F)1957 3476 y FG(11)2046 3464 y FA(\000)18 b FH(F)2182 3476 y FG(12)2253 3464 y FH(F)2306 3476 y FG(21)1474 3626 y FH(e)1513 3638 y FG(1)1633 3626 y FI(=)1790 3570 y FH(F)1843 3582 y FG(23)1914 3570 y FH(F)1967 3582 y FG(12)2056 3570 y FA(\000)g FH(F)2192 3582 y FG(22)2263 3570 y FH(F)2316 3582 y FG(13)p 1790 3607 597 4 v 2067 3683 a FH(g)1474 3846 y(e)1513 3858 y FG(2)1633 3846 y FI(=)1790 3790 y FH(F)1843 3802 y FG(13)1914 3790 y FH(F)1967 3802 y FG(21)2056 3790 y FA(\000)g FH(F)2192 3802 y FG(11)2263 3790 y FH(F)2316 3802 y FG(23)p 1790 3827 V 2067 3903 a FH(g)1474 4065 y(e)1513 4031 y FF(0)1513 4085 y FG(1)1633 4065 y FI(=)1790 4009 y FH(F)1843 4021 y FG(32)1914 4009 y FH(F)1967 4021 y FG(21)2056 4009 y FA(\000)g FH(F)2192 4021 y FG(22)2263 4009 y FH(F)2316 4021 y FG(31)p 1790 4046 V 2067 4122 a FH(g)1474 4284 y(e)1513 4250 y FF(0)1513 4305 y FG(2)1633 4284 y FI(=)1790 4228 y FH(F)1843 4240 y FG(31)1914 4228 y FH(F)1967 4240 y FG(12)2056 4228 y FA(\000)g FH(F)2192 4240 y FG(11)2263 4228 y FH(F)2316 4240 y FG(32)p 1790 4265 V 2067 4341 a FH(g)2397 4284 y(:)515 4502 y FQ(The)f(non-linear)e (minimization)h(is)j(performed)c(\(using)h(Po)n(well')-5 b(s)19 b(technique)c([59)o(]\))j(on)f(the)g(se)n(v)o(en)515 4602 y(independent)27 b(parameters:)42 b FH(a;)14 b(b;)g(c;)g(e)1648 4614 y FG(1)1684 4602 y FH(;)g(e)1760 4614 y FG(2)1797 4602 y FH(;)g(e)1873 4572 y FF(0)1873 4623 y FG(1)1910 4602 y FH(;)g(e)1986 4572 y FF(0)1986 4623 y FG(2)2053 4602 y FQ(and)29 b(once)f(a)i(minimum)e(is)j(attained)d(the)i FL(F)515 4702 y FQ(matrix)19 b(may)h(be)g(reco)o(v)o(ered)e(from:)1191 4882 y FH(F)1244 4894 y FG(11)1397 4882 y FI(=)83 b FH(b)1191 5006 y(F)1244 5018 y FG(12)1397 5006 y FI(=)g FH(a)p eop %%Page: 35 37 35 36 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(35)1191 523 y FH(F)1244 535 y FG(13)1397 523 y FI(=)83 b FA(\000)p FH(ae)1693 535 y FG(2)1748 523 y FA(\000)18 b FH(be)1906 535 y FG(1)1191 648 y FH(F)1244 660 y FG(21)1397 648 y FI(=)83 b FA(\000)p FH(d)1191 772 y(F)1244 784 y FG(22)1397 772 y FI(=)g FH(ce)1620 784 y FG(2)1675 772 y FI(+)18 b FH(de)1840 784 y FG(1)1191 897 y FH(F)1244 909 y FG(23)1397 897 y FI(=)83 b FH(ce)1620 909 y FG(2)1675 897 y FI(+)18 b FH(de)1840 909 y FG(1)1191 1021 y FH(F)1244 1033 y FG(31)1397 1021 y FI(=)83 b FH(de)1627 987 y FF(0)1627 1042 y FG(2)1683 1021 y FA(\000)18 b FH(be)1841 987 y FF(0)1841 1042 y FG(1)1191 1146 y FH(F)1244 1158 y FG(32)1397 1146 y FI(=)83 b FH(ce)1620 1111 y FF(0)1620 1166 y FG(2)1675 1146 y FA(\000)18 b FH(ae)1841 1111 y FF(0)1841 1166 y FG(1)1191 1270 y FH(F)1244 1282 y FG(33)1397 1270 y FI(=)83 b FA(\000)p FH(ce)1685 1236 y FF(0)1685 1291 y FG(2)1721 1270 y FH(e)1760 1282 y FG(2)1816 1270 y FA(\000)18 b FH(de)1981 1236 y FF(0)1981 1291 y FG(2)2018 1270 y FH(e)2057 1282 y FG(1)2112 1270 y FI(+)g FH(ae)2278 1282 y FG(2)2315 1270 y FH(e)2354 1236 y FF(0)2354 1291 y FG(1)2409 1270 y FI(+)h FH(be)2568 1282 y FG(1)2604 1270 y FH(e)2643 1236 y FF(0)2643 1291 y FG(1)2680 1270 y FH(:)515 1433 y FQ(Luong')-5 b(s)28 b(parameterization)g(breaks)h(do)n(wn)g(when)g(the)h(epipoles)f(are)h (at)h(in\002nity)-5 b(,)31 b(because)e(at)515 1533 y(in\002nity)e FH(ad)d FA(\000)g FH(bc)37 b FI(=)g(0)p FQ(.)49 b(T)-7 b(o)28 b(o)o(v)o(ercome)d(this)j(the)g(coordinate)e(system)j(is)f (changed.)47 b(Prior)27 b(to)515 1632 y(reparameterization)g(of)j FL(F)h FQ(the)g(epipoles)e(in)i(image)f(one)g(and)g(tw)o(o)g(are)h (calculated)e(from)h(the)515 1732 y(nullspace)19 b(of)h FL(F)h FQ(and)e FL(F)1221 1696 y FF(>)1297 1732 y FQ(by)h(observing:) 1806 1895 y FL(Fe)83 b FI(=)g FL(0)1724 2020 y(F)1784 1983 y FF(>)1840 2020 y FL(e)1884 1960 y Fn(0)1993 2020 y FI(=)g FL(0)p FH(:)515 2182 y FQ(If)28 b(the)h(epipole)f(is)i(on)e (or)g(near)h(the)f(line)h(at)h(in\002nity)e(all)h(the)g(coordinates)e (in)i(each)f(image)h(are)515 2282 y(transformed)16 b(by)j(an)f (projecti)n(v)o(e)g(transformation)e(that)j(tak)o(es)g(the)g(epipoles)f (well)h(a)o(w)o(ay)g(from)f(the)515 2382 y(line)27 b(at)h(in\002nity)-5 b(.)45 b(Gi)n(v)o(en)26 b(the)i(epipole)e(is)i(at)g FL(e)36 b FI(=)f(\()p FH(e)2108 2394 y FG(1)2146 2382 y FH(;)14 b(e)2222 2394 y FG(2)2258 2382 y FH(;)g FI(0\))2369 2352 y FF(>)2453 2382 y FQ(a)28 b(rotation)e FL(A)i FQ(about)e(the)h(axis) 515 2481 y FI(\()p FH(e)586 2493 y FG(2)623 2481 y FH(;)14 b FA(\000)p FH(e)764 2493 y FG(1)800 2481 y FH(;)g FI(0\))21 b FQ(by)1045 2449 y FB(\031)p 1045 2463 41 4 v 1049 2510 a FG(2)1117 2481 y FQ(de)o(grees)e(is)i(used,)1015 2752 y FL(A)i FI(=)1208 2696 y(1)p 1208 2733 42 4 v 1208 2809 a(2)1273 2585 y FC(2)1273 2734 y(4)1370 2651 y FI(1)18 b(+)g FH(e)1552 2621 y FG(2)1552 2672 y(2)1607 2651 y FA(\000)g FH(e)1729 2621 y FG(2)1729 2672 y(1)1918 2651 y FA(\000)p FI(2)p FH(e)2064 2663 y FG(1)2100 2651 y FH(e)2139 2663 y FG(2)2435 2651 y FA(\000)p FI(2)p FH(e)2581 2663 y FG(1)1439 2751 y FA(\000)p FI(2)p FH(e)1585 2763 y FG(1)1621 2751 y FH(e)1660 2763 y FG(2)1849 2751 y FI(1)g FA(\000)g FH(e)2031 2721 y FG(2)2031 2772 y(2)2086 2751 y FI(+)g FH(e)2208 2721 y FG(2)2208 2772 y(1)2435 2751 y FA(\000)p FI(2)p FH(e)2581 2763 y FG(2)1509 2851 y FI(2)p FH(e)1590 2863 y FG(1)1988 2851 y FI(2)p FH(e)2069 2863 y FG(2)2328 2851 y FI(1)g FA(\000)g FH(e)2510 2820 y FG(2)2510 2871 y(2)2566 2851 y FA(\000)g FH(e)2688 2820 y FG(2)2688 2871 y(1)2766 2585 y FC(3)2766 2734 y(5)2856 2752 y FH(;)299 b FQ(\(4.40\))515 3023 y(which)21 b(tak)o(es)i FL(e)976 2992 y FF(>)1059 3023 y FA(7!)j FI(\(0)p FH(;)14 b FI(0)p FH(;)g FI(1\))1432 2992 y FF(>)1487 3023 y FQ(.)31 b(If)22 b(the)g(image)f(coordinates)g(are)h(transformed) d(in)j(image)g(1)g(by)515 3122 y FL(A)f FQ(and)e(in)i(image)e(2)i(by)e FL(A)1296 3086 y FF(0)1340 3122 y FQ(then)1591 3285 y FI(\()p FL(A)1695 3249 y FF(0)1718 3285 y FL(x)1769 3251 y FF(0)1792 3285 y FI(\))1824 3251 y FF(>)1880 3285 y FL(F)1940 3249 y FF(0)1964 3285 y FI(\()p FL(Ax)q FI(\))k(=)g(0)874 b FQ(\(4.41\))515 3448 y(where)30 b FL(F)809 3412 y FF(0)874 3448 y FI(=)42 b FL(A)1053 3412 y FF(0>)1129 3448 y FL(FA)1261 3412 y FF(\000)p FG(1)1350 3448 y FQ(.)57 b(The)30 b(minimization)f(on) h FL(F)2236 3412 y FF(0)2291 3448 y FQ(is)h(conducted)d(in)j(the)g (transformed)515 3548 y(coordinate)18 b(system.)639 3647 y(The)24 b(non-linear)e(minimization)g(using)i(the)g(parameterization)d (in)j(\(4.39\))e(is)j(referred)e(to)h(as)515 3747 y FL(N)590 3759 y FK(1)631 3747 y FQ(,)30 b(and)d(using)h(the)f(parameterization)f (in)h(\(4.38\))f(is)j(referred)d(to)i(as)g FL(N)2716 3759 y FK(2)2758 3747 y FQ(.)48 b(Method)27 b FL(N)3189 3759 y FK(3)3258 3747 y FQ(is)i(a)515 3847 y(gradient)21 b(descent)h(minimization)f(on)h(the)g(epipolar)g(distance,)g(without)g (enforcing)e FA(j)p FL(F)p FA(j)27 b FI(=)g(0)p FQ(.)32 b(It)515 3946 y(is)21 b(pro)o(vided)d(as)i(a)h(benchmark)d(with)i (which)g(to)g(compare)f FL(N)2302 3958 y FK(1)2364 3946 y FQ(and)h FL(N)2580 3958 y FK(2)2621 3946 y FQ(.)639 4046 y(Gradient)e(descent)h(methods)f(are)h(vulnerable)e(to)i(local)g (minima)f(if)h(the)g(starting)g(point)f(is)i(not)515 4146 y(near)g(the)h(actual)f(solution.)26 b(The)20 b(starting)h(point)f (we)h(use)g(is)g(the)g(output)f(of)g(the)h(linear)f(algorithm)515 4245 y FL(O)587 4257 y FK(2)651 4245 y FQ(described)h(abo)o(v)o(e,)f (follo)n(wed)h(by)h(the)g(procrustean)e(con)m(v)o(ersion)f(of)j(the)g (estimate)h(to)f(a)g(rank)515 4345 y(tw)o(o)e(matrix)g(using)f(SVD.)515 4621 y FD(4.9)119 b(Constrained)31 b(Estimation)d(of)i Fo(F)515 4807 y FQ(Matlab)22 b(pro)o(vides)e(a)j(generic)f(constrained) e(estimator)i(for)g FL(F)h FO(fmincon)p FQ(.)31 b(This)22 b(allo)n(ws)h(for)f(the)515 4907 y(minimization)f(of)h(functions)f (subject)i(to)g(constraints)f(on)g(the)h(parameters,)f(thus)g (minimization)515 5006 y(of)c FL(F)h FQ(can)g(be)g(undertak)o(en)d (subject)i(to)h FA(j)p FL(F)p FA(j)k FI(=)g(0)c FQ(and)f(the)h (Frobenius)e(norm)h(of)g(the)h(top)f FI(2)13 b FA(\002)g FI(2)18 b FQ(=)h(1.)p eop %%Page: 36 38 36 37 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(36)515 523 y Fs(4.9.1)99 b(Constrained)19 b(Estimator)g(function:)30 b Fr(torr)p 2465 523 30 4 v 35 w(nonlinf)p 2920 523 V 34 w(mincon2x2)515 679 y FO(function)48 b(f)i(=)f(torr_nonlinf_mincon2x2\(f_init,)c (nx1,ny1,nx2,)515 778 y(ny2,)k(no_matches,)f(m3\))515 961 y FQ(with)26 b(most)h(parameters)e(de\002ned)g(as)i(in)g(Section)f (4.1.1,)g(e)o(xcept)g FO(f)p 2517 961 25 4 v 29 w(init)h FQ(which)f(is)h(an)f(initial)515 1061 y(estimate)j(of)f FL(f)39 b FQ(furnished)27 b(by)h(some)h(other)f(estimator)-5 b(.)51 b(The)28 b(initial)h(estimate)g(should)f(satisfy)515 1160 y FA(j)p FL(F)p FA(j)23 b FI(=)g(0)c FQ(otherwise)f(instability)h (may)f(result.)25 b(Note)19 b FO(torr)p 2266 1160 V 29 w(nonlcon)p 2645 1160 V 28 w(f2x2)g FQ(is)h(the)f(call)g(back)515 1260 y(function)f(for)i FO(torr)p 1135 1260 V 29 w(nonlinf)p 1514 1260 V 28 w(mincon2x2)g FQ(that)g(e)o(xpresses)f(the)i(tw)o(o)f (constraints:)515 1443 y FO(function)48 b([c,ceq])h(=)g (torr_nonlcon_f2x2\(f,)d(nx1,ny1,nx2,ny2,)h(m3\))515 1542 y(\045c)i(=)h(...)248 b(\045)50 b(Compute)e(nonlinear)h (inequalities)e(at)j(f.)515 1642 y(\045ceq)f(=)g(...)149 b(\045)50 b(Compute)e(nonlinear)h(equalities)f(at)h(f.)515 1841 y(\045g\(1\))g(=)g(norm\(f\))g(-1.0;)515 1941 y(\045g\(2\))g(=)g (f\(1\))g(*)h(\(f\(5\))f(*)g(f\(9\))h(-)f(f\(6\))g(*)h(f\(8\)\))f(-)g (f\(2\))g(*)h(\(f\(4\))f(*)h(f\(9\))f(-)515 2040 y(f\(6\))g(*)g (f\(7\)\))g(+)h(f\(3\))f(*)h(\(f\(4\))f(*)g(f\(8\))g(-)h(f\(5\))f(*)h (f\(7\)\);)515 2140 y(c)f(=)h([];)515 2240 y(\045what)f(norm)g(should)f (we)i(use!)515 2339 y(ceq\(1\))e(=)i(sqrt\(f\(1\)\2102)e(+)h (f\(2\)\2102)g(+)h(f\(4\)\2102)e(+)i(f\(5\)\2102\)-)e(1;)515 2439 y(\045ceq\(1\)=)g(norm\(f\))h(-1.0;)515 2538 y(ceq\(2\))f(=)i (f\(1\))f(*)h(\(f\(5\))f(*)g(f\(9\))g(-)h(f\(6\))f(*)h(f\(8\)\))f(-)g (f\(2\))g(*)h(\(f\(4\))f(*)g(f\(9\))g(-)515 2638 y(f\(6\))g(*)g (f\(7\)\))g(+)h(f\(3\))f(*)h(\(f\(4\))f(*)g(f\(8\))g(-)h(f\(5\))f(*)h (f\(7\)\);)515 2918 y FD(4.10)119 b(Thoughts)29 b(on)i(T)-11 b(esting)29 b(Estimators)515 3104 y FQ(W)m(ithin)21 b(this)g(section)g (tests)i(a)e(testing)g(methodology)d(is)k(described)e(for)g(e)n(v)n (aluating)g(ho)n(w)g(good)g(a)515 3203 y(method)f(for)g(\002tting)h FL(F)h FQ(is.)26 b(F)o(or)20 b(synthetic)f(data,)h(where)g(the)g (ground)e(truth)i(is)h(kno)n(wn,)d(an)i(empir)n(-)515 3303 y(ical)j(measure)e(of)h(the)h(goodness)e(of)h(\002t)h(is)h(achie)n (v)o(ed)d(by)h(calculating)f(the)h(reprojection)f(error)g(of)515 3403 y(the)k Ft(actual)f FQ(noise)i(free)e(projections)g(of)h(the)h (synthetic)e(w)o(orld)h(points)g(to)g FL(F)h FQ(pro)o(vided)d(by)i (each)515 3502 y(estimator)-5 b(.)36 b(T)m(raditionally)22 b(the)i(goodness)e(of)i(\002t)h(has)f(been)f(assessed)i(by)e(seeing)h (ho)n(w)f(well)i(the)515 3602 y(parameters)17 b(\002t)i(the)g Ft(observed)f FQ(data.)24 b(But)19 b(we)g(point)f(out)g(that)h(this)g (is)g(the)g(wrong)e(criterion)g(as)j(the)515 3702 y(aim)i(is)h(to)f (\002nd)g(the)g(set)g(of)g(parameters)f(that)h(best)g(\002t)h(the)f (\(unkno)n(wn\))d Ft(true)j(data)p FQ(.)29 b(The)22 b(parame-)515 3801 y(ters)17 b(of)g(the)f(fundamental)f(matrix)h(themselv)o(es)g(are) h(not)f(of)h(primary)e(importance,)g(rather)h(it)i(is)g(the)515 3901 y(structure)f(of)i(the)f(corresponding)d(epipolar)i(geometry)-5 b(.)23 b(Consequently)16 b(it)k(mak)o(es)e(little)h(sense)g(to)515 4000 y(compare)e(tw)o(o)h(solutions)g(by)g(directly)g(comparing)e (corresponding)f(parameters)i(in)i(their)f(funda-)515 4100 y(mental)i(matrices;)h(one)f(must)h(rather)f(compare)f(the)i(the)g (dif)n(ference)e(in)i(the)f(associated)h(epipolar)515 4200 y(geometry)j(weighted)h(by)g(the)h(density)g(of)f(the)h(gi)n(v)o (en)f(matching)f(points.)42 b(This)26 b(error)f(metric)h(is)515 4299 y(the)j(\002rst)h(order)e(approximation)e(of)i(the)h(reprojection) e(error)h(of)h(the)g(noise)g(free)g(points)g(to)g FL(F)p FQ(:)515 4399 y FA(E)559 4411 y FG(1)626 4399 y FI(=)720 4337 y FC(P)808 4357 y FB(n)808 4424 y(i)p FG(=1)920 4399 y FI(\()p FH(w)p 952 4412 62 4 v 1014 4420 a FB(i)1041 4399 y FL(f)1079 4362 y FF(>)1135 4399 y FL(z)1178 4411 y FB(i)p 1135 4425 71 4 v 1206 4399 a FI(\))1238 4369 y FG(2)1275 4399 y FQ(.)37 b(The)23 b(second)g(statistic)i FA(E)2063 4411 y FG(2)2125 4399 y FQ(is)g(the)f(a)n(v)o(erage)e (distance)i(in)g(pix)o(els)g(from)515 4499 y(the)c(true)g(epipole)f(in) h(each)g(image)g(to)g(that)g(yielded)g(by)f(the)i(estimate)f(of)g FL(F)p FQ(.)515 4736 y Fs(4.10.1)98 b(T)-9 b(est)26 b(script:)31 b Fr(torr)p 1615 4736 30 4 v 35 w(evalFsc)515 4892 y FQ(This)e(script)h(allo)n(ws)f(the)h(user)f(to)h(estimate)f FL(F)h FQ(with)g(a)g(method)e(determined)f(by)i(the)h(v)n(ariable)515 4991 y FO(method)19 b FQ(and)h(display)g FA(E)1280 5003 y FG(1)1337 4991 y FQ(and)g FA(E)1522 5003 y FG(2)1559 4991 y FQ(.)26 b(The)19 b(code)h(is)h(simple:)p eop %%Page: 37 39 37 38 bop 515 232 a FJ(CHAPTER)21 b(4.)45 b(DETERMIN)m(A)-9 b(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)308 b FQ(37)515 523 y FS(\045profile)43 b(on)515 706 y(m3)h(=)h(256;)515 797 y(\045user)f(chooses)f(the)i(appropriate)e (method)515 888 y(method)h(=)g(3;)515 980 y(total_sse)f(=)i(0;)515 1071 y(epipole_distance)d(=)j(0;)515 1162 y(\045)515 1254 y(\045)f(randn\('state',0\))515 1345 y(\045)g(rand\('state',0\)) 515 1436 y(no_tests)f(=)i(1;)515 1528 y(for\(i)f(=)g(1:no_tests\))515 1619 y(\045)224 b([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2])39 b(=)44 b(...)515 1710 y(\045)403 b(torr_gen_2view_matches\(foc,)40 b(no_matches,)j(noise_sigma,)g(trans-)515 1802 y(lation_mult,)g (translation_adder,)e(...)515 1893 y(\045)403 b(rotation_multplier,)42 b(min_Z,Z_RAN,m3\);)694 1984 y ([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2,true)o(_C,tr)o(ue_R,t)o(rue_t,)c (true_E])44 b(=)515 2076 y(torr_gen_2view_matches;)560 2167 y(true_epipole)e(=)j(torr_get_right_epipole\(true_F,m3\);)560 2350 y(no_matches)e(=)h(length\(nx1\);)560 2532 y(matches)f(=)i ([nx1,ny1,nx2,ny2];)560 2623 y(perfect_matches)d(=)j([x1,y1,x2,y2];)560 2715 y(set_rank2)e(=)i(0;)560 2897 y(\045first)e(estimate)h(F)560 2989 y([f,)g(e1,)g(n_inliers,inlier_index,nF])d(=)515 3080 y(torr_estimateF\()h(matches,)i(m3,)g([],)g(method,)g (set_rank2\);)560 3263 y(\045check)f(errors)h(vs)h(the)f(true)g (\(noise)g(free\))g(points)560 3354 y(groundtrutherrors)e(=)i (torr_errf2\(f,)f(x1,y1,x2,y2,)g(no_matches,)g(m3\);)560 3445 y(total_sse)g(=)i(total_sse)e(+)i(sum\(groundtrutherrors\);)560 3719 y(\045calc)f(noisy)g(epipole)560 3811 y(noisy_epipole)e(=)j (torr_get_right_epipole\(nF,m3\);)560 3902 y(epipole_distance)d(=)i (epipole_distance)f(+)h(sqrt\(norm\(true_epipole)d(-noisy_epipole\)\);) 515 3993 y(end)515 4176 y(disp\(')j(the)g(average)g(sse)g(vs)g(the)h (noise)f(free)g(points)g(is'\))515 4267 y(total_sse/no_tests)515 4359 y(\045profile)f(off)515 4541 y(disp\('RMS)g(distance)h(between)f (true)h(and)h(estimated)e(right)h(epipole)g(is'\))515 4633 y(epipole_distance)p eop %%Page: 38 40 38 39 bop 515 1146 a FE(Chapter)44 b(5)515 1561 y FN(Rob)l(ust)51 b(Estimation)i(of)f Fi(F)515 1993 y FQ(Rob)n(ust)29 b(parameter)f 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Ft(et)j(al.)48 b FQ([32)n(])24 b(it)g(is)g(sho)n(wn)f(that)g(linear)g (methods)g(produce)e(a)j(biased)f(solution,)g(and,)g(fur)n(-)515 2061 y(thermore)17 b(selecting)h(more)h(points)f(e)o(xponentially)e (increases)j(the)g(chance)f(that)h(the)g(set)g(contains)515 2160 y(an)j(outlier)-5 b(.)30 b(T)-7 b(o)22 b(illustrate)g(this)h(the)f (performance)d(of)i(the)h(tw)o(o)h(approaches)d(w)o(as)i(monitored)e (for)515 2260 y(100)f(trials)h(each)f(of)h(200)f(synthetically)f (generated)g(matches)h(with)h(added)f(noise,)g(using)h(7)f(and)g(8)515 2359 y(point)d(samplings.)23 b(The)17 b(a)n(v)o(erage)e(v)n(ariance)h (of)g(the)h(distance)g(of)f(point)g(to)h(the)g(estimated)g(epipolar)515 2459 y(lines)j(w)o(as)h(1.4)f(pix)o(els)g(for)f(7)i(point)e(sampling)g (and)h(12.7)f(pix)o(els)h(for)f(8)i(point)e(sampling!)639 2559 y(In)g(order)e(to)i(determine)f(whether)f(or)i(not)f(a)h(feature)f (pair)h(is)g(consistent)g(with)g(a)g(gi)n(v)o(en)e(funda-)515 2658 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4800 y(In)h(general)f(if)h(the)g(se)n(v)o(en)f (correspondence)d(sample)k(has)g(an)g(insuf)n(\002cient)e(spread)i(of)f (dispar)n(-)515 4900 y(ities)28 b(then)f(the)h(estimate)f(of)h FL(F)g FQ(obtained)e(from)g(that)i(sample)f(might)g(not)g(be)h(unique.) 45 b(This)28 b(is)515 5000 y(an)c(e)o(xample)f(of)h(de)o(generac)o(y)-5 b(.)33 b(Consider)24 b(the)g(se)n(v)o(en)f(correspondences)e(sho)n(wn)j (in)g(Figure)f(5.2.)p eop %%Page: 42 44 42 43 bop 515 232 a FJ(CHAPTER)21 b(5.)45 b(R)m(OB)o(UST)22 b(ESTIMA)-9 b(TION)18 b(OF)j FL(F)1224 b FQ(42)p 515 452 3240 4 v 515 2087 4 1636 v 664 616 a(1.)41 b(Repeat)20 b(for)g FH(m)g FQ(samplings)g(as)h(determined)d(in)i(T)-7 b(able)21 b(5.2:)817 782 y(\(a\))41 b(Select)24 b(a)g(random)d(sample)j (of)f(the)g(minimum)f(number)g(of)h(data)h(points)f(to)g(mak)o(e)g(a)h (pa-)951 882 y(rameter)19 b(estimate)h FL(F)p FQ(.)812 1015 y(\(b\))41 b(Calculate)20 b(the)g(distance)g FH(d)1743 1027 y FB(i)1792 1015 y FQ(of)f(each)h(feature)g(to)g(the)g(epipolar)f 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FO(T)p FQ(,)21 b(threshold)d(for)i (inliers.)515 3947 y FM(Output:)639 4113 y FA(\017)41 b FO(f)21 b FQ(de\002ned)e(as)i(in)f(Section)g(4.1.1..)639 4279 y FA(\017)41 b FO(f)p 777 4279 V 30 w(sq)p 907 4279 V 30 w(errors)19 b FQ(the)h(squared)f(error)g(for)h(each)g(match.)639 4445 y FA(\017)41 b FO(n)p 777 4445 V 30 w(inliers)19 b FQ(the)i(number)d(of)i(inliers!)639 4611 y FA(\017)41 b FO(inlier)p 1027 4611 V 29 w(index)20 b FQ(1)g(if)h(a)f(match)g(is)h (inlying,)e(0)h(otherwise.)p eop %%Page: 44 46 44 45 bop 515 232 a FJ(CHAPTER)21 b(5.)45 b(R)m(OB)o(UST)22 b(ESTIMA)-9 b(TION)18 b(OF)j FL(F)1224 b FQ(44)515 523 y Fs(5.2.3)99 b(Least)25 b(Median)g(Estimator)515 679 y FQ(Surprisingly)-5 b(,)31 b(RANSA)m(C)h(originated)e(in)h(the)g (\002eld)h(of)e(computer)g(vision)g(and)h(it)h(w)o(as)g(a)g(fe)n(w)515 778 y(years)18 b(later)h(that)f(a)h(similar)g(highly)e(rob)n(ust)h (estimator)g(w)o(as)h(de)n(v)o(eloped)d(in)j(the)f(\002eld)h(of)f (statistics,)515 878 y(namely)h(Rousseeuw')-5 b(s)20 b(least)h(median)e(square)h(\(LMS\))f(estimator)h([40)o(].)25 b(The)20 b(algorithms)f(dif)n(fer)515 978 y(slightly)f(in)h(that)g(the) g(solution)e(gi)n(ving)h(least)h(median)f(is)i(selected)e(as)i(the)e (estimate)h(in)g([40)o(].)25 b(Both)515 1077 y(estimators)19 b(perform)e(v)o(ery)i(well,)g(RANSA)m(C)i(gi)n(v)o(es)e(better)f (performance)f(for)i(up)g(to)g FI(35\045)g FQ(abo)o(v)o(e)515 1177 y(which)g(LMS)i(gi)n(v)o(es)f(a)g(mar)o(ginally)e(better)i (performance.)515 1454 y FD(5.3)119 b(Maximum)24 b(Lik)o(elihood)i (Estimation)d(in)j(the)f(Pr)n(esence)f(of)784 1604 y(Outliers:)40 b FW(M)6 b(A)g(P)g(S)g(A)q(C)517 1789 y FQ(M)t(A)t(P)t(S)t(A)q(C)23 b(stands)c(for)g(Maximum)f(a)i(posteriori)e(sample)i(consensus,)e(and)h (is)i(a)f(generalization)515 1889 y(of)25 b(my)g(pre)n(vious)f (algorithm,)j(M)t(L)t(E)t(S)t(A)q(C)h(to)e(the)f(Bayesian)g(case,)i (although)d(when)h(there)g(is)h(a)515 1988 y(uniform)14 b(prior)i(on)g(the)g(parameters)g(the)g(tw)o(o)h(are)g(the)f(same.)24 b(W)m(ithin)16 b(this)h(section)g(the)f(maximum)515 2088 y(lik)o(elihood)g(formulation)g(is)k(gi)n(v)o(en)d(for)g(computing)f (an)o(y)i(of)g(the)g(multiple)g(vie)n(w)g(relations,)g(as)h(the)515 2188 y(prior)f(on)i FL(F)g FQ(is)h(not)e(o)o(v)o(erly)f(important)g(in) i(the)f(case)i(of)e(estimation,)g(and)g(the)j(M)t(L)t(E)g(deri)n(v)n (ation)c(is)515 2287 y(much)h(simpler)h(to)g(follo)n(w)g(than)f(the)k (M)t(A)t(P)g(deri)n(v)n(ation)18 b(e.g.)24 b([50)o(,)d(62)o(].)639 2387 y(In)f(the)h(follo)n(wing)d(we)j(mak)o(e)f(the)g(assumption,)f (without)h(loss)h(of)f(generality)-5 b(,)18 b(that)i(the)h(noise)515 2486 y(in)i(the)g(tw)o(o)h(images)f(is)h(Gaussian)f(on)g(each)g(image)g (coordinate)e(with)i(zero)g(mean)g(and)g(uniform)515 2586 y(standard)14 b(de)n(viation)h FH(\033)s FQ(.)24 b(Thus)16 b(gi)n(v)o(en)e(a)j(true)e(correspondence)d(the)k (probability)e(density)h(function)515 2686 y(of)20 b(the)g(noise)g (perturbed)e(data)i(is)695 2904 y FI(Pr)o(\()p FL(D)q FA(j)p FL(M)p FI(\))j(=)1197 2825 y FC(Y)1147 3002 y FB(i)p FG(=1)p FB(:::)o(n)1368 2787 y FC(\022)1524 2848 y FI(1)p 1439 2885 212 4 v 1439 2901 a FA(p)p 1508 2901 92 4 v 69 x FI(2)p FH(\031)s(\033)1661 2787 y FC(\023)1722 2804 y FB(n)1781 2904 y FH(e)1820 2856 y FF(\000)1872 2795 y FC(\000)1909 2800 y(P)1997 2887 y Fl(j)r Fq(=1)p Fl(;)p Fq(2)2146 2856 y FG(\()p FB(x)p 2172 2869 38 4 v -28 x Fl(j)2210 2877 y(i)2241 2856 y FF(\000)p FB(x)2331 2828 y Fl(j)2331 2877 y(i)2361 2856 y FG(\))2387 2831 y Fq(2)2420 2856 y FG(+\()p FB(y)p 2497 2880 36 4 v 2532 2831 a Fl(j)2532 2888 y(i)2563 2856 y FF(\000)p FB(y)2651 2828 y Fl(j)2649 2877 y(i)2682 2856 y FG(\))2708 2831 y Fq(2)2740 2795 y FC(\001)2779 2856 y FB(=)p FG(\(2)p FB(\033)p 2872 2869 41 4 v 2912 2831 a Fq(2)2945 2856 y FG(\))3016 2904 y FH(;)181 b FQ(\(5.4\))515 3142 y(where)24 b FH(n)i FQ(is)g(the)f(number)e(of)h(correspondences)e(and)j FL(M)g FQ(is)h(the)f(appropriate)e(2)i(vie)n(w)g(relation,)515 3242 y(e.g.)d(the)h(fundamental)d(matrix)i(or)g(projecti)n(vity)-5 b(,)21 b(and)h FL(D)i FQ(is)g(the)e(set)i(of)e(matches.)32 b(The)23 b(ne)o(gati)n(v)o(e)515 3341 y(log)d(lik)o(elihood)e(of)i(all) h(the)f(correspondences)d FL(x)1939 3301 y FG(1)p FB(;)p FG(2)1939 3364 y FB(i)2029 3341 y FQ(,)j FH(i)j FI(=)f(1)p FH(::n)f FQ(:)644 3535 y FA(\000)766 3456 y FC(X)723 3633 y FB(i)p FG(=1)p FB(:::)o(n)944 3535 y FI(log\(Pr)o(\()p FL(x)1255 3495 y FG(1)p FB(;)p FG(2)1255 3558 y FB(i)1345 3535 y FA(j)p FL(M)p FH(;)14 b(\033)s FI(\)\))24 b(=)1765 3456 y FC(X)1721 3633 y FB(i)p FG(=1)p FB(:::n)1967 3456 y FC(X)1943 3633 y FB(j)s FG(=1)p FB(;)p FG(2)2125 3443 y FC(\020)2174 3535 y FI(\()p FH(x)p 2206 3548 48 4 v 2254 3495 a FB(j)2254 3558 y(i)2307 3535 y FA(\000)18 b FH(x)2437 3495 y FB(j)2437 3558 y(i)2473 3535 y FI(\))2505 3501 y FG(2)2561 3535 y FI(+)g(\()p FH(y)p 2676 3565 44 4 v 2720 3501 a FB(j)2720 3572 y(i)2773 3535 y FA(\000)g FH(y)2900 3495 y FB(j)2897 3558 y(i)2935 3535 y FI(\))2967 3501 y FG(2)3004 3443 y FC(\021)3068 3535 y FH(;)129 b FQ(\(5.5\))515 3785 y(discounting)24 b(the)i(constant)g(term.)42 b(Observing)25 b(the)h(data,)i(we)e(infer)g(that)g(the)g(true)g (relation)f FL(M)515 3884 y FQ(minimizes)20 b(this)i(log)e(lik)o (elihood.)26 b(This)21 b(inference)e(is)j(called)e(\223Maximum)g(Lik)o (elihood)f(Estima-)515 3984 y(tion\224.)639 4084 y(Gi)n(v)o(en)f(tw)o (o)h(vie)n(ws)g(with)g(associated)f(relation)g(for)g(each)g (correspondence)e FL(x)2890 4053 y FG(1)p FB(;)p FG(2)2999 4084 y FQ(the)j(task)g(be-)515 4183 y(comes)i(that)g(of)g(\002nding)f (the)h(maximum)e(lik)o(elihood)h(estimate,)2406 4182 y FI(^)2401 4183 y FL(x)2452 4145 y FG(1)p FB(;)p FG(2)2564 4183 y FQ(of)g(the)i(true)e(position)h FL(x)p 3218 4196 51 4 v -30 x FG(1)p FB(;)p FG(2)3358 4183 y FQ(,)515 4322 y(such)f(that)838 4321 y FI(^)834 4322 y FL(x)884 4284 y FG(1)p FB(;)p FG(2)995 4322 y FQ(satis\002es)i(the)f(relation)f (and)g(minimizes)2185 4260 y FC(P)2273 4347 y FB(j)s FG(=1)p FB(;)p FG(2)2458 4230 y FC(\020)2513 4322 y FI(^)-47 b FH(x)2555 4282 y FB(j)2555 4345 y(i)2609 4322 y FA(\000)18 b FH(x)2739 4282 y FB(j)2739 4345 y(i)2774 4230 y FC(\021)2824 4247 y FG(2)2880 4322 y FI(+)2963 4230 y FC(\020)3018 4322 y FI(^)-47 b FH(y)3057 4282 y FB(j)3054 4345 y(i)3110 4322 y FA(\000)18 b FH(y)3237 4282 y FB(j)3234 4345 y(i)3271 4230 y FC(\021)3321 4247 y FG(2)3358 4322 y FQ(.)515 4441 y(The)i(MLE)g(error)f FH(e)1082 4453 y FB(i)1130 4441 y FQ(for)g(the)h FH(i)p FQ(th)g(point)g(is)h(then)1313 4646 y FH(e)1352 4612 y FG(2)1352 4667 y FB(i)1412 4646 y FI(=)1524 4567 y FC(X)1500 4744 y FB(j)s FG(=1)p FB(;)p FG(2)1681 4554 y FC(\020)1736 4646 y FI(^)-47 b FH(x)1778 4606 y FB(j)1778 4669 y(i)1832 4646 y FA(\000)18 b FH(x)1962 4606 y FB(j)1962 4669 y(i)1997 4554 y FC(\021)2047 4571 y FG(2)2102 4646 y FI(+)2185 4554 y FC(\020)2241 4646 y FI(^)-48 b FH(y)2279 4606 y FB(j)2276 4669 y(i)2332 4646 y FA(\000)18 b FH(y)2459 4606 y FB(j)2456 4669 y(i)2494 4554 y FC(\021)2543 4571 y FG(2)3220 4646 y FQ(\(5.6\))515 4907 y(Thus)698 4844 y FC(P)786 4932 y FB(i)p FG(=1)p FB(:::)o(n)1011 4907 y FH(e)1050 4876 y FG(2)1050 4928 y FB(i)1104 4907 y FQ(pro)o(vides)d(the)i(error)f(function)e(for)i(the) h(point)f(data,)h(and)f FL(M)h FQ(for)f(which)3174 4844 y FC(P)3262 4932 y FB(i)3303 4907 y FH(e)3342 4876 y FG(2)3342 4928 y FB(i)515 5006 y FQ(is)k(a)g(minimum)e(is)i(the)f (maximum)f(lik)o(elihood)f(estimate)j(of)f(the)g(relation)g (\(fundamental)d(matrix,)p eop %%Page: 45 47 45 46 bop 515 232 a FJ(CHAPTER)21 b(5.)45 b(R)m(OB)o(UST)22 b(ESTIMA)-9 b(TION)18 b(OF)j FL(F)1224 b FQ(45)515 526 y(or)19 b(projecti)n(vity\).)k(Hartle)o(y)c(and)g(Sturm)g([23)o(])h (sho)n(w)f(ho)n(w)g FH(e)p FQ(,)2287 525 y FI(^)2283 526 y FL(x)h FQ(and)2498 525 y FI(^)2494 526 y FL(x)2544 463 y Fn(0)2591 526 y FQ(may)f(be)h(found)d(as)k(the)f(so-)515 626 y(lution)g(of)h(a)h(de)o(gree)d(6)j(polynomial.)j(A)d (computationally)c(ef)n(\002cient)j(\002rst)h(order)e(approximation)515 726 y(to)g(these)g(is)i(gi)n(v)o(en)c(in)j(T)-7 b(orr)19 b Ft(et)i(al.)f FQ([54)o(,)g(55)o(,)h(56)o(].)639 825 y(The)h(abo)o(v)o(e)e(deri)n(v)n(ation)g(assumes)i(that)f(the)h(errors) f(are)h(Gaussian,)g(often)e(ho)n(we)n(v)o(er)g(features)515 925 y(are)26 b(mismatched)g(and)g(the)h(error)e(on)h FL(m)g FQ(is)i(not)e(Gaussian.)44 b(Thus)27 b(the)f(error)g(is)i (modeled)d(as)i(a)515 1025 y(mixture)19 b(model)g(of)h(Gaussian)g(and)g (uniform)e(distrib)n(ution:-)1157 1241 y FI(Pr)o(\()p FH(e)p FI(\))23 b(=)1460 1124 y FC(\022)1521 1241 y FH(\015)1682 1185 y FI(1)p 1578 1222 249 4 v 1578 1238 a FA(p)p 1648 1238 180 4 v 1648 1311 a FI(2)p FH(\031)s(\033)1790 1287 y FG(2)1851 1241 y FI(exp\()p FA(\000)2111 1185 y FH(e)2150 1154 y FG(2)p 2085 1222 130 4 v 2085 1298 a FI(2)p FH(\033)2177 1274 y FG(2)2224 1241 y FI(\))18 b(+)g(\(1)h FA(\000)f FH(\015)5 b FI(\))2624 1185 y(1)p 2623 1222 44 4 v 2623 1298 a FH(v)2676 1124 y FC(\023)3220 1241 y FQ(\(5.7\))515 1468 y(where)19 b FH(\015)26 b FQ(is)21 b(the)f(mixing)f(parameter)g (and)g FH(v)24 b FQ(is)d(just)g(a)g(constant,)e FH(\033)24 b FQ(is)d(the)f(standard)f(de)n(viation)g(of)515 1559 y(the)d(error)g(on)g(each)h(coordinate.)k(T)-7 b(o)17 b(correctly)e(determine)h FH(\015)21 b FQ(and)16 b FH(v)21 b FQ(entails)c(some)f(kno)n(wledge)f(of)515 1650 y(the)22 b(outlier)f(distrib)n(ution;)h(here)g(it)h(is)g(assumed)e(that)h(the)h (outlier)e(distrib)n(ution)g(is)i(uniform,)d(with)515 1742 y FA(\000)590 1709 y FB(v)p 589 1723 36 4 v 590 1770 a FG(2)635 1742 y FH(::)h FI(+)797 1709 y FB(v)p 797 1723 V 798 1770 a FG(2)866 1742 y FQ(being)i(the)g(pix)o(el)g (range)g(within)g(which)g(outliers)g(are)h(e)o(xpected)d(to)j(f)o(all)g (\(for)f(feature)515 1833 y(matching)c(this)j(is)g(dictated)e(by)h(the) g(size)h(of)f(the)g(search)f(windo)n(w)g(for)h(matches\).)27 b(Therefore)19 b(the)515 1924 y(error)g(minimized)g(is)i(the)f(ne)o (gati)n(v)o(e)e(log)i(lik)o(elihood:)515 2164 y Ff(\000)p Fe(L)h FR(=)g Ff(\000)802 2087 y FC(X)851 2262 y Fd(i)934 2164 y FR(log)1046 2024 y FC( )1112 2164 y Fe(\015)1169 2049 y FC(\022)1319 2116 y FR(1)p 1240 2147 196 4 v 1240 2164 a Ff(p)p 1304 2164 86 4 v 61 x FR(2)p Fe(\031)s(\033)1445 2049 y FC(\023)1507 2066 y Fd(n)1562 2164 y FR(exp)1692 2024 y FC( )1758 2164 y Ff(\000)1830 2024 y FC( )1913 2087 y(X)1896 2262 y Fd(j)s Fc(=1)p Fd(;)p Fc(2)2051 2164 y FR(\()p Fe(x)p 2081 2177 44 4 v 2124 2128 a Fd(j)2124 2185 y(i)2174 2164 y Ff(\000)c Fe(x)2295 2128 y Fd(j)2295 2184 y(i)2327 2164 y FR(\))2357 2128 y Fc(2)2408 2164 y FR(+)g(\()p Fe(y)p 2515 2192 41 4 v 2555 2128 a Fd(j)2555 2200 y(i)2605 2164 y Ff(\000)g Fe(y)2723 2128 y Fd(j)2720 2184 y(i)2755 2164 y FR(\))2785 2128 y Fc(2)2819 2024 y FC(!)2898 2164 y Fe(=)p FR(\(2)p Fe(\033)3051 2128 y Fc(2)3086 2164 y FR(\))f(+)h(\(1)h Ff(\000)e Fe(\015)t FR(\))3456 2116 y(1)p 3455 2147 40 4 v 3455 2215 a Fe(v)3505 2024 y FC(!!)3678 2164 y Fe(:)3236 2339 y FT(\(5.8\))532 2439 y FQ(Gi)n(v)o(en)f(a)i(suitable)f(initial)g(estimate)h(there)f (are)g(se)n(v)o(eral)g(w)o(ays)g(to)h(estimate)f(the)g(parameters)g(of) g(the)515 2538 y(mixture)i(model,)h(most)h(prominent)e(being)h(the)i(E) t(M)i(algorithm)18 b([10)o(,)i(34)o(],)g(b)n(ut)g(gradient)e(descent) 515 2638 y(methods)e(could)g(also)h(be)g(used.)24 b(Because)17 b(of)f(the)h(presence)f(of)h(outliers)g(in)g(the)g(data)g(the)g (standard)515 2738 y(method)k(of)h(least)i(squares)e(estimation)g(is)i (often)d(not)i(suitable)f(as)h(an)g(initial)g(estimate,)g(and)f(it)i (is)515 2837 y(better)16 b(to)h(use)g(a)g(rob)n(ust)g(estimate)g(such)f (as)k(R)t(A)t(N)t(S)t(A)q(C)h(which)16 b(is)i(described)d(in)i(the)g (ne)o(xt)f(section.)515 3117 y FD(5.4)119 b(The)30 b(r)n(ob)n(ust)f (estimators:)38 b FW(M)6 b(A)g(P)g(S)g(A)q(C)515 3303 y FQ(The)22 b(R)t(A)t(N)t(S)t(A)q(C)i(algorithm)19 b(has)h(pro)o(v)o (en)e(v)o(ery)h(successful)h(for)g(rob)n(ust)g(estimation,)f(b)n(ut)i (ha)n(ving)515 3403 y(de\002ned)d(the)h(rob)n(ust)f(ne)o(gati)n(v)o(e)f (log)i(lik)o(elihood)f(function)f FA(\000)p FH(L)i FQ(as)h(the)f (quantity)f(to)h(be)g(minimized)515 3502 y(it)i(becomes)e(apparent)f (that)23 b(R)t(A)t(N)t(S)t(A)q(C)h(can)c(be)g(impro)o(v)o(ed)d(on.)639 3602 y(One)f(of)f(the)g(problems)f(with)j(R)t(A)t(N)t(S)t(A)q(C)j(is)c (that)f(if)h(the)f(threshold)f FH(T)27 b FQ(for)15 b(considering)e (inliers)515 3701 y(is)23 b(set)h(too)e(high)g(then)g(the)h(rob)n(ust)f (estimate)h(can)f(be)g(v)o(ery)g(poor)-5 b(.)31 b(Consideration)21 b(of)k(R)t(A)t(N)t(S)t(A)q(C)515 3801 y(sho)n(ws)20 b(that)g(in)h(ef)n (fect)e(it)i(\002nds)f(the)h(minimum)d(of)i(a)h(cost)f(function)f (de\002ned)g(as)1618 3984 y FH(C)30 b FI(=)1794 3905 y FC(X)1843 4082 y FB(i)1928 3984 y FH(\032)1985 3916 y FC(\000)2023 3984 y FH(e)2062 3949 y FG(2)2062 4004 y FB(i)2099 3916 y FC(\001)3220 3984 y FQ(\(5.9\))515 4244 y(where)19 b FH(\032)p FI(\(\))j FQ(is)1378 4393 y FH(\032)p FI(\()p FH(e)1492 4359 y FG(2)1530 4393 y FI(\))h(=)1673 4276 y FC(\032)1776 4342 y FI(0)318 b FH(e)2175 4312 y FG(2)2235 4342 y FH(<)23 b(T)2384 4312 y FG(2)1776 4442 y FQ(constant)82 b FH(e)2175 4412 y FG(2)2235 4442 y FA(\025)23 b FH(T)2384 4412 y FG(2)2441 4442 y FH(:)3178 4393 y FQ(\(5.10\))515 4587 y(In)i(other)f(w)o(ords)h (inliers)g(score)g(nothing)f(and)g(each)h(outlier)g(scores)g(a)h (constant)e(penalty)-5 b(.)39 b(Thus)515 4686 y(the)22 b(higher)f FH(T)935 4656 y FG(2)994 4686 y FQ(is)i(the)g(more)e (solutions)h(with)g(equal)g(v)n(alues)g(of)g FH(C)29 b FQ(tending)21 b(to)h(poor)f(estimation)515 4786 y(e.g.)i(if)h FH(T)36 b FQ(were)23 b(suf)n(\002ciently)g(lar)o(ge)g(then)g(all)i (solutions)e(w)o(ould)g(ha)n(v)o(e)h(the)f(same)h(cost)h(as)f(all)h (the)515 4886 y(matches)i(w)o(ould)g(be)h(inliers.)47 b(In)27 b(T)-7 b(orr)28 b(and)f(Zisserman)g([55)n(])h(it)h(w)o(as)f (sho)n(wn)f(that)h(at)g(no)f(e)o(xtra)p eop %%Page: 46 48 46 47 bop 515 232 a FJ(CHAPTER)21 b(5.)45 b(R)m(OB)o(UST)22 b(ESTIMA)-9 b(TION)18 b(OF)j FL(F)1224 b FQ(46)p 515 452 3240 4 v 515 2562 4 2111 v 664 616 a(1.)41 b(Detect)20 b(corner)f(features)h(using)f(the)i(Harris)f(corner)f(detector)g([18)n (].)664 782 y(2.)41 b(Putati)n(v)o(e)19 b(matching)f(of)h(corners)f(o)o (v)o(er)g(the)i(tw)o(o)f(images)g(using)g(proximity)f(and)h(cross)g (correla-)768 882 y(tion.)664 1048 y(3.)41 b(Repeat)25 b(until)f FO(no)p 1306 1048 25 4 v 29 w(samp)h FQ(samples)f(ha)n(v)o(e) g(been)g(tak)o(en)g(or)g(\223jump)f(out\224)h(occurs)g(as)h(described) 768 1148 y(in)20 b(Section)g(5.2.)817 1314 y(\(a\))41 b(Select)29 b(a)g(random)e(sample)i(of)f(the)h(minimum)f(number)f(of)h (correspondences)e FH(S)3394 1326 y FB(m)3496 1314 y FI(=)951 1413 y FA(f)p FL(x)1043 1373 y FG(1)p FB(;)p FG(2)1043 1436 y FB(i)1133 1413 y FA(g)p FQ(.)812 1546 y(\(b\))41 b(Estimate)18 b(the)g(image)g(relation)g FL(M)g FQ(consistent)g(with)h(this)g(minimal)e(set)j(using)d(the)i(meth-)951 1646 y(ods)h(described)f(in)h(Section)g(5.2.)817 1778 y(\(c\))41 b(Calculate)20 b(the)g(error)f FH(e)1628 1790 y FB(i)1676 1778 y FQ(for)h(each)g(datum.)812 1911 y(\(d\))41 b(Calculate)20 b FH(C)1344 1923 y FG(2)1382 1911 y FQ(.)664 2077 y(4.)41 b(Select)21 b(the)g(best)g(solution)f(o)o(v)o(er)f(all)i (the)g(samples)g(i.e.)f(that)h(with)g(lo)n(west)g FH(C)2973 2089 y FG(2)3011 2077 y FQ(.)27 b(Store)20 b(the)h(set)g(of)768 2177 y(correspondences)c FH(S)1393 2189 y FB(m)1477 2177 y FQ(that)j(ga)n(v)o(e)f(this)i(solution.)664 2343 y(5.)41 b(Minimize)28 b(rob)n(ust)f(cost)i(function)d(o)o(v)o(er)h(all)i (correspondences,)e(using)h(iterati)n(v)o(e)f(non-linear)768 2443 y(methods,)19 b(as)i(described)e(in)h(Section)g(4.9.)p 3751 2562 4 2111 v 515 2565 3240 4 v 995 2714 a(T)-7 b(able)20 b(5.4:)25 b Ft(A)20 b(brief)h(summary)f(of)g(all)h(the)f(sta) o(g)o(es)g(of)h(estimation)515 3063 y FQ(cost)h(this)g(undesirable)e (situation)i(can)f(be)h(remedied.)28 b(Rather)22 b(than)f(minimizing)g FH(C)28 b FQ(a)23 b(ne)n(w)e(cost)515 3163 y(function)d(can)i(be)g (minimized)1584 3262 y FH(C)1643 3274 y FG(2)1704 3262 y FI(=)1791 3183 y FC(X)1840 3360 y FB(i)1925 3262 y FH(\032)1968 3274 y FG(2)2019 3195 y FC(\000)2057 3262 y FH(e)2096 3228 y FG(2)2096 3283 y FB(i)2133 3195 y FC(\001)3178 3262 y FQ(\(5.11\))515 3483 y(where)f(the)i(rob)n(ust)e (error)g(term)h FH(\032)1485 3495 y FG(2)1543 3483 y FQ(is)1450 3711 y FH(\032)1493 3723 y FG(2)1530 3711 y FI(\()p FH(e)1601 3677 y FG(2)1638 3711 y FI(\))j(=)1781 3594 y FC(\032)1885 3660 y FH(e)1924 3630 y FG(2)2065 3660 y FH(e)2104 3630 y FG(2)2164 3660 y FH(<)g(T)2313 3630 y FG(2)1885 3760 y FH(T)1946 3730 y FG(2)2065 3760 y FH(e)2104 3730 y FG(2)2164 3760 y FA(\025)g FH(T)2313 3730 y FG(2)2370 3760 y FH(:)3178 3711 y FQ(\(5.12\))515 3938 y(This)j(is)g(a)g(simple,)h(redescending)c(M-estimator)h([26)o(].) 41 b(It)26 b(can)g(be)f(seen)h(that)g(outliers)f(are)g(still)515 4038 y(gi)n(v)o(en)d(a)j(\002x)o(ed)e(penalty)g(b)n(ut)g(no)n(w)h (inliers)g(are)f(scored)g(on)h(ho)n(w)f(well)h(the)o(y)g(\002t)g(the)g (data.)36 b(W)-7 b(e)25 b(set)515 4137 y FH(T)34 b FI(=)22 b(1)p FH(:)p FI(96)p FH(\033)d FQ(so)d(that)f(Gaussian)h(inliers)f(are) h(only)e(incorrectly)g(rejected)h(\002)n(v)o(e)g(percent)g(of)g(the)h (time.)515 4237 y(The)30 b(implementation)e(of)i(this)h(ne)n(w)f (method)f(\(one)h(incarnation)e(of)i(the)h(general)g(M)t(A)t(P)t(S)t(A) q(C)515 4336 y(principle\))21 b(yields)h(a)h(modest)f(to)h(hefty)f (bene\002t)g(to)h(all)g(rob)n(ust)f(estimations)h(with)f(absolutely)g (no)515 4436 y(additional)28 b(computational)e(b)n(urden.)51 b Ft(Once)29 b(this)g(is)i(under)o(stood)c(ther)m(e)i(is)i(no)d(r)m (eason)h(to)g(use)517 4536 y FQ(R)t(A)t(N)t(S)t(A)q(C)f Ft(in)d(pr)m(efer)m(ence)e(to)i(this)g(method.)37 b FQ(Similar)24 b(schemes)g(for)g(rob)n(ust)g(estimation)g(using)515 4635 y(random)18 b(sampling)h(and)h(M-estimators)f(were)i(also)f (proposed)e(in)i([51)o(])h(and)e([46)o(].)p eop %%Page: 47 49 47 48 bop 515 232 a FJ(CHAPTER)21 b(5.)45 b(R)m(OB)o(UST)22 b(ESTIMA)-9 b(TION)18 b(OF)j FL(F)1224 b FQ(47)515 523 y FD(5.5)119 b(Standard)31 b(De)n(viation)515 709 y FQ(This)22 b(section)f(gi)n(v)o(es)g(a)h(method)e(for)h(rob)n(ustly)g(estimating)g (the)h(standard)f(de)n(viation)f(of)h(the)h(error)515 808 y(term.)44 b(Rob)n(ust)27 b(techniques)f(to)h(eliminate)f(outliers) h(are)g(all)g(founded)e(upon)g(some)i(kno)n(wledge)515 908 y(of)c(the)h(standard)f(de)n(viation)f FH(\033)28 b FQ(of)c(the)f(error)-5 b(.)36 b(Generally)-5 b(,)23 b(gi)n(v)o(en)f FH(\033)s FQ(,)k(outliers)d(are)h(calculated)f(as)515 1007 y(follo)n(ws:)1301 1137 y FH(z)j FI(=)1454 1020 y FC(\032)1558 1087 y FQ(non)19 b(outlier)82 b FA(j)p FH(d)p FA(j)23 b(\024)g FH(t)g FI(=)f(1)p FH(:)p FI(96)p FH(\033)1558 1186 y FQ(outlier)227 b(otherwise)n FH(;)3178 1137 y FQ(\(5.13\))515 1331 y(where)26 b FH(t)36 b FI(=)f(1)p FH(:)p FI(96)p FH(\033)30 b FQ(is)e(a)f(user)g(de\002ned)f(threshold.) 44 b(In)26 b(the)h(case)h(of)f(the)g(fundamental)d(matrix)515 1431 y(there)k(are)g(tw)o(o)h(errors)f(for)f(each)h (correspondence\227the)d(epipolar)i(distances)h FH(d)2934 1443 y FG(1)2972 1431 y FQ(,)j FH(d)3067 1443 y FG(2)3133 1431 y FQ(in)e(each)515 1530 y(image.)d(There)21 b(are)g(tw)o(o)g (options:)26 b(either)20 b(both)g(can)h(be)g(tested)g(by)g(rule)f (\(5.13\))g(and)g(if)h(either)g FH(d)3321 1542 y FG(1)3358 1530 y FQ(,)515 1630 y FH(d)558 1642 y FG(2)623 1630 y FQ(is)28 b(greater)e(than)g FH(t)h FQ(then)g(the)g(correspondence)c (is)28 b(considered)d(outlying;)k(or)m(,)f(the)f(tw)o(o)g(may)515 1729 y(be)d(combined)e(for)h(a)h(single)g(test.)37 b(The)24 b(latter)g(approach)e(is)j(follo)n(wed,)e(noting)g(that)h FH(d)3074 1699 y FG(2)3074 1750 y(1)3133 1729 y FI(+)d FH(d)3262 1699 y FG(2)3262 1750 y(2)3324 1729 y FQ(is)515 1829 y(approximated)14 b(by)k(a)g FH(\037)1198 1799 y FG(2)1253 1829 y FQ(v)n(ariable)e(with)i(tw)o(o)g(de)o(grees)f(of)g (freedom)f(leads)h(to)h(the)g(follo)n(wing)e FI(95\045)515 1929 y FQ(con\002dence)i(test:)1184 2141 y FH(z)1223 2153 y FB(i)1273 2141 y FI(=)1361 2024 y FC(\032)1464 2091 y FQ(non)i(outlier)82 b FH(d)1953 2061 y FG(2)1953 2111 y(1)2008 2091 y FI(+)18 b FH(d)2134 2061 y FG(2)2134 2111 y(2)2195 2091 y FA(\024)23 b FH(t)g FI(=)f(5)p FH(:)p FI(99)p FH(\033)2622 2061 y FG(2)1464 2190 y FQ(outlier)228 b(otherwise)n FH(;)3178 2141 y FQ(\(5.14\))639 2368 y(The)30 b(standard)g(de)n(viation)e(is)k(related)e(to)g(the)h(characteristics)e (of)i(the)f(image,)i(the)f(feature)515 2468 y(detector)17 b(and)h(the)g(matcher)-5 b(.)24 b(Often)18 b(the)g(v)n(alue)g(of)g FH(\033)k FQ(is)d(unkno)n(wn,)d(in)j(which)f(case)g(it)i(must)e(be)g (es-)515 2567 y(timated)f(from)g(the)g(data.)24 b(If)18 b(there)f(are)h(no)f(outliers)g(in)h(the)g(data)f(the)h FH(\033)k FQ(can)17 b(be)h(estimated)f(directly)515 2667 y(as)22 b(the)g(standard)e(de)n(viation)g(of)h(the)h(residuals)f(of)g (a)h(non-linear)e(least)i(squares)f(minimization\227)515 2767 y(e.g.)c FL(N)728 2779 y FK(3)769 2767 y FQ(.)25 b(If)17 b(there)g(are)g(outliers)g(and)g(the)o(y)g(are)g(in)g(the)h (minority)-5 b(,)16 b(a)i(\002rst)g(estimate)f(of)g(the)h(v)n(ariance) 515 2866 y(can)26 b(be)h(deri)n(v)o(ed)e(from)h(the)h(median)f(squared) g(error)g(of)g(the)h(chosen)f(parameter)g(\002t)h([40)o(].)45 b(It)28 b(is)515 2966 y(kno)n(wn)c(that)i(med)1059 2978 y FB(i)1086 2966 y FA(j)p FH(d)1152 2978 y FB(i)1180 2966 y FA(j)p FH(=)p FI(\010)1305 2936 y FF(\000)p FG(1)1394 2966 y FI(\(0)p FH(:)p FI(75\))f FQ(is)i(an)e(asymptotically)g (consistent)g(estimator)g(of)h FH(\033)j FQ(when)515 3066 y(the)f FH(d)688 3078 y FB(i)744 3066 y FQ(are)g(distrib)n(uted)f (lik)o(e)h FH(N)9 b FI(\(0)p FH(;)14 b(\033)1645 3035 y FG(2)1682 3066 y FI(\))p FQ(,)30 b(where)e FI(\010)g FQ(is)h(the)f(cumulati)n(v)o(e)e(distrib)n(ution)g(function)515 3165 y(for)18 b(the)i(Gaussian)f(probability)e(density)i(function)2000 3135 y Fz(1)2031 3165 y FQ(.)25 b(Empirically)18 b(it)i(has)f(been)g (sho)n(wn)f([40)o(])h(that)515 3287 y(when)d FH(n)23 b FA(\031)g FI(2)p FH(p)17 b FQ(the)h(correction)d(f)o(actor)i(of)1746 3195 y FC(\020)1795 3287 y FI(1)h(+)1995 3254 y FG(5)p 1948 3268 128 4 v 1948 3316 a FB(n)p FF(\000)p FB(p)2085 3195 y FC(\021)2153 3287 y FQ(impro)o(v)o(es)d(the)j(estimate)f(of)g (the)h(standard)515 3417 y(de)n(viation.)23 b(Noting)d FI(1)p FH(=)p FI(\010)1264 3387 y FF(\000)p FG(1)1352 3417 y FI(\(0)p FH(:)p FI(75\))i(=)h(1)p FH(:)p FI(4826)18 b FQ(the)i(estimate)h(of)e FH(\033)25 b FQ(is)1332 3649 y FH(\033)i FI(=)22 b(1)p FH(:)p FI(4826)1740 3532 y FC(\022)1799 3649 y FI(1)c(+)2028 3593 y(5)p 1952 3630 194 4 v 1952 3706 a FH(n)g FA(\000)g FH(p)2155 3532 y FC(\023)2230 3649 y FQ(med)2373 3661 y FB(i)2401 3649 y FA(j)p FH(d)2467 3661 y FB(i)2494 3649 y FA(j)j FH(:)617 b FQ(\(5.15\))515 3876 y(The)22 b(LMS)h(algorithm)e(is)j(used)f(to)g (get)f(the)h(estimate)g(of)g(the)f(median.)32 b(The)22 b(standard)g(de)n(viation)515 3976 y(can)e(be)g(estimated)g(between)f (each)h(pair)g(of)g(images)g(and)g(the)g(results)h(\002ltered)e(o)o(v)o (er)g(time.)26 b(Image)515 4075 y(pairs)c(that)g(gi)n(v)o(e)f(rise)i (to)f(unusually)f(high)g(standard)g(de)n(viations)g(might)h(possess)h (independently)515 4175 y(mo)o(ving)f(objects.)37 b(Gi)n(v)o(en)23 b(random)g(perturbations)e(of)j(the)h(image)e(correspondences)e(with)k (unit)515 4275 y(standard)16 b(de)n(viation)h(then)g(the)h(estimate)g (of)f(the)h(standard)f(de)n(viation)f(of)h FL(F)i FQ(w)o(as)f(found)f (to)g(be)h FI(1)p FH(:)p FI(3)p FQ(,)515 4374 y(this)i(being)g(a)g (con\003ation)f(of)h(the)g(image)g(error)f(and)h(the)g(error)f(in)h (the)g(estimator)-5 b(.)p 515 4927 1146 4 v 605 4983 a Fy(1)634 5006 y Fw(N)7 b Fv(\(0)p Fw(;)12 b(\033)833 4983 y Fc(2)869 5006 y Fv(\))17 b FP(signi\002es)h(a)f(Gaussian)h(or)f (Normal)g(distrib)o(ution)j(mean)e Fv(0)e FP(and)i(v)n(ariance)i Fw(\033)2677 4983 y Fc(2)2712 5006 y FP(.)p eop %%Page: 48 50 48 49 bop 515 1146 a FE(Chapter)44 b(6)515 1561 y FN(Rematching)515 1993 y FQ(In)20 b(v)o(ersion)f(1)h(this)h(is)g(not)f(implemented,)e (hopefully)g(I)i(will)h(get)f(round)f(to)h(it!)515 2214 y FM(Epipolar)g(Geometry)f(and)i(F)n(eatur)o(e)d(Matching)515 2369 y FQ(Once)34 b(the)g(epipolar)f(geometry)g(has)i(been)e(estimated) i(this)f(can)h(be)f(used)g(to)h(aid)f(matching.)515 2469 y(In)20 b([11)o(,)h(2)o(])g(the)g(epipolar)e(geometry)g(is)j(used)e(to) h(constrain)f(the)h(search)f(area)h(for)f(a)h(gi)n(v)o(en)e(match.)515 2568 y(Our)i(proposal)f(goes)h(further)f(in)i(that)g(we)f(aim)h(to)g (conjoin)e(the)i(estimation)f(of)g(epipolar)f(geome-)515 2668 y(try)k(and)f(matching.)35 b(As)25 b(the)f(match)g(may)f(be)h (incorrect,)g(it)g(is)h(desirable)f(that,)h(if)f(in)g(the)g(course)515 2768 y(of)j(the)h(estimation)g(process)g(it)g(is)h(disco)o(v)o(ered)d (that)i(the)g(feature)f(is)i(mismatched,)g(then)e(it)i(can)515 2867 y(be)23 b(rematched)f(to)h(another)f(feature.)34 b(In)23 b(order)f(to)i(achie)n(v)o(e)e(this)i(not)f(only)f(is)j(a)e (feature')-5 b(s)23 b(initial)515 2967 y(match)d(stored,)g(based)g(on)g (cross)h(correlation,)d(b)n(ut)j(all)g(its)g(candidate)f(matches)g (that)g(ha)n(v)o(e)g(a)h(sim-)515 3067 y(ilarity)i(score)g(o)o(v)o(er)g (a)h(user)f(de\002ned)g(threshold.)33 b(After)23 b(the)h(rob)n(ust)f (estimation)g(of)g(the)h(epipolar)515 3166 y(geometry)h(all)j(corners)e (are)i(rematched)e(to)h(the)g(candidate)g(with)g(smallest)h(epipolar)e (distance.)515 3266 y(The)20 b(epipolar)f(geometry)g(may)h(then)g(be)g (further)g(re\002ned.)25 b(In)20 b(all)h(the)g(e)o(xamples)e(that)i (follo)n(w)f(the)515 3366 y(inliers)i(and)g(outliers)f(are)h(from)f (the)h(ra)o(w)g(matches)g(based)g(only)f(on)h(cross)g(correlation,)f (in)h(order)515 3465 y(to)e(demonstrate)f(the)h(success)h(of)e(a)i(gi)n (v)o(en)e(estimation.)1905 5255 y(48)p eop %%Page: 49 51 49 50 bop 515 1146 a FE(Chapter)44 b(7)515 1561 y FN(Self)51 b(Calibration,)i(establishing)g(a)515 1810 y(pr)l(ojecti)n(v)n(e)e (frame)515 2242 y FQ(Once)26 b(the)g(fundamental)e(matrix)i(is)h (estimated)f(the)g(projecti)n(v)o(e)f(structure)g(is)j(reco)o(v)o (ered.)40 b(This)515 2341 y(chapter)19 b(is)i(a)f(just)h(sk)o(etch)f (outline,)f(and)h(the)g(user)g(is)h(referred)d(for)i(more)f(details)i (to)f([13)o(,)g(25)o(,)h(33)o(].)515 2441 y(All)g(typos)e(spotted)h (please)g(mail)g(me!)639 2540 y(It)25 b(is)f(possible)g(to)g(obtain)f (only)g(a)i(projecti)n(v)o(e)d(reconstruction)f(from)i(the)h (fundamental)e(ma-)515 2640 y(trix,)33 b(as)e(sho)n(wn)f(in)g(Section)h (7.1.)55 b(A)31 b(Euclidean)e(reconstruction)f(w)o(ould)i(be)h (preferable.)54 b(In)515 2740 y(order)26 b(to)i(achie)n(v)o(e)e(this)i (the)g(process)f(has)h(to)f(stages)h(\(1\))f(is)i(to)e(reco)o(v)o(er)f (the)h(camera)g(positions)515 2839 y(and)f(motions)f(\(modulo)g(a)h (scaling\).)43 b(Once)27 b(this)g(is)g(done)e(step)i(\(2\))f(in)m(v)n (olv)o(es)f(triangulation)f(to)515 2939 y(reco)o(v)o(er)d(the)i(3D)h (points.)34 b(Stage)23 b(\(1\))g(requires)f(some)i(sort)f(of)g (calibration)f(to)i(con)m(v)o(ert)d(from)i(the)515 3039 y(uncalibrated)h(fundamental)g(matrix)i(to)g(the)g(essential)h(matrix.) 43 b(Here)26 b(a)h Ft(self)g(calibr)o(ation)e FQ(pro-)515 3138 y(cess)20 b(is)h(implemented,)d(described)g(in)i(Section)g(7.4.1,) e(although)g(there)i(are)f(other)g(options)g(\(such)515 3238 y(as)24 b(use)h(of)e(a)i(calibration)d(grid)i(etc.\).)36 b(It)24 b(is)h(assumed)e(only)g(the)h(focal)g(length)f(is)i(unkno)n (wn,)d(and)515 3337 y(the)c(Sturm)g([47)o(])g(self)h(calibration)e (method)g(used.)25 b(Note)18 b(that)g(the)h(focal)f(length)f(cannot)h (be)g(deter)n(-)515 3437 y(mined)d(from)f(a)j(pure)e(translation.)22 b(Once)16 b(the)g(essential)g(matrix)f(is)i(reco)o(v)o(ered,)d(if)i (the)g(\002rst)g(camera)515 3537 y(is)j(assumed)f(to)h(be)f(at)h(the)g (origin)e(of)i(the)f(coordinate)f(system,)h(then)h(it)g(is)g(a)g (simple)g(matter)f(to)h(cal-)515 3636 y(culate)k(the)h(rotation)e(and)h (translation)g(of)g(the)h(second)f(camera)g(relati)n(v)o(e)f(to)i(the)g (\002rst,)h(described)515 3736 y(in)20 b(Section)f(7.4.3.)24 b(T)-7 b(o)20 b(impro)o(v)o(e)e(the)i(estimates)g(of)g(the)g(camera)f (intrinsic)g(and)h(e)o(xtrinsic)f(param-)515 3836 y(eters)26 b(a)f(non)g(linear)g(optimization)f(is)i(used,)h(see)f(Section)f (7.4.5.)39 b(Once)26 b(these)f(parameters)g(are)515 3935 y(estimated)e(then)f(the)i(camera)e(projection)f(matrices)i(may)g(be)g (reco)o(v)o(ered)e(and)h(used)h(to)g(estimate)515 4035 y(the)h(structure)g(using)f(generic)h(projecti)n(v)o(e)e(methods)i(as)h (described)e(in)h(Section)g(7.1.)37 b(The)24 b(same)515 4134 y(method)d(of)i(triangulation)d(can)j(be)g(used)f(for)g(projecti)n (v)o(e)f(or)i(Euclidean)e(reconstruction)g(as)i(it)h(is)515 4234 y(the)c(image)g(error)f(that)h(is)h(to)f(be)h(minimized.)515 4514 y FD(7.1)119 b(Reco)o(v)o(ery)29 b(of)h(Pr)n(ojection)g(Matrices) 515 4700 y FQ(As)d(laid)g(out)f(in)h(Section)f(4.3,)h(perspecti)n(v)o (e)e(projection)g(from)h(3D)g(to)h(2D)g(by)f(a)h FI(3)c FA(\002)g FI(4)j FQ(camera)515 4800 y(matrix)19 b FL(P)1482 4982 y(x)24 b FI(=)e FL(PX)84 b FQ(and)e FL(x)2117 4948 y FF(0)2163 4982 y FI(=)23 b FL(P)2316 4946 y FF(0)2339 4982 y FL(X)1905 5255 y FQ(49)p eop %%Page: 50 52 50 51 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(50)515 523 y(Thus)k(if)g FL(P)p FH(;)14 b FL(P)945 487 y FF(0)991 523 y FQ(are)22 b(kno)n(wn)f(then)h FL(X)h FQ(may)f(be)g(reco)o(v)o(ered)d(from)j(from)f(the)h(matches)g FL(x)q FH(;)14 b FL(x)3163 493 y FF(0)3186 523 y FQ(.)32 b(F)o(or)n(-)515 623 y(tunately)-5 b(,)23 b(it)h(is)h(well)f(kno)n(wn)f (that)h(gi)n(v)o(en)e FL(F)i FQ(then)g FL(P)p FH(;)14 b FL(P)2162 586 y FF(0)2210 623 y FQ(can)23 b(be)h(estimated.)36 b(One)23 b(solution)g(for)515 722 y(establishing)c(a)i(projecti)n(v)o (e)d(frame)i(is)h(as)g(follo)n(ws)e([3])619 888 y(1.)40 b(Set)21 b FL(P)i FI(=)g([)p FL(I)p FA(j)p FL(0)p FI(])619 1054 y FQ(2.)40 b(Compute)19 b FL(F)q FQ(,)h(Compute)f FL(e)1518 1024 y FF(0)1562 1054 y FQ(such)h(that)g FL(e)1924 1024 y FF(0>)1999 1054 y FL(F)j FI(=)g FL(0)p FQ(,)d(let)h FL(M)i FI(=)f([)p FL(e)2631 1024 y FF(0)2654 1054 y FI(])2677 1066 y FF(\002)2734 1054 y FL(F)p FQ(.)619 1220 y(3.)40 b(Set)21 b(the)f(second)g(projection)e(matrix)i(as)1681 1403 y FL(P)1746 1367 y FF(0)1792 1403 y FI(=)j([)p FL(M)18 b FI(+)g FL(e)2139 1369 y FF(0)2162 1403 y FL(b)2215 1366 y FF(>)2271 1403 y FA(j)p FH(c)p FL(e)2374 1369 y FF(0)2397 1403 y FI(])800 b FQ(\(7.1\))722 1586 y(where)28 b FL(b)g FQ(and)f FH(c)i FQ(are)f(an)f(arbitrary)g(3-v)o(ector)e(and)j (scalar)g(respecti)n(v)o(ely)-5 b(,)27 b(thus)h(there)g(are)722 1685 y(four)d(de)o(grees)g(of)h(freedom)g(D)t(O)t(F)j(in)d(this)g (choice)f(\(as)i(there)e(are)h(11)h(D)t(O)t(F)i(in)d FL(P)h FQ(and)e(7)724 1785 y(D)t(O)t(F)f(in)c FL(F)p FQ(:)26 b(11-7)19 b(=)h(4\).)619 1951 y(4.)40 b(Normalize)17 b(so)h(that)f FI(det)1466 1930 y(~)1454 1951 y FL(P)24 b FH(>)e FI(0)p FQ(,)c(where)1944 1930 y FI(~)1932 1951 y FL(P)g FQ(is)h(the)e(\002rst)i FI(3)8 b FA(\002)g FI(3)17 b FQ(matrix)g(of)g FL(P)p FQ(,)i(this)f(is)g(useful)722 2051 y(for)i(determining)e(which)i(side)g(of)g(the)g(camera)g(points)f (are)h(on)g(in)h(section)f(7.4.3.)515 2217 y(This)i(leads)g(to)g (projecti)n(v)o(e)e(reconstruction)f(of)j(the)g(w)o(orld,)f(so)h (called)g(because)f(a)h(3D)g(projecti)n(v)o(e)515 2316 y(transformation)16 b(of)j(the)f(w)o(orld)h(coordinates)e FL(X)1919 2280 y FF(0)1965 2316 y FI(=)23 b FL(HX)p FQ(,)c(w)o(ould)g (lead)f(to)h(the)g(same)g(fundamen-)515 2416 y(tal)i(matrix.)j(The)c (follo)n(wing)e(algorithm)h(e)o(xtracts)h(the)g FL(P)p FQ(:)515 2654 y Fs(7.1.1)99 b Fp(P)o Fs(,)p Fp(P)992 2612 y Fb(0)1040 2654 y Fs(fr)n(om)25 b Fp(F)515 2809 y FO(function)48 b([P1,P2])h(=)g(torr_PfromF\(FMat,m3\))515 3030 y FM(Input:)83 b FQ(Input)19 b(parameters)g(de\002ned)g(as)i(in)f (Section)g(4.1.1.)515 3251 y FM(Output:)82 b FO(P1,)49 b(P2)20 b FQ(the)h(tw)o(o)f FI(3)e FA(\002)g FI(4)i FQ(projection)f (matrices.)515 3532 y FD(7.2)119 b(Reco)o(v)o(ery)29 b(of)h(Pr)n(ojecti)o(v)o(e)f(Structur)n(e)515 3717 y FQ(Once)h(the)g FL(P)h FQ(matrices)f(are)g(reco)o(v)o(ered)d(the)j (structure)f FL(X)i FQ(may)f(be)g(reco)o(v)o(ered)e(by)h(triangula-)515 3817 y(tion)h([24)o(])g(ho)n(we)n(v)o(er)f(obtaining)f(an)j(optimal)e (solution)h(can)g(be)h(costly)-5 b(.)55 b(This)30 b(is)i(because)e(the) 515 3916 y(optimal)20 b(estimate)g(w)o(ould)g(minimize)g(the)h (reprojection)d(error)h(of)i(the)f(3D)h(points)f(i.e.)26 b(minimize)515 4016 y(the)h(sum)h(of)f(squares)g(of)h(Euclidean)e (distance)h(between)g(the)h(observ)o(ed)d(point)i(in)h(each)f(image)515 4116 y(and)19 b(the)i(reprojection)c(using)j(the)g(projection)f (matrices)h(and)f(putati)n(v)o(e)g(3D)i(structure)e(i.e)1388 4298 y FI(min)1421 4369 y FL(X)1540 4298 y FH(e)1579 4310 y FB(u)1622 4298 y FI(\()p FL(x)q FH(;)14 b FL(PX)p FI(\))1911 4264 y FG(2)1967 4298 y FI(+)k FH(e)2089 4310 y FB(u)2132 4298 y FI(\()p FL(x)2215 4264 y FF(0)2238 4298 y FH(;)c FL(P)2341 4262 y FF(0)2364 4298 y FL(X)p FI(\))2468 4264 y FG(2)3220 4298 y FQ(\(7.2\))515 4531 y(where)i FH(e)774 4543 y FB(u)817 4531 y FI(\()p FL(a)q FH(;)e FL(b)p FI(\))k FQ(is)g(the)g(Euclidean)d(distance)i(between)g FL(a)h FQ(and)f FL(b)p FQ(.)24 b(This)18 b(is)g(equi)n(v)n(alent)d(to)j (\002nding)515 4631 y FI(\()5 b(^)-47 b FH(x;)20 b FI(^)-48 b FH(y)s(;)19 b FI(^)-47 b FH(x)759 4600 y FF(0)783 4631 y FH(;)19 b FI(^)-47 b FH(y)864 4600 y FF(0)887 4631 y FI(\))21 b FQ(such)f(that)1018 4751 y FC(X)1151 4830 y FH(e)j FI(=)g(\()p FH(x)c FA(\000)k FI(^)-47 b FH(x)p FI(\))1561 4796 y FG(2)1617 4830 y FI(+)18 b(\()p FH(y)k FA(\000)i FI(^)-48 b FH(y)s FI(\))1954 4796 y FG(2)2010 4830 y FI(+)18 b(\()5 b(^)-47 b FH(x)2172 4796 y FF(0)2214 4830 y FA(\000)18 b FH(x)2344 4796 y FF(0)2368 4830 y FI(\))2400 4796 y FG(2)2456 4830 y FI(+)g(\()6 b(^)-48 b FH(y)2615 4796 y FF(0)2657 4830 y FI(+)18 b FH(y)2784 4796 y FF(0)2807 4830 y FI(\))2839 4796 y FG(2)3220 4830 y FQ(\(7.3\))p eop %%Page: 51 53 51 52 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(51)515 523 y(is)j(a)f(minimum)f(and)h FI(\()5 b(^)-47 b FH(x)q(;)19 b FI(^)-47 b FH(y)r(;)19 b FI(^)-47 b FH(x)1377 493 y FF(0)1401 523 y FH(;)19 b FI(^)-47 b FH(y)1482 493 y FF(0)1505 523 y FI(\))21 b FQ(satis\002es)1767 675 y FI(^)1762 676 y FL(x)1813 637 y FF(>)1869 676 y FL(F)1933 675 y FI(^)1929 676 y FL(x)i FI(=)g(0)1088 b FQ(\(7.4\))515 828 y(where)19 b FL(x)24 b FI(=)e(\()p FH(x;)14 b(y)s(;)g FI(1\))1170 798 y FF(>)1247 828 y FQ(and)20 b FL(x)1438 798 y FF(0)1485 828 y FI(=)i(\()p FH(x)1651 798 y FF(0)1675 828 y FH(;)14 b(y)1756 798 y FF(0)1779 828 y FH(;)g FI(1\))1890 798 y FF(>)1946 828 y FQ(.)639 928 y(This)21 b(is)h(a)f(computationally)c(e)o(xpensi)n (v)o(e)i(thing)h(to)h(do)f(thus)g(I)h(ha)n(v)o(e)f(implemented)f(a)i (simpler)515 1027 y(scheme:)j(\(a\))c(correct)f(the)h(point)f(matches,) g(using)g(a)i(\002rst)f(order)f(correction)f(based)h(on)h(Sampson)515 1127 y(\(b\))k(use)g(a)h(SVD)g(method)e(to)i(estimate)f FL(X)i FQ(from)d FL(x)p FQ(.)38 b(First)26 b(the)e(SVD)h(method)e(is)j (described.)36 b(In)515 1227 y(section)22 b(7.3)f(the)i(correction)d (is)j(described.)30 b(Let)23 b FL(p)2019 1197 y FG(1)p FF(\000)p FG(3)2164 1227 y FQ(be)f(the)g(three)g(ro)n(ws)g(of)g FL(P)i FQ(and)d FL(p)3136 1197 y FF(0)p FG(1)p FF(\000)p FG(3)3301 1227 y FQ(be)515 1326 y(the)f(three)g(ro)n(ws)g(of)g FL(P)1159 1290 y FF(0)1203 1326 y FQ(then)g(it)h(can)f(be)g(seen)1570 1479 y FH(x)p FL(p)1670 1445 y FG(3)p FF(>)1759 1479 y FL(X)f FA(\000)f FL(p)1986 1445 y FG(1)2023 1479 y FL(X)84 b FI(=)e(0)1574 1603 y FH(y)s FL(p)1670 1569 y FG(3)p FF(>)1759 1603 y FL(X)19 b FA(\000)f FL(p)1986 1569 y FG(2)2023 1603 y FL(X)84 b FI(=)e(0)1526 1728 y FH(x)p FL(p)1627 1694 y FG(2)p FF(>)1716 1728 y FL(X)18 b FA(\000)g FH(y)s FL(p)1986 1694 y FG(1)2023 1728 y FL(X)84 b FI(=)e(0)515 1881 y FQ(thus)20 b(an)g(equation)f(of)h(the)g (form)f FL(AX)k FI(=)g FL(0)e FQ(may)e(be)h(written)g(with)1443 2189 y FL(A)k FI(=)1626 1972 y FC(2)1626 2118 y(6)1626 2168 y(6)1626 2221 y(4)1746 2030 y FH(x)p FL(p)1846 2000 y FG(3)p FF(>)2041 2030 y FA(\000)94 b FL(p)2253 2000 y FG(1)p FF(>)1748 2131 y FH(y)s FL(p)1844 2100 y FG(3)p FF(>)2041 2131 y FA(\000)g FL(p)2253 2100 y FG(1)p FF(>)1723 2239 y FH(x)1770 2209 y FF(0)1794 2239 y FL(p)1847 2184 y Fn(0)1869 2209 y FG(3)p FF(>)2041 2239 y FA(\000)83 b FL(p)2242 2184 y Fn(0)2264 2209 y FG(1)p FF(>)1725 2347 y FH(y)1769 2317 y FF(0)1792 2347 y FL(p)1845 2292 y Fn(0)1867 2317 y FG(3)p FF(>)2041 2347 y FA(\000)g FL(p)2242 2292 y Fn(0)2264 2317 y FG(2)p FF(>)2395 1972 y FC(3)2395 2118 y(7)2395 2168 y(7)2395 2221 y(5)3220 2189 y FQ(\(7.5\))515 2495 y(thus)22 b(once)f FL(A)i FQ(is)h(found)c FL(X)j FQ(can)f(be)g(solv)o(ed)g(for)f(using)h FO(torr)p 2345 2495 25 4 v 29 w(ls)p FQ(,)h(the)f(function)f(that)h (does)g(this)515 2595 y(is:)38 b FO(torr)p 836 2595 V 29 w(triangulate)24 b FQ(which)i(is)h(described)e(ne)o(xt.)42 b(Ho)n(we)n(v)o(er)m(,)25 b(in)i(the)f(noisy)f(case,)j(this)515 2694 y(should)19 b(not)h(be)g(applied)f(until)h(the)g(image)g (coordinates)e(ha)n(v)o(e)i(been)g(corrected.)515 2926 y Fs(7.2.1)99 b(Quick)25 b(triangulation)g(function)h Fr(torr)p 2308 2926 30 4 v 35 w(triangulate)515 3082 y FO(function)48 b(X)i(=)f(torr_triangulate\(matches,)c(m3,)50 b(P1,)f(P2\))515 3298 y FM(Input:)619 3440 y FQ(1.)40 b FO(matches)20 b FH(n)e FA(\002)g FI(4)i FQ(array)g(of)g(matches.)619 3594 y(2.)40 b FO(m3)21 b FQ(third)e(homogeneous)e(coordinate.)619 3748 y(3.)40 b FO(P1,)50 b(P2)20 b FQ(the)g(tw)o(o)h FI(3)d FA(\002)g FI(4)i FQ(projection)e(matrices.)515 3963 y FM(Output:)82 b FO(X)20 b FI(4)e FA(\002)g FH(n)j FQ(array)e(of)h(homogeneous)d(structure)i(points.)515 4238 y FD(7.3)119 b(Corr)n(ection)30 b(of)g(the)g(matches)515 4424 y FQ(Sampson')-5 b(s)21 b(\002rst)i(order)e(correction)f(has)j (been)e(studied)h(in)g(detail)g(in)h([25)n(,)g(29)o(],)g(it)g(is)g (stated)f(here)515 4523 y(without)e(proof)g(\(until)h(I)h(get)g(time)f (to)h(put)f(one)g(in\))g(that)h(the)f(\002rst)h(order)f(correction)e (to)j(a)g(point)f(is)515 4623 y(simply:)1254 4653 y FC(0)1254 4799 y(B)1254 4848 y(B)1254 4902 y(@)1386 4719 y FI(^)-48 b FH(x)1388 4819 y FI(^)g FH(y)1374 4918 y FI(^)h FH(x)1416 4888 y FF(0)1376 5018 y FI(^)f FH(y)1414 4988 y FF(0)1481 4653 y FC(1)1481 4799 y(C)1481 4848 y(C)1481 4902 y(A)1577 4869 y FI(=)1664 4653 y FC(0)1664 4799 y(B)1664 4848 y(B)1664 4902 y(@)1790 4719 y FH(x)1792 4819 y(y)1778 4918 y(x)1825 4888 y FF(0)1780 5018 y FH(y)1824 4988 y FF(0)1891 4653 y FC(1)1891 4799 y(C)1891 4848 y(C)1891 4902 y(A)1982 4869 y FA(\000)2160 4813 y FH(r)p 2075 4850 211 4 v 2075 4926 a FI(\()p FA(r)p FH(r)r FI(\))2247 4902 y FG(2)2309 4653 y FC(0)2309 4799 y(B)2309 4848 y(B)2309 4902 y(@)2435 4719 y FH(r)2472 4731 y FB(x)2436 4819 y FH(r)2473 4831 y FB(y)2423 4918 y FH(r)2460 4930 y FB(x)2498 4914 y Fn(0)2424 5018 y FH(r)2461 5030 y FB(y)2497 5014 y Fn(0)2567 4653 y FC(1)2567 4799 y(C)2567 4848 y(C)2567 4902 y(A)3220 4869 y FQ(\(7.6\))p eop %%Page: 52 54 52 53 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(52)515 523 y(contrary)j(to)j(the)f(statement)g(in)h([25)n(])g(that)f (this)h(\002rst)g(order)e(projection)g(w)o(as)i(only)f(v)n(alid)f(for)h (one)515 623 y(pix)o(el)29 b(noise,)j(I)e(ha)n(v)o(e)g(found)e(it)j(to) f(gi)n(v)o(e)f(good)g(results)h(o)o(v)o(er)f(a)i(wide)f(range)f(of)g (noise)h(v)n(alues.)515 722 y(Once)h(the)g(points)g(are)h(corrected)e (the)o(y)g(should)h(satisfy)2245 721 y FI(^)2241 722 y FL(x)2291 684 y FF(0)2314 722 y FL(F)2379 721 y FI(^)2374 722 y FL(x)45 b FI(=)e(0)32 b FQ(thus)f(the)h(triangulation)515 822 y(method)18 b(gi)n(v)o(en)g(abo)o(v)o(e)g(should)h(gi)n(v)o(e)g (equal)g(results)h(to)g(that)f(of)h([24)n(])g(with)g(only)f(a)h (fraction)e(of)i(the)515 922 y(computational)e(cost.)27 b(The)21 b(intuition)e(here)i(is)g(to)g(tak)o(e)g(a)g(tangent)f(plane)g (to)h(the)g(manifold)e(of)i(the)515 1021 y(fundamental)d(matrix)h(and)h (project)f(points)h(orthogonally)d(do)n(wn)i(onto)g(the)i(manifold.)515 1257 y Fs(7.3.1)99 b(Corr)n(ection)25 b(function)i Fr(torr)p 1927 1257 30 4 v 35 w(correctx4F)515 1413 y FQ(This)20 b(function)f(generates)g(the)h(corrected)f(matches)g(as)i(described)e (in)i(the)f(pre)n(vious)e(subsection.)515 1587 y FO(function)48 b([corrected_matches,sq_errors])515 1687 y(=)h(torr_correctx4F\(f,)e (nx1,ny1,nx2,ny2,)g(no_matches,)g(m3\))515 1906 y FM(Input:)83 b FQ(Input)19 b(parameters)g(de\002ned)g(as)i(in)f(Section)g(4.1.1.)515 2126 y FM(Output:)619 2285 y FQ(1.)40 b FO(corrected)p 1177 2285 25 4 v 28 w(matches)20 b FQ(the)g(set)h(of)f FH(n)e FA(\002)g FI(4)j FQ(corrected)d(matches.)619 2448 y(2.)40 b FO(sq)p 827 2448 V 30 w(errors)19 b FQ(the)i FH(n)d FA(\002)g FI(1)i FQ(array)f(of)h(squared)f(errors.)515 2684 y Fs(7.3.2)99 b(T)-9 b(esting)19 b(the)h(tw)o(o)e(view)h(match)h (corr)n(ection:)29 b Fr(torr)p 2703 2684 30 4 v 35 w(test)p 2978 2684 V 35 w(correct)p 3433 2684 V 34 w(sc)515 2840 y FQ(This)21 b(simple)f(script)h(generates)f(some)g(random)f(matches,)h (estimates)i(the)e(fundamental)e(matrix,)515 2939 y(and)29 b(then)g(corrects)f(them)h(to)h(lie)g(on)f(the)g(fundamental)f(matrix,) i(note)f(ho)n(w)g(near)g(to)h(zero)f(the)515 3039 y(resulting)19 b(residuals)h(are.)515 3213 y FO(\045torr_test_correct_sc.m)515 3313 y(m3)49 b(=)h(256;)515 3412 y(method)e(=)i(2;)515 3711 y([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny)o(2,tru)o(e_C,tr)o(ue_R,)o (true_)o(t,)44 b(true_E])k(=)565 3811 y(torr_gen_2view_matches;)515 4010 y(no_matches)g(=)h(length\(nx1\);)515 4209 y(matches)f(=)i ([nx1,ny1,nx2,ny2];)515 4309 y(set_rank2)e(=)h(0;)515 4608 y(\045first)f(estimate)h(F)515 4707 y([f,)g(e1,)g (n_inliers,inlier_index,nF])c(=)565 4807 y(torr_estimateF\()i(matches,) h(m3,)h([],)g(method,)g(set_rank2\);)p eop %%Page: 53 55 53 54 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(53)515 523 y FO(\045next)49 b(correct)f(the)h(points)g(so)h(that)f (they)g(lie)g(on)g(a)h(fundamental)e(matrix)515 623 y ([corrected_matches)e(error2])j(=)565 722 y(torr_correctx4F\(f,)d (nx1,ny1,nx2,ny2,)h(no_matches,)h(m3\);)515 922 y(\045check)g(errors)h (\(should)g(be)g(near)g(zero\))515 1021 y(e)g(=)h(torr_errf2\(f,)d (corrected_matches\(:,1\),)f(corrected_matches\(:,2\),)515 1121 y(corrected_matches\(:,3\),)f (corrected_matches\(:,4\),no_matches,)f(m3\))515 1501 y FD(7.4)119 b(Self)30 b(Calibration)515 1686 y FQ(In)22 b(section)f(7.1)h(it)h(w)o(as)g(sho)n(wn)e(that)h(there)g(is)h(a)f (four)f(de)o(gree)g(of)h(freedom)e(ambiguity)g(in)j(the)f(re-)515 1786 y(co)o(v)o(ery)16 b(of)h(the)h(projection)e(matrices.)24 b(Ideally)17 b(when)h(resolving)e(this)j(ambiguity)d(we)i(w)o(ould)f (lik)o(e)515 1885 y(to)i(choose)g(the)h(projection)e(matrices)h(so)h (that)g(the)o(y)f(w)o(ould)f(be)i(as)g(close)g(to)g(the)f(true)h (\(Euclidean\))515 1985 y(matrices)i(as)i(possible.)33 b(In)22 b(order)g(to)h(do)f(that)h(a)h(self)f(calibration)f(method)f (is)j(used,)f(e)n(v)o(en)f(if)h(it)h(is)515 2085 y(not)e(totally)h (accurate)f(it)h(should)f(help)h(to)g(produce)d(projection)i(matrices)g (that)h(look)f(near)h(to)g(the)515 2184 y(Euclidean)18 b(ones,)i(and)g(hence)f(structure)g(that)i(looks)e(reasonable.)639 2284 y FM(Road)h(map)619 2450 y FQ(1.)40 b(Reco)o(v)o(er)19 b FL(F)p FQ(,)i(see)g(last)g(chapter)-5 b(.)619 2616 y(2.)40 b(Self)21 b(Calibrate)f(to)g(get)g FL(C)p FQ(,)h(Section)f (7.4.1.)619 2782 y(3.)40 b(Reco)o(v)o(er)19 b FL(E)k FI(=)g FL(C)1260 2746 y FF(>)1316 2782 y FL(E)o(C)1447 2746 y FG(1)619 2948 y FQ(4.)40 b(Decompose)19 b FL(E)h FQ(into)g FL(R)h FQ(and)e FL(t)q FQ(.)619 3114 y(5.)40 b(Optimize)20 b FL(C)p FQ(,)h FL(R)f FQ(and)g FL(t)p FQ(.)515 3352 y Fs(7.4.1)99 b(Reco)o(v)o(ery)25 b(of)g Fp(C)515 3507 y FQ(Once)g(the)h(fundamental)e(matrix)h(has)h(been)g (reco)o(v)o(ered)d(the)j(camera)f(calibration)g(matrix)g(\(4.3\))515 3607 y(can)15 b(be)h(reco)o(v)o(ered.)21 b(At)c(present)e(only)g(the)h (case)g(of)g(self)g(calibration)f(with)h(unkno)n(wn)d(focal)i(length) 515 3707 y(is)26 b(considered)d(implemented.)38 b(This)25 b(is)h(because)e(it)i(can)f(generally)e(be)i(assumed,)h(for)e(modern) 515 3806 y(well)e(engineered)e(cameras,)h(that)h(the)g(principal)e (point)h FI(\()p FH(p)2237 3818 y FB(x)2279 3806 y FH(;)14 b(p)2358 3818 y FB(y)2397 3806 y FI(\))23 b FQ(is)g(at)f(the)g(centre)f (of)g(the)h(image)515 3906 y(and)17 b(that)g(the)g(aspect)h(ratio)f(is) h(one.)24 b(It)17 b(is)i(also)e(assumed)g(that)h(the)f(calibration)f (matrix)h(is)h(the)f(same)515 4006 y(between)22 b(the)h(images)g (\(i.e.)33 b(no)23 b(zoom\).)32 b(Peter)23 b(Sturm)g(recently)f (presented)f(a)j(ne)n(w)f(method)e(for)515 4105 y(self)f(calibration)f (in)i(this)f(case)h([47)o(])1580 4075 y Fz(1)1613 4105 y FQ(.)26 b(His)21 b(analysis)f(is)h(based)f(on)g(the)g FL(G)h FQ(matrix,)e(which)h(is)h(half)515 4205 y(w)o(ay)f(between)f (the)i(essential)f(and)g(fundamental)e(matrices:)1113 4484 y FL(G)23 b FI(=)1299 4317 y FC(2)1299 4467 y(4)1448 4384 y FH(a)187 b FI(0)h(0)1449 4484 y(0)g(1)135 b FA(\000)p FH(p)1963 4496 y FB(y)1396 4583 y FA(\000)p FH(p)1503 4595 y FB(x)1627 4583 y FA(\000)p FH(p)1734 4595 y FB(y)1863 4583 y FI(1)p FH(=f)2044 4317 y FC(3)2044 4467 y(5)2113 4484 y FL(F)2187 4317 y FC(2)2187 4467 y(4)2284 4384 y FH(a)83 b FI(0)f FA(\000)p FH(p)2642 4396 y FB(x)2285 4484 y FI(0)i(1)f FA(\000)p FH(p)2643 4496 y FB(y)2285 4583 y FI(0)h(0)90 b(1)p FH(=f)2725 4317 y FC(3)2725 4467 y(5)3220 4484 y FQ(\(7.7\))515 4761 y(and)1480 4861 y(diag)o FI(\(1)p FH(;)14 b FI(1)p FH(;)g(f)9 b FI(\))p FL(E)19 b FQ(diag)o FI(\(1)p FH(;)14 b FI(1)p FH(;)g(f)9 b FI(\))p FH(:)805 b FQ(\(7.8\))p 515 4914 1146 4 v 605 4969 a Fy(1)634 4993 y FP(It)17 b(is)g(important)i(to)f(note)g(that)g (the)g(focal)g(length)h(cannot)f(be)g(determined)h(in)e(the)h(case)g (of)f(pure)g(translation.)p eop %%Page: 54 56 54 55 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(54)515 523 y(At)30 b(the)g(moment)e(the)h(reader)g(is)i(referred)c (to)j(Sturm)f(for)g(the)h(details)g(of)f(the)h(self)g(calibration)515 623 y(method.)25 b(One)c(f)o(ailure)f(of)h(the)g(Sturm)f(method)f(is)j (that)f(we)g(are)g(\002tting)g(a)g(6)g(DOF)g FL(G)h FQ(to)f(a)g(7)g (DOF)515 722 y FL(F)e FQ(which)g(means)g(that)g(the)g FL(E)g FQ(is)i(not)d(guaranteed)f(to)i(ha)n(v)o(e)g(tw)o(o)g(equal)g (singular)f(v)n(alues,)h(in)g(order)515 822 y(to)h(\002x)g(this,)g (once)f FL(E)h FQ(is)g(reco)o(v)o(ered)d(the)j(tw)o(o)g(non)f(zero)g (singular)g(v)n(alues)g(are)h(set)g(to)g(be)g(equal)f(and)515 922 y FL(E)o FQ(.)515 1159 y Fs(7.4.2)99 b(Sturm)26 b(Self)f (Calibration)g(function)515 1315 y FO(function)48 b([focal_length,)f (E,)j(CC_out])e(=)i(torr_self_calib_f\(F,CC\))515 1536 y FM(Input:)619 1702 y FQ(1.)40 b FO(F)p FQ(:)21 b(the)f FI(3)e FA(\002)g FI(3)i FQ(fundamental)e(matrix)i FL(F)p FQ(.)619 1868 y(2.)40 b FO(CC)p FQ(:)21 b(the)f(estimated)g (calibration)f(matrix.)515 2089 y FM(Output:)619 2255 y FQ(1.)40 b FO(focal)p 977 2255 25 4 v 29 w(length)p FQ(:)25 b(estimate)20 b(of)g(focal)g(length.)619 2421 y(2.)40 b FO(E)p FQ(:)21 b(estimate)f(of)g(essential)h(matrix.)619 2587 y(3.)40 b FO(CC)p 827 2587 V 30 w(out)p FQ(:)25 b(estimate)c(of)e(calibration)g(matrix.)515 2825 y Fs(7.4.3)99 b(Reco)o(v)o(ery)25 b(of)g Fp(R)f Fs(and)i Fp(t)515 2980 y FQ(Gi)n(v)o(en)21 b(an)h(Essential)h(matrix)f(it)h(can)f(be)g (decomposed)e(as)j(follo)n(ws)e([20)o(,)i(33)o(].)31 b(Suppose)21 b(that)i(the)517 3080 y(S)t(V)t(D)g(of)d FL(E)g FQ(is)i FL(E)g FI(=)h FL(U)p FI(\003)p FL(V)1342 3044 y FF(>)1398 3080 y FQ(.)i(De\002ne)1065 3343 y FL(W)f FI(=)1276 3176 y FC(2)1276 3325 y(4)1373 3242 y FI(0)82 b FA(\000)p FI(1)g(0)1373 3342 y(1)115 b(0)f(0)1373 3441 y(0)h(0)f(1)1769 3176 y FC(3)1769 3325 y(5)1908 3343 y FQ(and)82 b FL(Z)23 b FI(=)2280 3176 y FC(2)2280 3325 y(4)2409 3242 y FI(0)115 b(1)82 b(0)2377 3342 y FA(\000)p FI(1)g(0)g(0)2409 3441 y(0)115 b(0)82 b(0)2773 3176 y FC(3)2773 3325 y(5)515 3620 y FQ(The)20 b(tw)o(o)g(possible)g(f)o (actorization)e FL(E)23 b FI(=)g FL(TR)1853 3632 y FB(j)1909 3620 y FQ(as)e(follo)n(ws:)1728 3812 y FL(T)i FI(=)g FL(UZ)q(U)2110 3775 y FF(>)3220 3812 y FQ(\(7.9\))515 3994 y(and)1267 4177 y FL(R)1339 4189 y FG(1)1399 4177 y FI(=)g FL(UW)q(V)1734 4140 y FF(>)1873 4177 y FQ(or)82 b FL(R)2097 4189 y FG(2)2157 4177 y FI(=)22 b FL(U)q(W)2418 4140 y FF(>)2474 4177 y FL(V)2547 4140 y FF(>)2604 4177 y FH(:)515 4359 y FQ(Let)30 b FL(t)g FQ(be)g(the)g(right)f(epipole)g (of)g FL(E)p FQ(,)j(gi)n(v)o(en)d(by)g(the)h(third)f(column)g(of)g FL(U)q FQ(.)54 b(Then,)31 b(if)f(the)g(\002rst)515 4459 y(projection)18 b(matrix)i(is)h(set)g(to)1748 4559 y FL(P)i FI(=)g FL(C)p FI([)p FL(I)p FA(j)p FL(0)p FI(])1032 b FQ(\(7.10\))515 4708 y(there)19 b(are)i(four)e(choices)g(for)h FL(P)1435 4672 y FF(0)1458 4708 y FQ(:)688 4891 y FL(P)753 4854 y FF(0)800 4891 y FI(=)i([)p FL(R)982 4903 y FG(1)1019 4891 y FA(j)p FL(t)p FI(])42 b FQ(or)f FL(P)1320 4854 y FF(0)1366 4891 y FI(=)23 b([)p FL(R)1549 4903 y FG(1)1586 4891 y FA(j)18 b(\000)g FL(t)p FI(])42 b FQ(or)f FL(P)1988 4854 y FF(0)2034 4891 y FI(=)23 b([)p FL(R)2217 4903 y FG(2)2254 4891 y FA(j)p FL(t)p FI(])42 b FQ(or)e FL(P)2555 4854 y FF(0)2601 4891 y FI(=)23 b([)p FL(R)2783 4903 y FG(2)2820 4891 y FA(j)c(\000)f FL(t)p FI(])173 b FQ(\(7.11\))p eop %%Page: 55 57 55 56 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(55)515 523 y(It)j(can)g(be)g(seen)g(that)h(tw)o(o)f(pairs)g(of)g (the)g(solutions)g(simply)g(ha)n(v)o(e)f(the)h(translation)g(re)n(v)o (ersed.)26 b(The)515 623 y(other)i(tw)o(o)h(pairs)g(are)g(called)g (\223twisted)g(pairs\224)g(and)g(in)g(these)g(the)g(camera)g(is)h (rotated)e(by)g(180)515 722 y(de)o(grees)19 b(about)g(the)h(line)h (joining)e(the)h(camera)f(centres.)25 b(Richard)20 b(Hartle)o(y)f([21)o (,)i(25)o(])f(proposed)e(a)515 822 y(scheme)e(to)h(resolv)o(e)g(the)g (ambiguity)e(by)i(looking)e(at)i(which)g(side)g(of)g(the)g(camera)f(a)i (reconstructed)515 922 y(point)f(lay)-5 b(.)24 b(F)o(or)18 b(only)f(one)h(of)g(the)g(four)f(putati)n(v)o(e)g(solutions)h(for)g FL(P)2432 885 y FF(0)2474 922 y FQ(will)h(the)f(reconstructed)e(point) 515 1021 y FL(X)32 b FQ(lie)h(in)f(front)e(of)i(the)g(camera)f(in)h (both)f(vie)n(ws.)60 b(In)31 b(order)g(to)h(determine)e(this)i(the)g (camera)515 1121 y(matrices)c(must)h(\002st)g(be)g(normalized)d(so)j (that)g(the)g(determinate)e(of)h(the)g(\002rst)i FI(3)24 b FA(\002)g FI(3)29 b FQ(matrix)f(is)515 1220 y(greater)19 b(than)i(zero.)k(Let)c FL(x)h FQ(and)e FL(x)1516 1190 y FF(0)1561 1220 y FQ(be)h(the)f(homogeneous)e(image)i(coordinate)f (projections)g(of)h(a)515 1320 y(point)f FL(X)p FQ(,)i(then)f FL(X)h FQ(is)g(in)f(front)f(of)h(both)f(cameras)h(if)1455 1495 y FL(x)1505 1507 y FG(3)1542 1495 y FL(X)1614 1507 y FG(4)1675 1495 y FH(>)i FI(0)83 b FQ(and)f FL(x)2140 1461 y FF(0)2140 1516 y FG(3)2177 1495 y FL(X)2250 1507 y FG(4)2310 1495 y FH(>)23 b FI(0)639 1670 y FQ(Once)d(the)g(rotation)g (matrix)f(is)i(found)e(it)i(can)f(written)g(in)g(terms)g(of)g(a)h (minimal)e(parametriza-)515 1770 y(tion)h(using)f(Rodrigues')g(formula) 1249 1945 y FL(R)j FI(=)h(cos)13 b FH(!)s FL(I)18 b FI(+)g(sin)c FH(!)s FI([)p FL(l)o FI(])1991 1957 y FF(\002)2066 1945 y FI(+)k(\(1)g FA(\000)g FI(cos)13 b FH(!)s FI(\))p FL(ll)2589 1908 y FF(>)3178 1945 y FQ(\(7.12\))515 2120 y(where)19 b FL(l)i FQ(is)g(the)f(axis)g(of)g(rotation,)f(which)h(may)g(be)g(reco) o(v)o(ered)d(from)i(the)i(elements)e(of)h FL(R)h FQ(as:)1571 2392 y FL(l)i FI(=)1709 2225 y FC(0)1709 2374 y(@)1823 2291 y FL(R)1894 2303 y FG(32)1983 2291 y FA(\000)18 b FL(R)2138 2303 y FG(23)1823 2391 y FL(R)1894 2403 y FG(13)1983 2391 y FA(\000)g FL(R)2138 2403 y FG(31)1823 2491 y FL(R)1894 2503 y FG(21)1983 2491 y FA(\000)g FL(R)2138 2503 y FG(12)2250 2225 y FC(1)2250 2374 y(A)3178 2392 y FQ(\(7.13\))515 2661 y(and)h FH(!)24 b FQ(is)d(the)f(angle)g(of)g (rotation,)1487 2883 y FH(!)26 b FI(=)d(arccos)1888 2766 y FC(\022)1960 2827 y FQ(trace)o FL(R)18 b FA(\000)g FI(1)p 1960 2864 376 4 v 2127 2940 a(2)2345 2766 y FC(\023)3178 2883 y FQ(\(7.14\))515 3159 y Fs(7.4.4)99 b(Function)25 b(f)n(or)g Fp(R)g Fs(and)g Fp(t)p Fs(;)g Fr(torr)p 2012 3159 30 4 v 35 w(linear)p 2407 3159 V 34 w(EtoPX)515 3314 y FO(function)48 b([P1,P2,R,t,rot_axis,rot_angle])515 3414 y(=)h(torr_linear_EtoPX\(E,matches,C,m3\))515 3634 y FM(Input:)619 3794 y FQ(1.)40 b FO(E)p FQ(:)21 b(Essential)f(matrix)g FL(E)p FQ(.)619 3957 y(2.)40 b FO(matches)49 b FQ(:)26 b FH(n)18 b FA(\002)g FI(4)i FQ(array)f(of)h(matches.)619 4120 y(3.)40 b FO(C)p FQ(:)21 b(calibration)e(matrix)g FL(C)p FQ(.)619 4283 y(4.)40 b FO(m3)p FQ(:)26 b(third)19 b(homogeneous)e(image)j(coordinate.)515 4502 y FM(Output:)619 4662 y FQ(1.)40 b FO(P1,P2)p FQ(:)25 b(estimated)20 b(projection)f (matrices)p FL(P)o FH(;)14 b FL(P)2165 4626 y FF(0)619 4825 y FQ(2.)40 b FO(R,t)p FQ(:)26 b(estimated)20 b(rotation)f(and)g (translation)g FL(R)p FH(;)14 b FL(t)619 4988 y FQ(3.)40 b FO(rot)p 877 4988 25 4 v 30 w(axis,rot)p 1307 4988 V 28 w(angle)p FQ(:)25 b(angle)19 b(and)h(axis)h(of)f(rotation)f FL(l)h FQ(and)f FH(!)p eop %%Page: 56 58 56 57 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(56)515 523 y Fs(7.4.5)99 b(Non-linear)25 b(Optimization)f(of)h Fp(G)515 679 y FQ(Once)20 b(e)o(xtrinsic)h(parameters)f FL(R)h FQ(and)f FL(t)i FQ(and)e(the)h(intrinsic)g(parameter)f FH(f)30 b FQ(ha)n(v)o(e)20 b(been)g(reco)o(v)o(ered,)515 778 y(the)o(y)26 b(may)g(be)h(further)e(optimized)g(by)i(gradient)e (descent.)44 b(There)26 b(are)h(6)g(DOF)g(in)g(this)g(formu-)515 878 y(lation,)32 b(a)e(good)f(minimal)h(parametrization,)g(represented) e(by)i(the)g FI(6)c FA(\002)f FI(1)30 b FQ(v)o(ector)f FL(g)j FQ(is)f FH(f)40 b FQ(the)515 978 y(focal)27 b(length,)i FL(t)g FQ(the)f(translation)g(v)o(ector)f(\(parametrized)e(in)k(2)f (spherical)f(coordinates\),)h FL(l)h FQ(the)515 1077 y(axis)23 b(of)f(rotation)f(\(parametrized)f(in)j(2)g(spherical)f (coordinates\),)e(and)i FH(!)k FQ(the)d(angle)f(of)g(rotation.)515 1177 y(Note)e(the)h(follo)n(wing)d(tw)o(o)j(functions)e(are)i(pro)o (vided)d(for)h(going)h(from)f(unit)h(v)o(ectors)g(to)g(spherical)515 1276 y(coordinates)e(and)i(vice)g(v)o(ersa)g FO(torr)p 1618 1276 25 4 v 29 w(unit2sphere)e FQ(and)i FO(torr)p 2556 1276 V 29 w(sphere2unit)p FQ(.)k(.)639 1376 y(Gi)n(v)o(en)15 b(these)g(six)h(parameters)e FL(g)k FQ(plus)d(the)g(calibration)f (matrix)h FL(C)p FQ(,)i(it)f(is)g(possible)f(to)h(calculate)515 1476 y FL(F)p FQ(,)h(the)f(function)e(pro)o(vided)f(to)j(do)f(this)h (is)h FO(torr)p 1927 1476 V 29 w(g2F)p FQ(.)e(Once)h FL(F)g FQ(is)h(obtained)d(the)i(Sampson)f(error)515 1575 y(may)i(be)h(computed)e(in)i(the)g(usual)f(w)o(ay)-5 b(.)24 b(The)18 b(function)e(pro)o(vided)f(to)j(do)g(this)g FO(torr)p 2976 1575 V 29 w(errg)p 3205 1575 V 30 w(sse)515 1675 y FQ(Thus)24 b(it)h(is)g(possible)f(to)g(do)g(gradient)f(descent)h (of)g(the)g(error)f(function)g(for)g FL(g)q FQ(.)38 b(The)24 b(function)e(to)515 1775 y(do)e(this)g(is)h FO(torr)p 1040 1775 V 30 w(nonlinG)p FQ(.)515 2012 y Fs(7.4.6)99 b(Non)25 b(Linear)g(minimization)g(of)g Fp(g)515 2168 y FO(function)48 b([g,f])h(=)565 2268 y (torr_nonlinG\(g_init,nx1,ny1,nx2,n)o(y2,)44 b(no_matches,)j(m3,)j(C\)) 515 2489 y FM(Input:)619 2655 y FQ(1.)40 b FO(nx1,ny1,nx2,ny2,)47 b(no)p 1674 2655 V 30 w(matches,)h(m3,)p FQ(:)25 b(de\002ned)19 b(as)i(in)g(Section)e(4.1.1.)619 2821 y(2.)40 b FO(g)p 777 2821 V 30 w(init)p FQ(:)25 b(initial)c(estimate)f(of)g FL(g)i FQ(obtained)c(from)i FO(torr)p 2416 2821 V 29 w(linear)p 2745 2821 V 29 w(EtoPX)p FQ(.)619 2987 y(3.)40 b FO(C)p FQ(:)21 b(the)f(calibration)f(matrix)h FL(C)o FQ(.)515 3208 y FM(Output:)619 3374 y FQ(1.)40 b FO(f)p FQ(:)26 b FL(f)9 b FQ(.)619 3540 y(2.)40 b FO(g)p FQ(:)26 b FL(g)q FQ(.)515 3820 y FD(7.5)119 b(T)-11 b(esting)29 b(Self)i(Calibration)g Fu(torr)p 2320 3820 36 4 v 42 w(test)p 2650 3820 V 42 w(calib)p 3052 3820 V 41 w(sc)515 4006 y FQ(This)20 b(script)g(generates)f(some)g(synthetic)g(matches)h (as)g(described)f(in)h(Chapter)f(8.)25 b(Then)19 b(a)h(funda-)515 4105 y(mental)i(matrix)g(is)h(estimated,)g(then)f(calibration)f(reco)o (v)o(ered)f(\(assuming)i(e)n(v)o(erything)e(is)j(kno)n(wn)515 4205 y(bar)f(the)g(focal)g(length\).)31 b(Then)21 b(Essential)i (matrix,)f(rotation)g(matrix)g(and)f(angle)h(and)g(axis)h(of)f(ro-)515 4304 y(tation)e(are)g(reco)o(v)o(ered.)i(The)e(script)g(prints)g(out)g (the)g(estimated)g(and)g(ground)e(truth)h(rotation)g(axis,)515 4404 y(angle,)25 b(translation)f(and)h(focal)g(length.)38 b(As)26 b(a)g(deb)n(ugging)c(e)o(x)o(ercise)j(set)g FO(no)p 2797 4404 25 4 v 30 w(noise)49 b(=)g(1)26 b FQ(on)515 4504 y(line)h(25.)47 b(The)27 b(script)h(runs)f(on)g(the)g(noise)h (free)f(data,)i(the)e(astute)h(reader)f(will)h(notice)f(that)h(the)515 4603 y(result)20 b(and)g(ground)e(truth)h(are)h(the)g(same.)515 4778 y FS(\045this)44 b(is)g(a)h(script)f(to)g(test)g(the)h(self)f (calibration)f(stuff)515 4869 y(\045torr_test_calib_sc.m)515 4960 y(\045main\(\))p eop %%Page: 57 59 57 58 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(57)515 523 y FS(\045profile)43 b(on)515 614 y(clear)h(all;)515 706 y(m3)g(=)h(256;)515 797 y(method)f(=)g('mapsac';)515 888 y(method)g(=)g('linear';)515 1162 y(\045)560 1254 y(randn\('state',0\))515 1345 y(rand\('state',0\))515 1436 y(no_test)f(=)i(1;)515 1528 y(for\(i)f(=)g(1:no_test\))515 1710 y(\045)224 b([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2])39 b(=)44 b(...)515 1802 y(\045)403 b(torr_gen_2view_matches\(foc,)40 b(no_matches,)j(noise_sigma,)560 1893 y(translation_mult,)f (translation_adder,)f(...)515 1984 y(\045)403 b(rotation_multplier,)42 b(min_Z,Z_RAN,m3\);)515 2076 y([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2,tr)o (ue_C,t)o(rue_R)o(,true_)o(TX,)d(true_E])k(=)515 2167 y(torr_gen_2view_matches;)515 2350 y(no_matches)g(=)i(length\(nx1\);) 515 2532 y(\045if)f(we)h(set)f(this)g(to)h(one)f(then)g(the)g(result)g (should)g(be)g(the)h(same)f(as)g(the)h(groundtruth...)515 2623 y(no_noise)e(=)i(0;)515 2715 y(if)f(\(no_noise\))694 2806 y(nx1)g(=)h(x1;)694 2897 y(nx2)f(=)h(x2;)694 2989 y(ny1)f(=)h(y1;)694 3080 y(ny2)f(=)h(y2;)515 3171 y(end)515 3537 y(matches)e(=)i([nx1,ny1,nx2,ny2];)515 3628 y(perfect_matches)d(=) j([x1,y1,x2,y2];)515 3719 y(set_rank2)e(=)i(1;)515 3993 y(\045first)f(estimate)f(F)515 4085 y([f,)h(e1,)g (n_inliers,inlier_index,nF])d(=)515 4176 y(torr_estimateF\()h(matches,) i(m3,)g([],)g(method,)g(set_rank2\);)515 4359 y(\045next)g(correct)f (the)i(points)f(so)g(that)g(they)g(lie)h(on)f(a)h(fundamental)e(matrix) 515 4450 y([corrected_matches)f(error2])h(=)515 4541 y(torr_correctx4F\(f,)f(nx1,ny1,nx2,ny2,)g(no_matches,)h(m3\);)515 4724 y(\045now)h(guess)g(the)g(camera)g(calibration)f(matrix)515 4815 y(CC)h(=)h(diag\(ones\(3,1\),0\);)515 4907 y(CC\(3,3\))e(=)i(1;)p eop %%Page: 58 60 58 59 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(58)515 523 y FS(\045next)44 b(self)g(calibrate)f(for)i(focal)f (length)515 614 y([focal_length,)e(nE,CCC])i(=)h (torr_self_calib_f\(nF,CC\);)515 797 y(\045now)f(we)g(have)h(an)f (Essential)f(matrix)h(we)h(can)f(establish)f(the)i(camera)e(frame...) 515 888 y([P1,P2,R,t,srot_axis,rot_angle,g])84 b(=)45 b(torr_linear_EtoPX\(nE,matches,CCC,m3\))o(;)515 1071 y(\045next)f(convert)f(the)i(6)f(parameters)g(of)g(g)h(to)f(a)h (fundamental)e(matrix)515 1162 y(f2)h(=)h(torr_g2F\(g,CCC\);)515 1254 y(disp\('error)e(before)h(non-linear)f(minimization'\))515 1345 y(e)h(=)h(torr_errf2\(f2,)e(nx1,ny1,nx2,ny2,)f(length\(nx1\),)g (m3\);)515 1436 y(norm\(e\))515 1619 y([g,f])i(=)g(torr_nonlinG\(g)f (,nx1,ny1,nx2,ny2,)f(no_matches,)h(m3,)h(CCC\))515 1893 y(disp\('error)f(after'\))560 1984 y(e2)h(=)h(torr_errf2\(f,)d (nx1,ny1,nx2,ny2,)h(length\(nx1\),)f(m3\);)515 2076 y(norm\(e2\))515 2350 y(\045the)i(question)f(now)i(arises:)e(how)i(good)f(is)g(the)h (fit?)f(compare)f(to)i(groundtruth)515 2441 y(true_rot_axis)d(=)j ([true_R\(3,2\)-true_R\(2,3\),)c(true_R\(1,3\))i(-)h(true_R\(3,1\),)f (true_R\(2,1\))g(-)515 2532 y(true_R\(1,2\)]';)515 2623 y(true_rot_axis)f(=)j(true_rot_axis)e(/norm\(true_rot_axis\);)515 2715 y(true_rot_angle)f(=)j(acos\()f(\(trace\(true_R\)-1\)/2\);)515 2897 y(true_t\(1\))f(=)i(-true_TX\(2,3\);)515 2989 y(true_t\(2\))e(=)i (true_TX\(1,3\);)515 3080 y(true_t\(3\))e(=)i(-true_TX\(1,2\);)515 3171 y(true_t)f(=)g(true_t/norm\(true_t\);)515 3445 y(disp\('true)f (camera)h(parameters'\))515 3537 y(true_t)515 3628 y(true_rot_axis)515 3719 y(true_rot_angle)515 3811 y(true_C)515 4085 y(rot_axis)f(=)i (torr_sphere2unit\([g\(2\))c(g\(3\)]\);)515 4176 y(tt)j(=)h (torr_sphere2unit\([g\(5\))c(g\(6\)]\);)515 4267 y(rot_angle)i(=)i (g\(4\);)515 4450 y(CCC\(3,3\))e(=)i(1/g\(1\);)515 4633 y(disp\('estimated)d(camera)i(parameters'\))515 4724 y(tt)515 4815 y(rot_axis)515 4907 y(rot_angle)515 4998 y(CCC)p eop %%Page: 59 61 59 60 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(59)515 523 y FS(end)515 852 y Fs(7.5.1)99 b(Displaying)24 b(Structur)n(e,)j Fr(torr)p 1996 852 30 4 v 35 w(display)p 2451 852 V 34 w(structure)515 1008 y FQ(An)g(e)o(xample)f(of)i (synthetic)e(matches)i(and)e(structure)h(generated)f(from)g(them)h(is)i (gi)n(v)o(en)d(in)i(\002g-)515 1107 y(ure)20 b(7.1.)k(The)c(3D)g (structure)f(can)h(be)g(displayed)f(using)560 1282 y FS(f1)44 b(=)h(torr_display_structure\(X,)40 b(P1,)45 b(P2,display_numbers,f1\).)639 1464 y FQ(The)30 b(test)h(diagnostics)e 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FO(X)p FQ(:)20 b(either)g FI(3)c FA(\002)g FH(n)k FQ(inhomogeneous)c(3D)j(coordinates) f(or)h FI(4)d FA(\002)g FH(n)k FQ(homogeneous)d(3D)i(coor)n(-)722 2549 y(dinate.)619 2715 y(2.)40 b FO(P1,)50 b(P2)p FQ(:)25 b(tw)o(o)20 b FI(3)e FA(\002)g FI(4)j FQ(projection)d(matrices.)619 2881 y(3.)40 b FO(f1)p FQ(:)26 b(\002gure)19 b(handle)g(of)h(the)g (\002gure)g(for)f(the)i(matches)e(to)i(be)f(displayed)f(in.)619 3047 y(4.)40 b FO(display)p 1077 3047 V 29 w(numbers)p FQ(:)24 b(if)d(set)g(to)f(1)g(then)g(displays)g(the)g(inde)o(x)f(of)h (each)g(match.)515 3268 y FM(Output:)619 3434 y FQ(1.)40 b FO(f1)p FQ(:)26 b(:)g(\002gure)19 b(handle)g(of)h(the)g(\002gure)g (for)f(the)h(matches)g(to)h(be)f(displayed)f(in.)515 3738 y Fs(7.5.2)99 b(An)25 b(example)g(script)g(f)n(or)g(3D)g(structur) n(e)i(generation)515 3894 y FQ(The)20 b(script)g FO(torr)p 1075 3894 V 29 w(test)p 1304 3894 V 29 w(SFMsc)g FQ(creates)h(some)f (synthetic)f(matches,)h(and)g(then)g(displays)g(the)515 3993 y(3D)j(structure.)32 b(The)22 b(\002rst)i(half)f(of)f(the)h (script)g(follo)n(ws)g FO(torr)p 2344 3993 V 29 w(test)p 2573 3993 V 29 w(calib)p 2852 3993 V 29 w(sc)p FQ(;)i(the)d(second)515 4093 y(half)e(is)h(self)f(e)o(xplanatory)e(and)h(is)j(listed)e(here:) 515 4359 y FS(\045next)44 b(convert)f(the)i(6)f(parameters)g(of)g(g)h (to)f(a)h(fundamental)e(matrix)515 4450 y(f2)h(=)h(torr_g2F\(g,CCC\);) 515 4633 y(\045next)f(correct)f(the)i(points)f(so)g(that)g(they)g(lie)h (on)f(the)g(fundamental)f(matrix)515 4724 y([corrected_matches)f (error2])h(=)i(torr_correctx4F\(f2,)c(nx1,ny1,nx2,ny2,)i(no_matches,)g (m3\);)515 4907 y(\045next)h(we)g(need)g(to)h(obtain)f(P1)g(&)h(P2)515 4998 y([P1,)f(P2])g(=)h(torr_g2FP\(g,CCC\);)p eop %%Page: 60 62 60 61 bop 515 232 a FJ(CHAPTER)18 b(7.)43 b(SELF)18 b(CALIBRA)-9 b(TION,)18 b(EST)-8 b(ABLISHING)18 b(A)g(PR)m(OJECTIVE)g(FRAME)p FQ(60)547 1492 y currentpoint currentpoint translate 0.3347 0.3347 scale neg exch neg exch translate 547 1492 a @beginspecial 67 @llx 212 @lly 551 @urx 589 @ury 4840 @rwi @setspecial %%BeginDocument: mateg1.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: D:\Matlab\AAAFaceGUI\Manual\mateg1.eps %%CreationDate: 05/16/2002 11:35:55 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK %%Pages: 1 %%BoundingBox: 67 212 551 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID 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fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile exch readhexstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto 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4953 1812 L 2124 3100 mt 2196 3100 L 2160 3064 mt 2160 3136 L 2517 1187 mt 2589 1187 L 2553 1151 mt 2553 1223 L 2432 1480 mt 2504 1480 L 2468 1444 mt 2468 1516 L 2044 2175 mt 2116 2175 L 2080 2139 mt 2080 2211 L 4795 766 mt 4867 766 L 4831 730 mt 4831 802 L 3972 2653 mt 4044 2653 L 4008 2617 mt 4008 2689 L 4209 2015 mt 4281 2015 L 4245 1979 mt 4245 2051 L 5426 2623 mt 5498 2623 L 5462 2587 mt 5462 2659 L 5291 2200 mt 5363 2200 L 5327 2164 mt 5327 2236 L 4246 2507 mt 4318 2507 L 4282 2471 mt 4282 2543 L 3705 3167 mt 3777 3167 L 3741 3131 mt 3741 3203 L 4535 1052 mt 4607 1052 L 4571 1016 mt 4571 1088 L 5083 2658 mt 5155 2658 L 5119 2622 mt 5119 2694 L 5128 1328 mt 5200 1328 L 5164 1292 mt 5164 1364 L 5250 1123 mt 5322 1123 L 5286 1087 mt 5286 1159 L 1812 2389 mt 1884 2389 L 1848 2353 mt 1848 2425 L 1670 1304 mt 1742 1304 L 1706 1268 mt 1706 1340 L 4683 1097 mt 4755 1097 L 4719 1061 mt 4719 1133 L 2651 2521 mt 2723 2521 L 2687 2485 mt 2687 2557 L 4391 1136 mt 4463 1136 L 4427 1100 mt 4427 1172 L 2976 3483 mt 3048 3483 L 3012 3447 mt 3012 3519 L 2339 2855 mt 2411 2855 L 2375 2819 mt 2375 2891 L 3594 2008 mt 3666 2008 L 3630 1972 mt 3630 2044 L 5114 1315 mt 5186 1315 L 5150 1279 mt 5150 1351 L 3204 2625 mt 3276 2625 L 3240 2589 mt 3240 2661 L 2244 1955 mt 2316 1955 L 2280 1919 mt 2280 1991 L 3002 3793 mt 3074 3793 L 3038 3757 mt 3038 3829 L 3983 2948 mt 4055 2948 L 4019 2912 mt 4019 2984 L 5301 1909 mt 5373 1909 L 5337 1873 mt 5337 1945 L 3759 3115 mt 3831 3115 L 3795 3079 mt 3795 3151 L 2695 1538 mt 2767 1538 L 2731 1502 mt 2731 1574 L 3025 2128 mt 3097 2128 L 3061 2092 mt 3061 2164 L 4689 1227 mt 4761 1227 L 4725 1191 mt 4725 1263 L 2767 3227 mt 2839 3227 L 2803 3191 mt 2803 3263 L 2276 4012 mt 2348 4012 L 2312 3976 mt 2312 4048 L 4194 1289 mt 4266 1289 L 4230 1253 mt 4230 1325 L 2436 3491 mt 2508 3491 L 2472 3455 mt 2472 3527 L 3975 1061 mt 4047 1061 L 4011 1025 mt 4011 1097 L 4681 887 mt 4753 887 L 4717 851 mt 4717 923 L 3839 3942 mt 3911 3942 L 3875 3906 mt 3875 3978 L 4036 1671 mt 4108 1671 L 4072 1635 mt 4072 1707 L 4696 1557 mt 4768 1557 L 4732 1521 mt 4732 1593 L 2951 2561 mt 3023 2561 L 2987 2525 mt 2987 2597 L 3696 3928 mt 3768 3928 L 3732 3892 mt 3732 3964 L 3748 2279 mt 3820 2279 L 3784 2243 mt 3784 2315 L 3364 1080 mt 3436 1080 L 3400 1044 mt 3400 1116 L 4245 1733 mt 4317 1733 L 4281 1697 mt 4281 1769 L 5247 2584 mt 5319 2584 L 5283 2548 mt 5283 2620 L 2739 1274 mt 2811 1274 L 2775 1238 mt 2775 1310 L 4554 3603 mt 4626 3603 L 4590 3567 mt 4590 3639 L 1878 1897 mt 1950 1897 L 1914 1861 mt 1914 1933 L 3968 3556 mt 4040 3556 L 4004 3520 mt 4004 3592 L 4129 820 mt 4201 820 L 4165 784 mt 4165 856 L 5226 3101 mt 5298 3101 L 5262 3065 mt 5262 3137 L 4430 2712 mt 4502 2712 L 4466 2676 mt 4466 2748 L 2319 3177 mt 2391 3177 L 2355 3141 mt 2355 3213 L 4249 2377 mt 4321 2377 L 4285 2341 mt 4285 2413 L 5340 2778 mt 5412 2778 L 5376 2742 mt 5376 2814 L 5344 3373 mt 5416 3373 L 5380 3337 mt 5380 3409 L 2259 3031 mt 2331 3031 L 2295 2995 mt 2295 3067 L 5454 2742 mt 5526 2742 L 5490 2706 mt 5490 2778 L 4189 3076 mt 4261 3076 L 4225 3040 mt 4225 3112 L 2106 3553 mt 2178 3553 L 2142 3517 mt 2142 3589 L 5331 1149 mt 5403 1149 L 5367 1113 mt 5367 1185 L 4691 3047 mt 4763 3047 L 4727 3011 mt 4727 3083 L 2717 2189 mt 2789 2189 L 2753 2153 mt 2753 2225 L 4516 776 mt 4588 776 L 4552 740 mt 4552 812 L 4276 826 mt 4348 826 L 4312 790 mt 4312 862 L 4448 3582 mt 4520 3582 L 4484 3546 mt 4484 3618 L 2607 2604 mt 2679 2604 L 2643 2568 mt 2643 2640 L 5425 1299 mt 5497 1299 L 5461 1263 mt 5461 1335 L 4472 2019 mt 4544 2019 L 4508 1983 mt 4508 2055 L 2788 3524 mt 2860 3524 L 2824 3488 mt 2824 3560 L 3308 3706 mt 3380 3706 L 3344 3670 mt 3344 3742 L 2926 895 mt 2998 895 L 2962 859 mt 2962 931 L 1787 884 mt 1859 884 L 1823 848 mt 1823 920 L 4611 1648 mt 4683 1648 L 4647 1612 mt 4647 1684 L 4471 1268 mt 4543 1268 L 4507 1232 mt 4507 1304 L 2251 2685 mt 2323 2685 L 2287 2649 mt 2287 2721 L 3672 3678 mt 3744 3678 L 3708 3642 mt 3708 3714 L 2449 3502 mt 2521 3502 L 2485 3466 mt 2485 3538 L 2789 1598 mt 2861 1598 L 2825 1562 mt 2825 1634 L 4737 2358 mt 4809 2358 L 4773 2322 mt 4773 2394 L 3830 2950 mt 3902 2950 L 3866 2914 mt 3866 2986 L 2361 2591 mt 2433 2591 L 2397 2555 mt 2397 2627 L 4154 3255 mt 4226 3255 L 4190 3219 mt 4190 3291 L 3513 2647 mt 3585 2647 L 3549 2611 mt 3549 2683 L 5539 1668 mt 5611 1668 L 5575 1632 mt 5575 1704 L 5509 2413 mt 5581 2413 L 5545 2377 mt 5545 2449 L gs 899 389 5357 4228 MR c np gr 0 sg 2957 1501 mt ( 1) s 3501 2467 mt ( 2) s 4526 4339 mt ( 3) s 2498 3592 mt ( 4) s 2148 2345 mt ( 5) s 5622 1248 mt ( 6) s 2640 1124 mt ( 7) s 4146 3683 mt ( 8) s 5702 822 mt ( 9) s 2975 3169 mt ( 10) s 5035 1963 mt ( 11) s 1297 3705 mt ( 12) s 2229 1172 mt ( 13) s 2003 1546 mt ( 14) s 1583 2347 mt ( 15) s 5016 750 mt ( 16) s 3864 2961 mt ( 17) s 4207 2196 mt ( 18) s 5554 3005 mt ( 19) s 5452 2537 mt ( 20) s 4157 2917 mt ( 21) s 3433 3707 mt ( 22) s 4645 1117 mt ( 23) s 5147 3002 mt ( 24) s 5349 1483 mt ( 25) s 5465 1212 mt ( 26) s 1414 2535 mt ( 27) s 1384 1257 mt ( 28) s 4830 1219 mt ( 29) s 2299 2778 mt ( 30) s 4475 1201 mt ( 31) s 2724 3773 mt ( 32) s 1673 3338 mt ( 33) s 3147 2501 mt ( 34) s 5315 1447 mt ( 35) s 3018 2848 mt ( 36) s 1595 2185 mt ( 37) s 2721 4133 mt ( 38) s 3816 3395 mt ( 39) s 5445 2102 mt ( 40) s 3615 3416 mt ( 41) s 2490 1585 mt ( 42) s 2727 2357 mt ( 43) s 4822 1320 mt ( 44) s 2455 3539 mt ( 45) s 1907 4331 mt ( 46) s 4216 1386 mt ( 47) s 2050 3834 mt ( 48) s 3957 1095 mt ( 49) s 4842 914 mt ( 50) s 3664 4273 mt ( 51) s 3981 1857 mt ( 52) s 4800 1750 mt ( 53) s 2157 3273 mt ( 54) s 3487 4307 mt ( 55) s 3650 2494 mt ( 56) s 3302 1079 mt ( 57) s 4272 1873 mt ( 58) s 5424 3229 mt ( 59) s 2375 1306 mt ( 60) s 4471 3947 mt ( 61) s 1380 2011 mt ( 62) s 3701 4258 mt ( 63) s 4166 800 mt ( 64) s 5274 3530 mt ( 65) s 4360 3198 mt ( 66) s 1708 3672 mt ( 67) s 4169 2766 mt ( 68) s 5538 3531 mt ( 69) s 5359 3719 mt ( 70) s 1435 3658 mt ( 71) s 5677 3440 mt ( 72) s 4102 3386 mt ( 73) s 1595 3960 mt ( 74) s 5560 1248 mt ( 75) s 4667 3363 mt ( 76) s 2325 2432 mt ( 77) s 4640 753 mt ( 78) s 4376 823 mt ( 79) s 4330 4096 mt ( 80) s 2134 2956 mt ( 81) s 5664 1433 mt ( 82) s 4494 2260 mt ( 83) s 2447 3891 mt ( 84) s 3078 4033 mt ( 85) s 2510 827 mt ( 86) s 1326 766 mt ( 87) s 4698 1778 mt ( 88) s 4517 1436 mt ( 89) s 1975 2846 mt ( 90) s 3479 4031 mt ( 91) s 1906 4015 mt ( 92) s 2473 1700 mt ( 93) s 4774 2761 mt ( 94) s 3701 3248 mt ( 95) s 2062 2773 mt ( 96) s 3971 3856 mt ( 97) s 3201 3111 mt ( 98) s 5733 1842 mt ( 99) s 5755 2941 mt (100) s gs 899 389 5357 4228 MR c np /c9 { 0.000000 0.000000 1.000000 sr} bdef c9 122 9 2957 1459 2 MP stroke 258 -301 3501 2425 2 MP stroke 140 -416 4526 4297 2 MP stroke 393 -338 2498 3550 2 MP stroke 572 -247 2148 2303 2 MP stroke -235 -70 5622 1206 2 MP stroke 582 10 2640 1082 2 MP stroke 130 -259 4146 3641 2 MP stroke -254 0 5702 780 2 MP stroke 245 -219 2975 3127 2 MP stroke -82 -145 5035 1921 2 MP stroke 863 -563 1297 3663 2 MP stroke 324 57 2229 1130 2 MP stroke 465 -24 2003 1504 2 MP stroke 497 -130 1583 2305 2 MP stroke -185 58 5016 708 2 MP stroke 144 -266 3864 2919 2 MP stroke 38 -139 4207 2154 2 MP stroke -92 -340 5554 2963 2 MP stroke -125 -295 5452 2495 2 MP stroke 125 -368 4157 2875 2 MP stroke 308 -498 3433 3665 2 MP stroke -74 -23 4645 1075 2 MP stroke -28 -302 5147 2960 2 MP stroke -185 -113 5349 1441 2 MP stroke -179 -47 5465 1170 2 MP stroke 434 -104 1414 2493 2 MP stroke 322 89 1384 1215 2 MP stroke -111 -80 4830 1177 2 MP stroke 388 -215 2299 2736 2 MP stroke -48 -23 4475 1159 2 MP stroke 288 -248 2724 3731 2 MP stroke 702 -441 1673 3296 2 MP stroke 483 -451 3147 2459 2 MP stroke -165 -90 5315 1405 2 MP stroke 222 -181 3018 2806 2 MP stroke 685 -188 1595 2143 2 MP stroke 317 -298 2721 4091 2 MP stroke 203 -405 3816 3353 2 MP stroke -108 -151 5445 2060 2 MP stroke 180 -259 3615 3374 2 MP stroke 241 -5 2490 1543 2 MP stroke 334 -187 2727 2315 2 MP stroke -97 -51 4822 1278 2 MP stroke 348 -270 2455 3497 2 MP stroke 405 -277 1907 4289 2 MP stroke 14 -55 4216 1344 2 MP stroke 422 -301 2050 3792 2 MP stroke 54 8 3957 1053 2 MP stroke -125 15 4842 872 2 MP stroke 211 -289 3664 4231 2 MP stroke 91 -144 3981 1815 2 MP stroke -68 -151 4800 1708 2 MP stroke 830 -670 2157 3231 2 MP stroke 245 -337 3487 4265 2 MP stroke 134 -173 3650 2452 2 MP stroke 98 43 3302 1037 2 MP stroke 9 -98 4272 1831 2 MP stroke -141 -603 5424 3187 2 MP stroke 400 10 2375 1264 2 MP stroke 119 -302 4471 3905 2 MP stroke 534 -72 1380 1969 2 MP stroke 303 -660 3701 4216 2 MP stroke -1 62 4166 758 2 MP stroke -12 -387 5274 3488 2 MP stroke 106 -444 4360 3156 2 MP stroke 647 -453 1708 3630 2 MP stroke 116 -347 4169 2724 2 MP stroke -162 -711 5538 3489 2 MP stroke 21 -304 5359 3677 2 MP stroke 860 -585 1435 3616 2 MP stroke -187 -656 5677 3398 2 MP stroke 123 -268 4102 3344 2 MP stroke 547 -365 1595 3918 2 MP stroke -193 -57 5560 1206 2 MP stroke 60 -274 4667 3321 2 MP stroke 428 -201 2325 2390 2 MP stroke -88 65 4640 711 2 MP stroke -64 45 4376 781 2 MP stroke 154 -472 4330 4054 2 MP stroke 509 -310 2134 2914 2 MP stroke -203 -92 5664 1391 2 MP stroke 14 -199 4494 2218 2 MP stroke 377 -325 2447 3849 2 MP stroke 266 -285 3078 3991 2 MP stroke 452 110 2510 785 2 MP stroke 497 160 1326 724 2 MP stroke -51 -88 4698 1736 2 MP stroke -10 -126 4517 1394 2 MP stroke 312 -119 1975 2804 2 MP stroke 229 -311 3479 3989 2 MP stroke 579 -471 1906 3973 2 MP stroke 352 -60 2473 1658 2 MP stroke -1 -361 4774 2719 2 MP stroke 165 -256 3701 3206 2 MP stroke 335 -140 2062 2731 2 MP stroke 219 -559 3971 3814 2 MP stroke 348 -422 3201 3069 2 MP stroke -158 -132 5733 1800 2 MP stroke -210 -486 5755 2899 2 MP stroke -42 8 39 13 3040 1455 3 MP stroke -44 121 127 -77 3632 2201 3 MP stroke 11 149 103 -125 4563 4006 3 MP stroke -83 145 176 -78 2715 3290 3 MP stroke -154 131 222 -33 2498 2089 3 MP stroke 87 3 -67 -43 5454 1179 3 MP stroke -193 47 191 53 3031 1039 3 MP stroke -7 97 78 -74 4198 3456 3 MP stroke 84 -22 -84 -22 5532 802 3 MP stroke -51 94 111 -52 3109 2960 3 MP stroke 47 41 -7 -55 4960 1831 3 MP stroke -207 260 362 -111 1798 3211 3 MP stroke -114 9 99 47 2454 1140 3 MP stroke -150 48 157 32 2311 1448 3 MP stroke -146 86 182 0 1898 2175 3 MP stroke 53 -35 -69 3 4900 763 3 MP stroke -11 100 84 -75 3924 2728 3 MP stroke 6 49 32 -42 4213 2057 3 MP stroke 77 104 17 -120 5445 2743 3 MP stroke 82 87 -1 -108 5328 2308 3 MP stroke 9 133 92 -111 4190 2618 3 MP stroke -33 191 170 -138 3571 3305 3 MP stroke 27 1 -21 -14 4592 1066 3 MP stroke 51 97 33 -102 5086 2760 3 MP stroke 77 21 -46 -53 5210 1381 3 MP stroke 66 0 -53 -31 5339 1154 3 MP stroke -129 72 157 3 1691 2386 3 MP stroke -119 -1 94 57 1612 1247 3 MP stroke 48 17 -26 -36 4745 1133 3 MP stroke -99 104 158 -38 2529 2559 3 MP stroke 19 3 -13 -12 4440 1148 3 MP stroke -60 107 129 -57 2883 3540 3 MP stroke -171 206 292 -85 2083 2940 3 MP stroke -97 191 222 -107 3408 2115 3 MP stroke 67 15 -42 -43 5192 1358 3 MP stroke -48 79 98 -41 3142 2666 3 MP stroke -200 121 252 -3 2028 1958 3 MP stroke -63 126 146 -71 2892 3864 3 MP stroke -11 151 123 -116 3896 3064 3 MP stroke 56 41 -15 -59 5352 1968 3 MP stroke -24 101 95 -70 3700 3185 3 MP stroke -79 22 80 19 2651 1519 3 MP stroke -84 91 136 -33 2925 2161 3 MP stroke 39 8 -25 -25 4750 1252 3 MP stroke -78 119 152 -60 2651 3287 3 MP stroke -96 126 172 -57 2140 4069 3 MP stroke 3 19 12 -17 4218 1306 3 MP stroke -98 136 181 -63 2291 3554 3 MP stroke -19 2 17 7 3994 1054 3 MP stroke 39 -16 -43 -6 4760 893 3 MP stroke -30 114 110 -77 3765 4019 3 MP stroke -10 56 50 -40 4022 1711 3 MP stroke 44 44 -2 -56 4734 1613 3 MP stroke -181 292 366 -150 2621 2711 3 MP stroke -35 132 128 -90 3604 4018 3 MP stroke -20 68 68 -45 3716 2324 3 MP stroke -38 -5 26 22 3374 1058 3 MP stroke 11 33 16 -31 4265 1764 3 MP stroke 130 187 37 -211 5246 2795 3 MP stroke -133 31 131 38 2644 1236 3 MP stroke 2 110 81 -89 4509 3692 3 MP stroke -166 70 186 22 1728 1875 3 MP stroke -9 244 191 -192 3813 3748 3 MP stroke -8 -20 -9 20 4174 800 3 MP stroke 58 127 49 -129 5213 3230 3 MP stroke 26 156 96 -138 4370 2850 3 MP stroke -151 205 276 -94 2079 3271 3 MP stroke 10 125 86 -105 4199 2482 3 MP stroke 152 221 44 -249 5332 3027 3 MP stroke 35 102 49 -98 5331 3471 3 MP stroke -203 267 365 -119 1930 3150 3 MP stroke 152 201 29 -233 5461 2975 3 MP stroke -4 99 78 -78 4147 3154 3 MP stroke -130 167 231 -73 1911 3626 3 MP stroke 72 2 -56 -35 5423 1184 3 MP stroke 18 96 58 -86 4669 3133 3 MP stroke -113 103 169 -30 2584 2219 3 MP stroke 20 -29 -38 14 4590 762 3 MP stroke 15 -20 -28 9 4340 817 3 MP stroke 14 169 116 -143 4368 3725 3 MP stroke -125 146 210 -59 2433 2663 3 MP stroke 80 13 -54 -48 5515 1347 3 MP stroke 23 67 32 -65 4476 2084 3 MP stroke -80 140 170 -75 2654 3599 3 MP stroke -48 117 127 -71 3217 3777 3 MP stroke -164 2 134 75 2828 820 3 MP stroke -186 -10 142 95 1681 789 3 MP stroke 29 25 -4 -34 4651 1682 3 MP stroke 21 41 14 -43 4493 1311 3 MP stroke -87 66 119 -13 2168 2698 3 MP stroke -33 122 118 -83 3590 3761 3 MP stroke -126 205 256 -106 2229 3608 3 MP stroke -108 50 124 10 2701 1588 3 MP stroke 50 119 49 -120 4724 2478 3 MP stroke -19 99 89 -71 3777 3021 3 MP stroke -91 75 130 -17 2267 2608 3 MP stroke 5 203 149 -166 4041 3421 3 MP stroke -57 169 173 -110 3376 2757 3 MP stroke 70 30 -34 -57 5609 1725 3 MP stroke 136 142 -2 -178 5547 2591 3 MP stroke gr c9 end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 547 1492 a currentpoint currentpoint translate 1 0.3347 div 1 0.3347 div scale neg exch neg exch translate 547 1492 a 1997 1492 a currentpoint currentpoint translate 0.33821 0.33821 scale neg exch neg exch translate 1997 1492 a @beginspecial 78 @llx 211 @lly 557 @urx 583 @ury 4790 @rwi @setspecial %%BeginDocument: struct1eg.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: D:\Matlab\AAAFaceGUI\Manual\struct1eg.eps %%CreationDate: 05/16/2002 11:33:37 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK %%Pages: 1 %%BoundingBox: 78 211 557 583 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile exch readhexstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 78 211 557 583 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 723 341 5749 4461 MR c np 86 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5187 PR 6 w 0 2335 -2326 -1069 0 -2336 6255 3794 4 MP PP 2326 1070 0 2335 -2326 -1069 0 -2336 6255 3794 5 MP stroke 2326 1070 3030 -822 -2325 -1070 3224 4616 4 MP PP -3031 822 2326 1070 3030 -822 -2325 -1070 3224 4616 5 MP stroke 0 2335 3030 -821 0 -2336 899 3546 4 MP PP -3030 822 0 2335 3030 -821 0 -2336 899 3546 5 MP stroke 4 w DO SO 6 w 0 sg 3224 4616 mt 6255 3794 L 3224 4616 mt 899 3546 L 899 3546 mt 899 1210 L 3224 4616 mt 3295 4648 L %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 3327 4774 mt (-6) s 3729 4479 mt 3800 4511 L 3832 4637 mt (-4) s 4234 4342 mt 4305 4375 L 4337 4500 mt (-2) s 4739 4205 mt 4811 4238 L 4842 4363 mt (0) s 5244 4068 mt 5316 4101 L 5347 4226 mt (2) s 5749 3931 mt 5821 3964 L 5852 4090 mt (4) s 6255 3794 mt 6326 3827 L 6357 3953 mt (6) s 3224 4616 mt 3148 4636 L 2979 4756 mt (-6) s 2836 4437 mt 2761 4458 L 2591 4578 mt (-4) s 2449 4259 mt 2373 4279 L 2204 4399 mt (-2) s 2061 4081 mt 1985 4101 L 1886 4221 mt (0) s 1674 3902 mt 1598 3923 L 1499 4043 mt (2) s 1286 3724 mt 1210 3744 L 1111 3864 mt (4) s 899 3546 mt 823 3566 L 723 3686 mt (6) s 899 3546 mt 827 3513 L 730 3542 mt (0) s 899 3212 mt 827 3179 L 730 3209 mt (1) s 899 2878 mt 827 2845 L 730 2875 mt (2) s 899 2544 mt 827 2512 L 730 2541 mt (3) s 899 2211 mt 827 2178 L 730 2207 mt (4) s 899 1877 mt 827 1844 L 730 1874 mt (5) s 899 1543 mt 827 1510 L 730 1540 mt (6) s 899 1210 mt 827 1177 L 730 1206 mt (7) s gs 899 389 5357 4228 MR c np gr 0 j 0 -58 -58 0 0 58 58 0 3374 2590 5 MP /c8 { 1.000000 0.000000 0.000000 sr} bdef c8 DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2777 2600 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2843 2507 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3071 2685 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3164 2572 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3877 1901 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2488 1986 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2633 1701 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3102 1598 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3301 3188 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3706 2344 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3479 2693 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3449 2418 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3807 2650 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4288 2243 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4563 941 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4552 2027 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3940 2646 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4089 1447 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4703 1806 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3817 2194 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4028 1667 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4069 2848 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3994 2717 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3866 2846 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3454 2424 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3522 2680 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3168 2282 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3204 2835 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3598 2874 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3715 1715 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3308 3017 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3506 2793 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3858 2649 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4814 2192 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4247 2891 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4506 2569 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3294 2866 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3413 2639 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3400 2103 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3465 2982 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4007 2581 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4765 2079 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4346 1986 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3829 2154 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3874 2270 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4720 1808 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3148 3055 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4129 2283 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3757 2270 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3588 2583 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3106 2562 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2921 2931 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3265 3178 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3363 2233 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 5054 1884 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3574 1787 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3561 2038 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3857 1805 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3573 1601 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3373 1984 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4506 2338 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3995 2125 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3788 1312 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3803 1155 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4071 2973 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3186 2316 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3353 1150 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3551 2201 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4063 2104 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3936 937 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3519 2868 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4131 2895 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3118 2921 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3541 2941 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3467 3142 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 5574 1578 5 MP DP gs 899 389 5357 4228 MR c np gr 0 sg 5603 1649 mt ( 70) s 5083 1955 mt ( 61) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3949 2431 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4184 2963 5 MP DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4979 2326 5 MP DP gs 899 389 5357 4228 MR c np gr 0 sg 5008 2397 mt ( 3) s 4843 2263 mt ( 65) s 4794 2150 mt ( 51) s 4749 1879 mt ( 76) s 4732 1877 mt ( 8) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4646 2125 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 4675 2196 mt ( 19) s 4592 1012 mt ( 99) s 4581 2098 mt ( 24) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4543 1065 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 4572 1136 mt ( 40) s 4535 2409 mt ( 55) s 4535 2640 mt ( 80) s 4375 2057 mt ( 73) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4343 2267 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 4372 2338 mt ( 91) s 4317 2314 mt ( 20) s 4276 2962 mt ( 72) s 4272 2780 mt (100) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 4243 2709 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 4213 3034 mt ( 69) s 4160 2966 mt ( 59) s 4158 2354 mt ( 85) s 4118 1518 mt ( 82) s 4100 3044 mt ( 63) s 4098 2919 mt ( 97) s 4092 2175 mt ( 41) s 4057 1738 mt ( 11) s 4036 2652 mt ( 94) s 4024 2196 mt ( 95) s 4023 2788 mt ( 66) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3985 1139 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 4014 1210 mt ( 75) s 3978 2502 mt ( 38) s 3969 2717 mt ( 39) s 3965 1008 mt ( 26) s 3906 1972 mt ( 6) s 3903 2341 mt ( 17) s 3895 2917 mt ( 22) s 3887 2720 mt ( 21) s 3886 1876 mt ( 35) s 3858 2225 mt ( 83) s 3846 2265 mt ( 25) s 3836 2721 mt ( 68) s 3832 1226 mt ( 88) s 3817 1383 mt ( 9) s 3786 2341 mt ( 32) s 3744 1786 mt ( 18) s 3735 2415 mt ( 53) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3636 2609 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 3665 2680 mt ( 84) s 3627 2945 mt ( 98) s 3617 2654 mt ( 46) s 3603 1858 mt ( 44) s 3602 1672 mt ( 58) s 3590 2109 mt ( 56) s 3580 2272 mt ( 10) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3548 3641 5 MP /c9 { 0.000000 1.000000 0.000000 sr} bdef c9 DP gs 899 389 5357 4228 MR c np gr 0 sg 3570 3012 mt ( 29) s 3551 2751 mt ( 4) s 3548 2939 mt ( 89) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3514 3327 5 MP c9 DP gs 899 389 5357 4228 MR c np gr 0 sg 3535 2864 mt ( 2) s 3508 2764 mt ( 16) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3469 2483 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 3498 2554 mt ( 45) s 3496 3213 mt ( 34) s 3494 3053 mt ( 92) s 3483 2495 mt ( 52) s 3478 2489 mt ( 23) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3427 3250 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 3456 3321 mt ( 54) s 3442 2710 mt ( 48) s 3429 2174 mt ( 36) s 3403 2661 mt ( 78) s 3402 2055 mt ( 31) s 3392 2304 mt ( 47) s 3382 1221 mt ( 50) s 3337 3088 mt ( 67) s 3330 3259 mt ( 71) s 3323 2937 mt ( 74) s 3294 3249 mt ( 12) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 3237 3052 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 3266 3123 mt ( 33) s 3233 2906 mt ( 81) s 3215 2387 mt ( 49) s 3197 2353 mt ( 64) s 3193 2643 mt ( 43) s 3177 3126 mt ( 7) s 3147 2992 mt ( 5) s 3135 2633 mt ( 30) s 3131 1669 mt ( 79) s 3100 2756 mt ( 77) s 2950 3002 mt ( 37) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2875 2782 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2904 2853 mt ( 86) s 2872 2578 mt ( 93) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2825 2238 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2854 2309 mt ( 96) s 2806 2671 mt ( 60) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2711 2073 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2740 2144 mt ( 90) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2689 2615 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2718 2686 mt ( 14) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2664 2603 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2693 2674 mt ( 15) s 2662 1772 mt ( 57) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2490 2604 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2519 2675 mt ( 62) s 2517 2057 mt ( 42) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2000 2394 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2430 2386 5 MP DP gs 899 389 5357 4228 MR c np gr 0 sg 2459 2457 mt ( 27) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2414 2259 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2443 2330 mt ( 13) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 2409 1329 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 2438 1400 mt ( 1) s 2029 2465 mt ( 87) s gs 899 389 5357 4228 MR c np gr 0 -58 -58 0 0 58 58 0 1410 1778 5 MP c8 DP gs 899 389 5357 4228 MR c np gr 0 sg 1439 1849 mt ( 28) s end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 1997 1492 a currentpoint currentpoint translate 1 0.33821 div 1 0.33821 div scale neg exch neg exch translate 1997 1492 a 1176 1591 a FQ(\(a\))1354 b(\(b\))515 1769 y(Figure)16 b(7.1:)24 b Ft(\(a\))16 b(A)i(set)g(of)f(synthetic)g(matc)o(hes)g(g)o(ener)o(ated)f(by)h FO(torr)p 2527 1769 25 4 v 29 w(gen)p 2706 1769 V 30 w(2view)p 2986 1769 V 29 w(matches)p Ft(,)515 1869 y(displayed)51 b(by)i FO(torr)p 1223 1869 V 29 w(display)p 1602 1869 V 29 w(matches)p Ft(.)122 b(\(b\))52 b(the)h(3D)g(structur)m(e)g (displayed)f(by)515 1968 y FO(torr)p 720 1968 V 29 w(display)p 1099 1968 V 28 w(structure)515 2322 y FS(\045now)44 b(use)g(P)h (matrices)e(and)i(corrected)e(matches)h(to)g(get)g(structure:)515 2413 y(X)g(=)h(torr_triangulate\(corrected_matches,)39 b(m3,)44 b(P1,)h(P2\);)515 2596 y(torr_display_structure\(X,)40 b(P1,)45 b(P2,)f(1\);)515 2870 y(\045test)515 2961 y(XX)g(=)h ([X\(1,:\))f(./)g(X\(4,:\))g(;)h(X\(2,:\))88 b(./)45 b(X\(4,:\))f(;)g(X\(3,:\))89 b(./)44 b(X\(4,:\))g(];)515 3144 y(disp\('ratio)f(of)h(estimated)g(and)g(true)g(X'\);)515 3235 y(XX)g(./true_X)515 3600 y(show_result)f(=)h(1;)515 3691 y(if)g(show_result)694 3783 y(disp\('look)f(at)i(reprojection)e (error)h(to)g(groundtruth)f(points'\))694 3874 y(x1_rp)h(=)h(P1)f(*)h (X;)694 3965 y(x1_rp\(1,:\))e(=)i(x1_rp\(1,:\))e(./)i(x1_rp\(3,:\))e(*) h(m3;)694 4057 y(x1_rp\(2,:\))f(=)i(x1_rp\(2,:\))e(./)i(x1_rp\(3,:\))e (*)h(m3;)694 4148 y(\(x1)g(-)h(x1_rp\(1,:\)'\)')694 4239 y(\(y1)f(-)h(x1_rp\(2,:\)'\)')694 4422 y(x2_rp)f(=)h(P2)f(*)h(X;)694 4513 y(x2_rp\(1,:\))e(=)i(x2_rp\(1,:\))e(./)i(x2_rp\(3,:\))e(*)h(m3;) 694 4605 y(x2_rp\(2,:\))f(=)i(x2_rp\(2,:\))e(./)i(x2_rp\(3,:\))e(*)h (m3;)694 4696 y(\(x2)g(-)h(x2_rp\(1,:\)'\)')694 4787 y(\(y2)f(-)h(x2_rp\(2,:\)'\)')515 4879 y(end)p eop %%Page: 61 63 61 62 bop 515 1146 a FE(Chapter)44 b(8)515 1561 y FN(Generating)51 b(Synthetic)g(Data)515 2017 y FQ(Data)24 b(is)i(randomly)c(generated)g (in)j FA(R)1627 1981 y FG(3)1689 2017 y FQ(so)g(that)f(the)g(imaged)f (points)h(lie)h(in)g(the)f(range)f(bounded)515 2117 y(by)h(the)h(the)g (lines)g FH(y)35 b FI(=)c FA(\000)p FI(256)p FQ(,)24 b FH(y)35 b FI(=)c(256)p FQ(,)25 b FH(x)32 b FI(=)g FA(\000)p FI(256)p FQ(,)24 b FH(x)32 b FI(=)f(256)24 b FQ(pix)o(els)h(for)f(tw)o (o)h(imaginary)515 2216 y(cameras.)35 b(The)23 b(intrinsic)g (parameters)g(of)g(the)h(synthetic)f(cameras)g(can)h(be)f(represented)f (by)i(the)515 2316 y(matrix)1445 2495 y FL(C)f FI(=)1624 2328 y FC(2)1624 2478 y(4)1721 2395 y FI(1)p FH(:)p FI(00)82 b(0)p FH(:)p FI(00)127 b(0)1721 2494 y(0)p FH(:)p FI(00)82 b(1)p FH(:)p FI(00)127 b(0)1721 2594 y(0)p FH(:)p FI(00)82 b(0)p FH(:)p FI(00)g(1)p FH(=f)2357 2328 y FC(3)2357 2478 y(5)2426 2495 y FH(;)771 b FQ(\(8.1\))515 2733 y(corresponding)22 b(to)j(an)g(aspect)g(ratio)g(of)g FI(1)p FQ(,)i(with)e(an)g(optic)g (centre)g(at)h(the)f(image)g(centre,)g(and)g(a)515 2833 y(focal)19 b(length)g(of)g FH(f)32 b FI(=)19 b FO(foc)p FQ(.)51 b(The)19 b(camera)g(motion)f(is)j(a)f(random)d(translation)i (and)g(rotation,)g(the)515 2933 y(translation)f(is)j(a)e(multiple)g(of) g(the)h(focal)f(length.)24 b(The)19 b(rotation)f(is)i(generated)e (using)h(Rodrigues')515 3032 y(formula)1249 3132 y FL(R)j FI(=)h(cos)13 b FH(!)s FL(I)18 b FI(+)g(sin)c FH(!)s FI([)p FL(l)o FI(])1991 3144 y FF(\002)2066 3132 y FI(+)k(\(1)g FA(\000)g FI(cos)13 b FH(!)s FI(\))p FL(ll)2589 3095 y FF(>)3220 3132 y FQ(\(8.2\))515 3276 y(where)19 b FL(l)h FQ(is)h(the)f(axis)g(of)f(rotation)g(\(a)h(random)e(unit)i(v)o (ector\),)e(and)h FH(!)k FQ(the)d(angle)f(of)h(rotation)f(about)515 3376 y(this)h(axis.)639 3475 y(The)j(synthetic)g(image)f(positions)h (are)g(perturbed)e(by)i(Gaussian)g(noise)g(standard)f(de)n(viation)515 3575 y(by)28 b(def)o(ault)g FI(1)p FH(:)p FI(0)g FQ(pix)o(els)g(and)g (then)g(quantized)e(to)j(the)g(nearest)f(pix)o(el,)h(simulating)f (quite)g(noisy)515 3675 y(image)19 b(conditions.)515 3911 y Fs(8.0.3)99 b(Synthetic)26 b(T)-7 b(w)o(o)25 b(view)f(match)i (function)515 4067 y FO(function)48 b ([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2,)o(true_)o(C,)515 4166 y(true_R,true_t,)f(true_E])h(=)i(...)565 4266 y (torr_gen_2view_matches\(foc,)44 b(no_matches,)k(noise_sigma,)515 4365 y(translation_mult,)e(translation_adder,)h(...)565 4465 y(rotation_multplier,)f(min_Z,Z_RAN,m3\))515 4684 y FM(Input:)619 4844 y FQ(1.)40 b FO(foc)p FQ(:)26 b(focal)19 b(length.)619 5006 y(2.)40 b FO(no)p 827 5006 25 4 v 30 w(matches)p FQ(:)24 b FH(n)d FQ(the)f(number)e(of)i(matches.)1905 5255 y(61)p eop %%Page: 62 64 62 63 bop 515 232 a FJ(CHAPTER)21 b(8.)45 b(GENERA)-9 b(TING)20 b(SYNTHETIC)g(D)m(A)-9 b(T)h(A)1034 b FQ(62)619 523 y(3.)40 b FO(noise)p 977 523 25 4 v 29 w(sigma)p FQ(:)32 b FH(\033)27 b FQ(of)d(noise)f FH(e)p FI(\()p FH(\033)s FI(\))i FQ(added)d(to)i(generated)e(matches)h(such)g(that)h FO(nx1)49 b(=)722 623 y(x1)p FQ(:)26 b FI(+)p FH(e)p FI(\()p FH(\033)s FI(\))21 b FQ(etc.)619 789 y(4.)40 b FO(translation)p 1277 789 V 28 w(adder,)49 b(translation)p 2204 789 V 28 w(mult)p FQ(:)g(the)32 b(translation)f(v)n(aries)h(uni-) 722 888 y(formly)14 b(between)g FO(translation)p 1811 888 V 28 w(adder)h FQ(and)g FO(translation)p 2790 888 V 28 w(adder)48 b(+)i(trans-)722 988 y(lation)p 1027 988 V 29 w(mult)p FQ(:.)619 1154 y(5.)40 b FO(rotation)p 1127 1154 V 29 w(multplier)p FQ(:)i(the)30 b(Euler)e(angles)h(of)h(the) f(rotation)f(go)h(between)g(0)g(and)722 1254 y FO(rotation)p 1127 1254 V 29 w(multplier)p FQ(:.)619 1420 y(6.)40 b FO(min)p 877 1420 V 30 w(Z,)49 b(Z)p 1106 1420 V 30 w(RAN)p FQ(:)15 b(the)g(depth)g(v)n(aries)f(uniformly)f(between)i FO(min)p 2644 1420 V 29 w(Z)h FQ(and)30 b(min)p 3020 1420 V 29 w(Z)16 b(+)f(Z)p 3229 1420 V 30 w(RAN:.)619 1586 y(7.)40 b FO(m3)21 b FQ(third)e(homogeneous)e(coordinate.)515 1807 y FM(Output:)619 1973 y FQ(1.)40 b FO(true)p 927 1973 V 29 w(F)p FQ(:)21 b(true)f FL(F)p FQ(.)619 2139 y(2.)40 b FO(x1,y1,x2,y2)19 b FQ(true)h FH(x;)14 b(y)s(;)g(x)1653 2109 y FF(0)1677 2139 y FH(;)g(y)1758 2109 y FF(0)1780 2139 y FQ(.)619 2305 y(3.)40 b FO(nx1,ny1,nx2,ny2)p FQ(:)23 b(noisy)d FH(x;)14 b(y)s(;)g(x)1931 2275 y FF(0)1955 2305 y FH(;)g(y)2036 2275 y FF(0)2059 2305 y FQ(.)619 2471 y(4.)40 b FO(true)p 927 2471 V 29 w(C,)50 b(true)p 1306 2471 V 29 w(R,true)p 1635 2471 V 29 w(t,)f(true)p 2013 2471 V 29 w(E)p FQ(:)21 b(true)f FL(C)p FQ(,)g FL(R)p FQ(,)g FL(t)h FQ(and)f FL(E)p FQ(.)515 2709 y Fs(8.0.4)99 b(A)24 b(Script)i(to)f(generate)h(and)g(display)e(synthetic)i(matches) 515 2864 y FQ(The)15 b(script)g FO(torr)p 1065 2864 V 30 w(test)p 1295 2864 V 29 w(mat)g FQ(generates)g(some)g(synthetic)g (matches)g(and)g(then)g(displays)g(them.)515 3039 y FS ([true_F,x1,y1,x2,y2,nx1,ny1,nx2,ny2,tr)o(ue_C,t)o(rue_R)o(,true_)o(t,) 39 b(true_E])560 3130 y(=)44 b(torr_gen_2view_matches;)515 3221 y(no_matches)f(=)i(length\(nx1\);)515 3313 y(matches)e(=)i ([nx1,ny1,nx2,ny2];)515 3495 y(\045displayes)e(matches)515 3587 y(torr_display_matches\(matches\))p eop %%Page: 63 65 63 64 bop 515 1146 a FE(Chapter)44 b(9)515 1561 y FN(The)51 b Fa(Torr)p 1419 1561 62 4 v 74 w(tool)g FN(GUI)515 1993 y FQ(The)27 b(G)t(U)t(I)h(for)d(the)i(S)t(A)t(M)i(library)24 b(is)j(the)e FO(torr)p 1986 1993 25 4 v 29 w(tool)p FQ(.)41 b(At)26 b(present)f(it)h(is)h(not)e(fully)g(\002nished)515 2092 y(and)d(so)i(the)f(documentation)e(and)h(help)h(in)h(this)f (chapter)f(is)j(not)e(yet)g(complete.)33 b(The)25 b(G)t(U)t(I)h(puts) 515 2192 y(together)h(all)i(the)g(functions)e(that)h(ha)n(v)o(e)g(been) g(described)f(so)i(f)o(ar)g(into)f(an)g(application,)h(which)515 2291 y(can)d(display)f(the)h(images)g(and)g(results.)43 b(A)27 b(rough)d(and)i(ready)f(help)h(can)g(be)g(obtained)e(by)i(typ-) 515 2391 y(ing)i FO(guide)49 b(torr)p 1154 2391 V 29 w(tool)29 b FQ(and)f(using)g(the)g(property)f(inspector)g(to)i(e)o (xamine)e(the)i(call)g(back)515 2491 y(routines)19 b(of)h(the)g(b)n (uttons.)639 2590 y(Some)g(things)g(to)g(note:)619 2725 y(1.)40 b(The)20 b(image)g(coordinate)e(system)j(is)g(chosen)e(with)h (origin)g(at)g(the)g(centre)g(of)g(the)g(image)g(by)722 2825 y(setting)g(the)h FO(XData)e FQ(and)h FO(YData)g FQ(properties)f FO(image)p FQ(.)619 2975 y(2.)40 b(The)20 b(demo)f(images)h(supplied)f(are)i FO(j1.bmp,)48 b(j2.bmp)p FQ(.)619 3126 y(3.)40 b FO(m3)21 b FQ(the)f(third)f(homogeneous)e (coordinate)i(is)i(256.)515 3399 y FD(9.1)119 b(Example)515 3584 y FQ(A)20 b(quick)g(tour)619 3719 y(1.)40 b(click)20 b(on)g(load)g(demo)f(images)h(this)h(loads)f FO(j1.bmp)f FQ(and)h FO(j2.bmp)619 3870 y FQ(2.)40 b(click)22 b(on)f(load)g (corners,)g(and)g(load)g FO(tab.cor)g FQ(this)h(displays)g(Harris)f (corners)g(\(detected)722 3969 y(by)f(pressing)g(Detect)g(Corners\).)k (Observ)o(e)19 b(the)h(corners)f(displayed)g(by)h(little)h(crosses.)619 4120 y(3.)40 b(click)24 b(on)g(load)f(matches,)h(and)f(load)h FO(table.matches)p FQ(,)e(this)j(displays)e(a)i(whole)e(load)722 4220 y(of)d(correlation)f(matches)g(\(detected)g(by)h(correlation)f (matching\))619 4370 y(4.)40 b(click)29 b(on)f(load)g(mat)g(and)g(F)-7 b(,)29 b(load)f FO(table.fmatches)f FQ(this)i(loads)f(a)h(whole)f(load) g(of)722 4470 y(inlying)19 b(matches)h(to)g(a)h(precalculated)d FL(F)j FQ(\(via)f(the)g(MAPSA)m(C)h(b)n(utton\))619 4620 y(5.)40 b(click)17 b(on)f(epipolar)f(geometry)-5 b(,)15 b(then)h(click)h(in)g(the)f(image)g(and)g(depress)g(return,)g(this)i (sho)n(ws)722 4720 y(the)i(point)g(click)o(ed)g(on)f(and)h(its)h (corresponding)c(epipolar)i(line.)619 4870 y(6.)40 b(click)25 b(on)g(SFM)h(to)f(perform)e(self)i(calibration)f(\(here)g(just)i(for)e (focal)h(length\))e(and)i(create)722 4970 y(3D)20 b(structure)g(for)f (the)i(inliers.)1905 5255 y(63)p eop %%Page: 64 66 64 65 bop 515 232 a FJ(CHAPTER)21 b(9.)45 b(THE)21 b FO(TORR)p 1410 232 25 4 v 29 w(TOOL)f FJ(GUI)1494 b FQ(64)447 3791 y currentpoint currentpoint translate 0.7438 0.7438 scale neg exch neg exch translate 447 3791 a @beginspecial 67 @llx 212 @lly 551 @urx 589 @ury 4840 @rwi @setspecial %%BeginDocument: mateg1.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: D:\Matlab\AAAFaceGUI\Manual\mateg1.eps %%CreationDate: 05/16/2002 11:35:55 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK %%Pages: 1 %%BoundingBox: 67 212 551 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap 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{pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad 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mt 2723 2521 L 2687 2485 mt 2687 2557 L 4391 1136 mt 4463 1136 L 4427 1100 mt 4427 1172 L 2976 3483 mt 3048 3483 L 3012 3447 mt 3012 3519 L 2339 2855 mt 2411 2855 L 2375 2819 mt 2375 2891 L 3594 2008 mt 3666 2008 L 3630 1972 mt 3630 2044 L 5114 1315 mt 5186 1315 L 5150 1279 mt 5150 1351 L 3204 2625 mt 3276 2625 L 3240 2589 mt 3240 2661 L 2244 1955 mt 2316 1955 L 2280 1919 mt 2280 1991 L 3002 3793 mt 3074 3793 L 3038 3757 mt 3038 3829 L 3983 2948 mt 4055 2948 L 4019 2912 mt 4019 2984 L 5301 1909 mt 5373 1909 L 5337 1873 mt 5337 1945 L 3759 3115 mt 3831 3115 L 3795 3079 mt 3795 3151 L 2695 1538 mt 2767 1538 L 2731 1502 mt 2731 1574 L 3025 2128 mt 3097 2128 L 3061 2092 mt 3061 2164 L 4689 1227 mt 4761 1227 L 4725 1191 mt 4725 1263 L 2767 3227 mt 2839 3227 L 2803 3191 mt 2803 3263 L 2276 4012 mt 2348 4012 L 2312 3976 mt 2312 4048 L 4194 1289 mt 4266 1289 L 4230 1253 mt 4230 1325 L 2436 3491 mt 2508 3491 L 2472 3455 mt 2472 3527 L 3975 1061 mt 4047 1061 L 4011 1025 mt 4011 1097 L 4681 887 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FB(i)1315 2906 y FA(g)p FH(;)14 b(i)35 b FI(=)h(1)p FH(;)14 b(:)g(:)g(:)f(N)t(;)28 b Ft(be)f(tr)o(ansformed)f(to)h(the)g(set)h FA(f)p FL(x)2841 2876 y FF(0)2841 2927 y FB(i)2869 2906 y FA(g)f Ft(on)g(the)g(ima)o(g)o (e)515 3005 y(plane)f(by)i(the)g(motion)f(par)o(ameter)o(s)g FA(f)p FL(R)o FH(;)14 b FL(t)p FA(g)28 b Ft(suc)o(h)f(that)g FL(t)37 b FA(6)p FI(=)g FL(0)p Ft(.)48 b(Then)27 b(ther)m(e)h(e)n (xists)h(a)e FI(3)d FA(\002)f FI(3)515 3105 y Ft(essential)d(matrix)h FL(E)h FI(=)h FL(R)p FI([)p FL(t)p FI(])1392 3117 y FF(\002)1469 3105 y Ft(suc)o(h)c(that)1736 3299 y FL(x)1786 3240 y Fn(0)1809 3265 y FF(>)1786 3320 y FB(i)1865 3299 y FL(Ex)1978 3311 y FB(i)2029 3299 y FI(=)j(0)1043 b FQ(\(A.1\))515 3482 y Ft(for)20 b(all)h FH(i)p Ft(.)515 3648 y FQ(It)j(can)g(also)h (be)f(sho)n(wn)g(that)g FA(k)p FL(E)o FA(k)30 b FI(=)1653 3580 y FA(p)p 1722 3580 42 4 v 68 x FI(2)24 b FQ(and)g(that)g(the)g (singular)g(v)n(alues)g(of)g FL(E)g FQ(are)g FI(1)p FQ(,)p FI(1)g FQ(and)g FI(0)p FQ(,)515 3748 y(and)c(so)h FA(j)p FL(E)p FA(j)j FI(=)g(0)p FQ(.)j(Gi)n(v)o(en)20 b(a)i(perfect)e(camera)g (one)g(can)h(reco)o(v)o(er)e(the)i(motion)e(parameters)h(purely)515 3847 y(from)k(point)h(correspondences)e(without)i(kno)n(wledge)e(of)i (the)h(scene)g(structure,)f(gi)n(v)o(en)g(that)h(the)515 3947 y(points)19 b(do)h(not)g(lie)h(in)f(special)h(con\002guration.)639 4047 y(The)e(analysis)h(for)f(the)g(calibrated)g(case)g(has)h(been)f(e) o(xtended)e(to)j(that)g(where)e(the)i(calibration)515 4146 y(is)26 b(unkno)n(wn.)37 b(It)26 b(is)g(a)f(remarkable)e(f)o(act)j (that)f(if)g(neither)g(the)g(internal)f(camera)h(calibration)e(nor)515 4246 y(the)28 b(camera)f(motion)g(are)i(kno)n(wn)d(then)i(there)g(is)h (still)g(a)g(set)g(of)f(linear)g(equations)f(linking)g(the)515 4346 y(points)e(in)g(the)g(tw)o(o)h(images.)40 b(The)25 b(case)h(for)e(algorithms)g(that)i(do)e(not)h(require)f(calibration)g (has)515 4445 y(been)19 b(strongly)f(made)h(in)h([12)o(].)25 b(Camera)19 b(calibration)g(is)i(at)f(best)g(dif)n(\002cult)f(possibly) g(introducing)515 4545 y(correlated)h(errors)h(into)g(the)h(system,)f (and)h(at)g(w)o(orst)f(it)i(is)f(often)f(impossible.)29 b(The)21 b(fundamental)515 4644 y(matrix)e(constraint)g(is)j(no)n(w)d (stated)i(and)e(pro)o(v)o(ed:)515 4827 y FM(Theor)o(em)h(5)g(\(F)n (augeras)f([12)n(],)i(Hartley)d([19)o(]\))41 b Ft(Given)29 b(an)f(uncalibr)o(ated)f(camer)o(a)h(let)i(the)f(set)515 4927 y(of)e(homo)o(g)o(eneous)e(ima)o(g)o(e)i(points)g FA(f)p FL(x)1642 4939 y FB(i)1670 4927 y FA(g)p FH(;)14 b(i)35 b FI(=)h(1)p FH(;)14 b(:)g(:)g(:)g(N)t(;)28 b Ft(be)f(tr)o(ansformed)g(to)g(the)h(set)g FA(f)p FL(x)3199 4897 y FF(0)3199 4948 y FB(i)3226 4927 y FA(g)g Ft(on)1905 5255 y FQ(67)p eop %%Page: 68 70 68 69 bop 515 232 a FJ(APPENDIX)20 b(A.)41 b(DERIV)-11 b(A)i(TION)19 b(OF)i(THE)f(FUND)m(AMENT)-8 b(AL)20 b(MA)-9 b(TRIX)446 b FQ(68)515 523 y Ft(the)21 b(ima)o(g)o(e)f(plane)g(by)h (the)g(motion)f(par)o(ameter)o(s)g FA(f)p FL(R)o FH(;)14 b FL(t)p FA(g)21 b Ft(suc)o(h)f(that)h FL(t)j FA(6)p FI(=)g FL(0)p Ft(.)j(Then)20 b(ther)m(e)h(e)n(xists)h(a)515 623 y FI(3)c FA(\002)g FI(3)i Ft(matrix)h FL(F)f Ft(suc)o(h)g(that)1737 722 y FL(x)1788 663 y Fn(0)1810 688 y FF(>)1788 743 y FB(i)1866 722 y FL(Fx)1977 734 y FB(i)2027 722 y FI(=)j(0)1044 b FQ(\(A.2\))515 855 y Ft(for)20 b(all)h FH(i)p Ft(.)515 1009 y(Pr)l(oof)p FQ(.)k(Let)20 b(a)g(real)g(w)o(orld)g(point)f FL(x)k FI(=)g(\()p FH(x;)14 b(y)s(;)g(z)t FI(\))23 b FA(2)g(R)2089 973 y FG(3)2147 1009 y FQ(and)d(let)g FL(x)h FQ(and)f FL(x)2653 979 y FF(0)2697 1009 y FQ(be)g(the)g(homogeneous)515 1109 y(coordinates)26 b(of)h(the)h(image)f(points)g(with)h(the)f (camera)g(at)h FL(0)g FQ(and)f FL(t)h FQ(respecti)n(v)o(ely)-5 b(.)45 b(If)28 b FL(C)g FQ(gi)n(v)o(es)515 1209 y(the)e(camera')-5 b(s)26 b(intrinsic)g(parameters)f(then)h(the)h(camera)e (transformations)f(are)j(gi)n(v)o(en)e(by)h FL(P)34 b FI(=)515 1308 y([)p FL(C)24 b FA(j)h FL(0)p FI(])c FQ(and)g FL(P)978 1278 y FF(0)1026 1308 y FI(=)j([)p FL(CR)h FA(j)f FL(Ct)p FI(])e FQ(before)e(and)g(after)h(the)g(motion.)26 b(As)c(before)e(we)h(determine)f(the)515 1408 y(epipolar)f(line)i(of)f FL(x)i FQ(in)e(the)h(second)f(image)g(by)g(looking)f(at)i(the)f(image)g (under)g FL(P)2882 1378 y FF(0)2926 1408 y FQ(of)h(the)f(camera)515 1508 y(centre)j FI(\(0)p FH(;)14 b FI(0)p FH(;)g FI(0)p FH(;)g FI(1\))22 b FQ(and)h(the)h(point)e(at)j(in\002nity)e FI(\()p FL(C)2021 1471 y FF(\000)p FG(1)2110 1508 y FL(x)p FH(;)14 b FI(0\))p FQ(.)36 b(The)23 b(images)g(of)g(these)h(tw)o(o)g (points)515 1607 y(under)19 b FL(P)790 1577 y FF(0)834 1607 y FQ(are)h FL(Ct)h FQ(and)e FL(CRC)1433 1571 y FF(\000)p FG(1)1522 1607 y FL(x)i FQ(respecti)n(v)o(ely)-5 b(.)23 b(Thus)d(the)g(epipolar)f(line)h FL(n)h FQ(is)g(gi)n(v)o(en)e(by)1587 1762 y FL(n)k FI(=)f FL(Ct)d FA(\002)f FL(CRC)2168 1725 y FF(\000)p FG(1)2257 1762 y FL(x)894 b FQ(\(A.3\))515 1916 y(Since)20 b FL(x)770 1886 y FF(0)814 1916 y FQ(lies)h(on)f(this)h (epipolar)1751 2025 y FL(x)1801 1966 y Fn(0)1824 1991 y FF(>)1880 2025 y FL(Fx)i FI(=)g(0)1058 b FQ(\(A.4\))515 2158 y(where)19 b(we)i(term)f FL(F)h FQ(the)f(fundamental)e(matrix)h (such)h(that)g FL(F)j FI(=)g FL(Ct)c FA(\002)f FL(CR)o(C)2811 2121 y FF(\000)p FG(1)2900 2158 y FQ(.)639 2257 y(If)i FL(a)h FQ(and)f FL(b)h FQ(are)f FI(3)p FQ(-dimensional)e(column)h(v)o (ectors)g(and)h FL(C)g FQ(is)h(a)g FI(3)d FA(\002)g FI(3)i FQ(matrix,)g(then)1536 2412 y FL(Ca)f FA(\002)f FL(Cb)23 b FA(\030)f FL(C)2054 2375 y FF(\003)2092 2412 y FI(\()p FL(a)e FA(\002)e FL(b)p FI(\))843 b FQ(\(A.5\))515 2566 y(where)24 b FL(C)812 2530 y FF(\003)875 2566 y FQ(is)h(the)g(adjoint)f (of)g FL(C)p FQ(.)38 b(This)25 b(f)o(act)g(allo)n(ws)g(us)f(to)h(write) g(the)f(fundamental)f(matrix)g(in)515 2666 y(terms)d(of)g(the)g (intrinsic)g(matrix)g(and)f(the)h(essential)h(matrix:)1663 2821 y FL(F)i FI(=)f FL(C)1902 2784 y FF(\000>)2010 2821 y FL(EC)2142 2784 y FF(\000)p FG(1)3201 2821 y FQ(\(A.6\))515 2975 y(thus)e(the)h(fundamental)e(matrix)h(at)h(most)g(of)f(rank)g(tw)o (o.)27 b(In)21 b(order)e(to)i(proceed)e(only)h(from)g(image)515 3075 y(measurements)d FL(F)i FQ(is)h(the)e(k)o(e)o(y)g(concept,)g(as)h (it)g(encapsulates)f(all)h(the)g(geometric)e(information)f(on)515 3174 y(camera)k(and)h(motion)f(contained)g(within)h(a)g(set)i(of)e (point)f(correspondences.)25 b(The)c(fundamental)515 3274 y(matrix)g(is)i(determined)d(by)i(and)f(determines)g(the)h (epipolar)e(transformation.)28 b(Theorem)20 b(5)i(leads)515 3374 y(to)e(the)g(follo)n(wing)f(lemma.)515 3528 y FM(Lemma)i(2)41 b Ft(If)25 b FL(F)h Ft(is)g(a)f(fundamental)e(matrix)i(corr)m (esponding)e(to)j(a)f(pair)g(of)g(ima)o(g)o(e)g(and)f FL(x)i Ft(is)g(a)515 3628 y(point)17 b(in)i(the)f(\002r)o(st)h(ima)o(g) o(e)o(,)g(then)e FL(Fx)j Ft(is)f(the)f(epipolar)f(line)i(in)f(the)g (second)g(ima)o(g)o(e)g(corr)m(esponding)515 3728 y(to)i FL(x)p Ft(.)639 3882 y FQ(From)k FL(F)g FQ(and)g(the)g(image)f (correspondences)e(it)k(is)g(straightforw)o(ard)d(to)i(reco)o(v)o(er)e Ft(pr)l(ojective)515 3982 y FQ(structure)d(as)i(has)f(been)g(pointed)f (out)g(by)h(F)o(augeras)f([12)o(])i(and)e(Hartle)o(y)h([19)n(]:)515 4136 y FM(Theor)o(em)g(6)g(\(F)n(augeras)f([12)n(],Hartley)g([19)n(]\)) 41 b Ft(Given)17 b(a)g(set)h(of)f(ima)o(g)o(e)f(corr)m(espondences)f (suf)o(\002-)515 4236 y(cient)k(to)g(determine)g(the)g(fundamental)e (matrix,)j(the)f(corr)m(esponding)e(world)i(space)g(coor)m(dinates)515 4336 y(ar)m(e)h(determined)f(up)h(to)g(a)h(collineation)d(of)j(pr)l (ojective)f FI(3)p Ft(-space)e FA(P)2484 4299 y FG(3)2521 4336 y Ft(.)639 4490 y(Pr)l(oof)p FQ(.)46 b(W)-7 b(e)28 b(shall)f(not)g(present)f(a)i(formal)d(proof,)i(b)n(ut)g(note)g(that)g (if)g(a)h(set)f(of)g(points)g FL(x)3251 4502 y FB(i)3314 4490 y FA(\032)515 4590 y(P)579 4554 y FG(3)643 4590 y FQ(are)f(visible)f(to)h(a)g(pair)g(of)f(camera)g(with)h(transform)e (matrices)i FL(P)g FQ(and)f FL(P)2822 4560 y FF(0)2846 4590 y FQ(,)i(and)f(if)g FL(G)g FQ(is)h(an)515 4689 y(arbitrary)18 b(non-singular)f FI(4)f FA(\002)g FI(4)k FQ(matrix,)f(then)g(replacing) f FL(x)2263 4701 y FB(i)2311 4689 y FQ(by)h FL(G)2489 4653 y FF(\000)p FG(1)2578 4689 y FL(x)2629 4701 y FB(i)2656 4689 y FQ(,)h FL(P)h FQ(by)e FL(P)q(G)h FQ(and)f FL(P)3252 4659 y FF(0)3296 4689 y FQ(by)515 4789 y FL(P)580 4759 y FF(0)603 4789 y FL(G)j FQ(preserv)o(es)e(the)i(object-point)d(to)j (image-space)d(correspondences.)26 b(As)c(may)f(be)g(seen,)h(the)515 4889 y(internal)d(parameters)g(of)h(one)g(of)g(the)g(cameras)g(may)g (be)g(changed)e(arbitrarily)-5 b(.)24 b(Thus)c(points)f(that)515 4988 y(are)h(consistent)g(with)g(Equation)f(\(A.4\))g(can)h(be)g(said)g (to)h(mo)o(v)o(e)e(rigidly)g(modulo)f(a)j(collineation.)p eop %%Page: 69 71 69 70 bop 515 1146 a FE(A)l(ppendix)42 b(B)515 1561 y FN(Singular)52 b(V)-19 b(alue)52 b(Decomposition)515 1810 y(and)f(Least)h(Squar)l(es)515 2242 y FQ(Let)20 b(the)g(singular)g(v)n(alue)f(decomposition)f(of)i FL(D)h FQ(\(e)o(g.)j([48)o(,)c(59)o(]\),)g(be)g(gi)n(v)o(en)f(by)1692 2424 y FL(D)24 b FI(=)e FL(V)r FI(\003)o FL(U)2081 2388 y FF(>)2178 2424 y FH(;)1005 b FQ(\(B.1\))515 2607 y(where)16 b FL(V)i FQ(is)g(a)f FH(n)5 b FA(\002)g FH(p)16 b FQ(matrix)g(whose)h (columns)e(are)i(the)f(left)h(hand)f(singular)f(v)o(ectors)h(of)g FL(D)p FQ(,)i FL(U)g FQ(is)f(a)515 2707 y FH(p)c FA(\002)g FH(p)k FQ(matrix)h(whose)g(columns)g(are)g(the)h(right)f(hand)f (singular)h(v)o(ectors)g(of)g FL(D)h FQ(and)f FI(\003)h FQ(is)h(the)e(diag-)515 2806 y(onal)i(matrix)h(of)f(the)i (corresponding)17 b(singular)j(v)n(alues)h(of)g FL(D)p FQ(:)27 b FI(\003)e(=)f FQ(diag)o FI(\()p FH(\025)2768 2754 y Fq(1)p 2769 2763 29 4 v 2769 2796 a(2)2811 2818 y FG(1)2848 2806 y FH(;)14 b(\025)2943 2754 y Fq(1)p 2943 2763 V 2943 2796 a(2)2986 2818 y FG(2)3023 2806 y FH(;)g(:)g(:)g(:)g(;)g(\025)3266 2754 y Fq(1)p 3266 2763 V 3266 2796 a(2)3308 2818 y FB(p)3347 2806 y FI(\))515 2917 y FQ(in)19 b(ascending)g(order)f(such)h(that)h FH(\025)1522 2865 y Fq(1)p 1522 2874 V 1522 2907 a(2)1565 2929 y FG(1)1622 2917 y FQ(is)g(the)g(minimum)e(singular)h(v)n(alue.)24 b FL(U)c FQ(is)h(ro)n(w)e(and)g(column)515 3017 y(orthogonal)h(i.e.)34 b FL(U)1108 2981 y FF(>)1164 3017 y FL(U)29 b FI(=)f FL(I)p FH(;)14 b FL(UU)1579 2981 y FF(>)1663 3017 y FI(=)28 b FL(I)q FQ(;)d FL(V)g FQ(is)f(column)e(orthogonal)e(such)j(that)g FL(V)3071 2981 y FF(>)3127 3017 y FL(V)30 b FI(=)e FL(I)p FQ(.)515 3117 y(The)21 b(SVD)i(gi)n(v)o(es)f(orthonormal)d(bases)j(for) g(the)g(null)g(space)g(and)f(range)g(of)h FL(D)p FQ(.)31 b(The)22 b(columns)f(of)515 3216 y FL(V)h FQ(corresponding)16 b(to)k(zero)f(singular)g(v)n(alues)g(form)g(a)h(basis)h(for)e(the)g (range)g(of)h FL(D)p FQ(.)25 b(The)20 b(columns)515 3316 y(of)g FL(U)h FQ(corresponding)c(to)j(zero)g(singular)f(v)n(alues)h (form)f(a)h(basis)h(for)f(the)g(null)g(space.)639 3416 y(It)33 b(is)g(clear)g(that)f(the)h(SVD)g(of)f FL(D)h FQ(is)g(v)o(ery)f(closely)g(related)g(to)g(the)h(eigensystem)e(of)i (the)515 3515 y(moment)19 b(matrix:)24 b FL(Z)1132 3479 y FF(>)1188 3515 y FL(Z)f FI(=)g FL(D)1430 3479 y FF(>)1486 3515 y FL(D)p FQ(,)1689 3698 y FL(D)83 b FI(=)g FL(V)q FI(\003)p FL(U)2197 3661 y FF(>)3206 3698 y FQ(\(B.2\))1453 3822 y FA(\))23 b FL(D)1633 3786 y FF(>)1689 3822 y FL(D)83 b FI(=)g FL(U)p FI(\003)p FL(V)2197 3786 y FF(>)2253 3822 y FL(V)r FI(\003)o FL(U)2458 3786 y FF(>)3206 3822 y FQ(\(B.3\))1845 3947 y FI(=)g FL(U)p FI(\003)2124 3911 y FG(2)2161 3947 y FL(U)2234 3910 y FF(>)3206 3947 y FQ(\(B.4\))1380 4071 y FA(\))23 b FL(D)1559 4035 y FF(>)1615 4071 y FL(DU)84 b FI(=)f FL(U)p FI(\003)2124 4035 y FG(2)2202 4071 y FH(:)981 b FQ(\(B.5\))515 4254 y(Thus)24 b(we)h(see)g(that)f (the)h(columns)e(of)i FL(U)g FQ(are)g(eigen)m(v)o(ectors)d(of)i FL(Z)2453 4218 y FF(>)2509 4254 y FL(Z)q FQ(,)i(and)e(that)g(the)h (squares)f(of)515 4354 y(the)g(singular)g(v)n(alues)h(of)f FL(D)i FQ(are)e(the)h(eigen)m(v)n(alues)e(of)h FL(Z)2181 4317 y FF(>)2237 4354 y FL(Z)p FQ(.)39 b(The)25 b(solution)f(to)h(the)f (orthogonal)515 4453 y(re)o(gression)18 b(problem:)1593 4553 y FL(f)33 b FI(=)22 b FQ(min)e FL(f)10 b FI(\()p FL(f)2001 4516 y FF(>)2057 4553 y FL(Z)2115 4516 y FF(>)2171 4553 y FL(Z)q(f)f FI(\))906 b FQ(\(B.6\))515 4702 y(is)18 b(well)g(kno)n(wn)e(to)i(be)f(the)g(eigen)m(v)o(ector)e(of)i FL(Z)1836 4666 y FF(>)1892 4702 y FL(Z)h FQ(corresponding)c(to)k(the)f (minimum)f(eigen)m(v)n(alue,)515 4802 y(this)24 b(is)g(simply)g FL(u)1042 4814 y FG(1)1103 4802 y FQ(the)g(\002rst)g(column)e(of)h FL(U)q FQ(,)i(which)e(is)h(the)g(shortest)f(principal)g(axis.)35 b(Thus)23 b(we)515 4902 y(ha)n(v)o(e)j(selected)i(the)f(direction)f(of) h(minimum)f(v)n(ariation)g(as)i(normal)e(to)h(our)g(\002tted)g(plane.) 46 b(The)515 5001 y(sum)18 b(of)h(squares)f(of)g(residuals)h(in)g(this) g(direction)e(is)j FH(\025)2092 5013 y FG(1)2130 5001 y FQ(,)f(similarly)-5 b(,)18 b(if)h(we)g(were)g(to)f(choose)g FL(u)3253 5013 y FG(2)3310 5001 y FQ(as)1905 5255 y(69)p eop %%Page: 70 72 70 71 bop 515 232 a FJ(APPENDIX)15 b(B.)32 b(SINGULAR)15 b(V)-11 b(ALUE)15 b(DECOMPOSITION)g(AND)g(LEAST)g(SQ)o(U)m(ARES)p FQ(70)515 523 y(our)j(solution)f(then)h(the)h(sum)f(of)h(squares)f(of)g (residuals)g(w)o(ould)g(be)h FH(\025)2502 535 y FG(2)2540 523 y FQ(.)24 b(Thus)19 b(if)g FH(\025)2889 535 y FG(2)2945 523 y FQ(is)h(close)f(to)f FH(\025)3341 535 y FG(1)515 623 y FQ(we)h(can)h(see)f(that)h(there)f(might)f(not)h(be)h(a)f(unique) f(solution.)24 b(The)19 b(residuals,)g(i.e.)25 b(the)19 b(projections)515 722 y(of)27 b(each)h(point)f(onto)h FL(u)1222 734 y FG(1)1287 722 y FQ(can)g(be)g(ef)n(\002ciently)f (calculated)g(by)h(noting)f(that)h FL(DU)38 b FI(=)f FL(V)r FI(\003)o FQ(,)31 b(thus)515 822 y FH(r)552 834 y FB(i)611 822 y FI(=)g FH(\025)765 769 y Fq(1)p 766 778 29 4 v 766 812 a(2)808 834 y FG(1)845 822 y FL(V)r FI(\()p FH(i;)14 b FI(1\))25 b FQ(where)f FL(V)r FI(\()p FH(i;)14 b FI(1\))24 b FQ(is)i(the)f(element)f(of)h FL(V)h FQ(on)f(the)g FH(i)p FQ(th)f(ro)n(w)g(and)h(\002rst)g(column.)515 930 y(Similarly)20 b FL(u)898 900 y FF(>)898 954 y FB(k)954 930 y FL(z)996 942 y FB(i)1047 930 y FI(=)i FH(\025)1192 878 y Fq(1)p 1193 887 V 1193 920 a(2)1235 942 y FB(k)1276 930 y FL(V)r FI(\()p FH(i;)14 b(k)s FI(\))p FQ(.)639 1039 y(W)-7 b(e)28 b(shall)f(prefer)e(the)h(use)h(of)f(the)h(SVD)g (rather)e(than)h(the)h(eigen-decomposition)22 b(of)k FL(Z)3265 1002 y FF(>)3321 1039 y FL(Z)515 1138 y FQ(for)31 b(numerical)f(reasons)i([4)o(]:)49 b(namely)31 b(that)h(the)g(SVD)g (relates)g(directly)f(to)i(the)e(data)h(matrix)515 1238 y(and)f(the)g(algorithms)f(in)i(e)o(xistence)f(for)g(its)h(computation) d(are)j(more)e(stable)i(than)f(those)g(that)515 1338 y(calculate)23 b(the)g(eigensystem)f(of)h FL(Z)1544 1301 y FF(>)1600 1338 y FL(Z)p FQ(,)i(especially)d(if)i FL(D)g FQ(is)g(ill-conditioned.)31 b(Furthermore,)21 b(in)515 1437 y(operating)16 b(directly)h(on)h(the)g FH(n)10 b FA(\002)g FH(p)19 b FQ(data)f(matrix)f(we)h(a)n(v)n(oid)g(the)g FH(np)2456 1407 y FG(2)2512 1437 y FQ(sums)g(and)f(products)g(needed) 515 1537 y(to)22 b(calculate)f FL(Z)977 1500 y FF(>)1033 1537 y FL(Z)q FQ(.)30 b(In)21 b(passing)h(we)g(note)f(that)h(in)g(the)g (case)g(of)f(classical)i(least)f(squares)g(the)g(hat)515 1637 y(matrix)d(is)i(easily)g(computed)d(from)h(the)h(SVD:)1753 1819 y FL(H)i FI(=)h FL(V)r(V)2085 1783 y FF(>)3206 1819 y FQ(\(B.7\))515 2002 y(this)d(form)g(of)g(calculation)f(is)i (preferable)d(especially)i(if)g FL(D)h FQ(is)g(of)f(less)h(than)f(full) g(rank.)p eop %%Page: 71 73 71 72 bop 515 1146 a FE(A)l(ppendix)42 b(C)515 1561 y FN(Orthogonal)52 b(Regr)l(ession\227af\002ne)515 1810 y(case)515 2242 y FQ(Consider)27 b(\002tting)g(a)h(hyperplane)d FL(f)46 b FI(=)36 b(\()p FH(f)1780 2254 y FG(1)1818 2242 y FH(;)14 b(f)1896 2254 y FG(2)1932 2242 y FH(;)g(:)g(:)g(:)g(;)g(f) 2158 2254 y FB(p)2196 2242 y FI(\))28 b FQ(through)e(a)i(set)g(of)g FH(n)g(p)p FQ(-dimensional)515 2341 y(points)c(with)g(homogeneous)d (coordinates)i FL(z)1851 2353 y FB(i)1910 2341 y FI(=)30 b(\()p FH(z)2076 2353 y FB(i)p FG(1)2137 2341 y FH(;)14 b(z)2213 2353 y FB(i)p FG(2)2273 2341 y FH(;)g(:)g(:)g(:)g(;)g(z)2497 2353 y FB(ip)p FF(\000)p FG(1)2643 2341 y FH(;)g FI(1\))p FQ(.)38 b(This)24 b(can)g(alterna-)515 2441 y(ti)n(v)o(ely)18 b(be)g(vie)n(wed)g(as)i(either)e(\002tting)h(a)g(hyperplane)d(in)i FH(p)p FQ(-dimensions)f(through)g(the)i(origin)e(using)515 2540 y(homogeneous)i(coordinates)h(or)i(\002tting)g(a)h(hyperplane)c (in)k FH(p)d FA(\000)f FI(1)j FQ(dimensions)g(not)f(through)g(the)515 2640 y(origin)26 b(using)h(inhomogeneous)d(coordinates)i FL(z)1957 2652 y FB(i)2021 2640 y FI(=)36 b(\()p FH(z)2193 2652 y FB(i)p FG(1)2254 2640 y FH(;)14 b(z)2330 2652 y FB(i)p FG(2)2390 2640 y FH(;)g(:)g(:)g(:)g(;)g(z)2614 2652 y FB(ip)p FF(\000)p FG(1)2760 2640 y FI(\))p FQ(.)48 b(The)28 b(best)g(\002tting)515 2740 y(hyperplane)16 b FL(f)29 b FQ(is)20 b(estimated)f(by)f(minimizing)g(the)h (perpendicular)d(sum)j(of)g(Euclidean)f(distances)515 2839 y(from)k(the)h(points)g(to)h(the)f(plane.)34 b(This)23 b(is)i(accomplished)c(by)i(minimizing)2747 2777 y FC(P)2834 2798 y FB(n)2834 2864 y(i)p FG(=1)2946 2839 y FI(\()p FL(f)3016 2802 y FF(>)3073 2839 y FL(z)3115 2851 y FB(i)3143 2839 y FI(\))3175 2809 y FG(2)3236 2839 y FQ(sub-)515 2952 y(ject)k(to)f(the)h(constraint)1211 2922 y Fz(1)1269 2889 y FC(P)1357 2910 y FB(p)p FF(\000)p FG(1)1357 2977 y FB(i)p FG(=1)1480 2952 y FI(\()p FL(f)10 b FI(\))1583 2922 y FG(2)1655 2952 y FI(=)34 b(1)p FQ(.)44 b(This)27 b(constraint)e(ensures)h(that)h(the)f(estimate)h(will)515 3051 y(be)20 b(in)m(v)n(ariant)f(to)i(equiform)e(transformation)f(of)i (the)h(inhomogeneous)c(coordinates.)24 b(F)o(or)d(e)o(xam-)515 3151 y(ple)j(the)g(best)h(\002tting)f(line)g(to)g(a)h FI(2)f FQ(dimensional)f(scatter)h FI(\()p FH(x)2266 3163 y FB(i)2294 3151 y FH(;)14 b(y)2372 3163 y FB(i)2400 3151 y FI(\))p FQ(,)25 b FH(i)30 b FI(=)g(1)14 b FH(:)g(:)g(:)f(n)25 b FQ(is)g(estimated)f(by)515 3251 y(minimizing)918 3188 y FC(P)1006 3209 y FB(n)1006 3276 y(i)p FG(=1)1118 3251 y FI(\()p FH(ax)19 b FI(+)f FH(by)j FI(+)d FH(c)p FI(\))1592 3221 y FG(2)1650 3251 y FQ(subject)i(to)g(the)g(constraint)g FH(a)2506 3221 y FG(2)2561 3251 y FI(+)e FH(b)2680 3221 y FG(2)2740 3251 y FI(=)23 b(1)d FQ([37)o(].)639 3361 y(Let)e FL(M)23 b FI(=)g FL(Z)1028 3325 y FF(>)1084 3361 y FL(Z)c FQ(be)f(the)g(moment)f(matrix)g(then)h(the)g(estimate)g FL(f)28 b FQ(minimizes)17 b FL(f)2940 3324 y FF(>)2996 3361 y FL(Mf)28 b FQ(subject)515 3461 y(to)c FL(f)643 3424 y FF(>)699 3461 y FL(Jf)40 b FI(=)30 b FQ(constant)o(,)c(where)e FL(J)31 b FI(=)f FQ(diag)o FI(\(1)p FH(;)14 b FI(1)p FH(;)g FI(1)p FH(;)g(:)g(:)g(:)f(;)h FI(1)p FH(;)g FI(0\))p FQ(.)37 b(This)24 b(estimate)h(is)g(gi)n(v)o(en)f(by)g(the)515 3561 y(eigen)m(v)o(ector)c(corresponding)g(to)k(the)f(minimum)f(eigen)m (v)n(alue)f(of)i(the)g(centred)f(moment)g(matrix.)515 3660 y(Centring)d(is)i(a)g(standard)e(statistical)i(technique)e(that)h (in)m(v)n(olv)o(es)f(shifting)g(the)h(coordinate)e(system)515 3760 y(of)25 b(the)g(data)g(points)g(so)g(that)g(the)h(centroid)d(lies) j(at)g(the)f(origin.)39 b(This)25 b(can)g(be)g(ef)n(fected)f(by)h(sub-) 515 3859 y(tracting)790 3842 y FH(~)796 3859 y FI(1)p 851 3814 43 4 v 13 w FH(z)894 3871 y FB(j)956 3859 y FQ(from)h(each)h(column)e(of)i FL(Z)p FQ(.)46 b(Where)2066 3842 y FH(~)2072 3859 y FI(1)27 b FQ(is)h(an)f FH(n)g FQ(dimensional)f(v)o(ector)g(such)g(that)508 3951 y FH(~)515 3969 y FI(1)c(=)h(\(1)p FH(;)14 b FI(1)p FH(;)g FI(1)p FH(;)g(:)g(:)g(:)e(;)i FI(1\))1156 3938 y FF(>)1233 3969 y FQ(and)p 1373 3923 V 19 w FH(z)1416 3981 y FB(j)1471 3969 y FQ(is)21 b(the)g(mean)e(of)h(that)g(column.)639 4068 y Ft(Pr)l(oof:)37 b FQ(Let)26 b(us)g(partition)f FL(f)36 b FQ(into)26 b FI(\()p FL(f)1726 4080 y FG(1)1763 4068 y FA(j)p FL(f)1825 4080 y FG(2)1862 4068 y FI(\))h FQ(with)f(components)e(of)h(length)g FH(p)e FA(\000)f FI(1)k FQ(and)g FI(1)g FQ(re-)515 4168 y(specti)n(v)o(ely)-5 b(,)18 b(and)i(let)h FL(M)f FQ(be)g(partitioned)f(in)h(a)h (corresponding)16 b(manner:)1519 4398 y FL(M)23 b FI(=)1721 4280 y FC(\024)1806 4347 y FL(M)1896 4359 y FG(11)2050 4347 y FL(M)2140 4359 y FG(12)1806 4447 y FL(M)1896 4459 y FG(21)2050 4447 y FL(M)2140 4459 y FG(22)2252 4280 y FC(\025)2351 4398 y FH(;)832 b FQ(\(C.1\))515 4624 y(then)1129 4724 y FL(f)1168 4687 y FF(>)1224 4724 y FL(M)o(f)33 b FI(=)22 b FL(f)1501 4687 y FF(>)1501 4744 y FG(1)1557 4724 y FL(M)1648 4736 y FG(11)1718 4724 y FL(f)1757 4736 y FG(1)1812 4724 y FI(+)c(2)p FL(f)1975 4687 y FF(>)1975 4744 y FG(1)2031 4724 y FL(M)2122 4736 y FG(12)2192 4724 y FL(f)2230 4736 y FG(2)2286 4724 y FI(+)g FL(f)2407 4687 y FF(>)2407 4744 y FG(2)2463 4724 y FL(M)2554 4736 y FG(22)2624 4724 y FL(f)2663 4736 y FG(2)2741 4724 y FH(:)442 b FQ(\(C.2\))p 515 4777 1146 4 v 605 4833 a Fy(1)634 4856 y FP(W)m(ith)19 b(non-af)n(\002ne)h (higher)f(order)g(surf)o(aces)g(the)g(constraint)i(will)e(be)f(more)g (comple)o(x)h(in)m(v)o(olving)i(higher)e(order)g(com-)515 4935 y(binations)g(of)e(the)h(coef)n(\002cients.)1905 5255 y FQ(71)p eop %%Page: 72 74 72 73 bop 515 232 a FJ(APPENDIX)20 b(C.)42 b(OR)-5 b(THOGON)m(AL)20 b(REGRESSION\227AFFINE)g(CASE)525 b FQ(72)515 523 y(F)o(or)20 b(an)o(y)f(\002x)o(ed)h FL(f)1013 486 y FF(>)1013 544 y FG(1)1069 523 y FQ(,)g FL(f)1149 486 y FF(>)1205 523 y FL(M)o(f)30 b FQ(is)21 b(minimal)f(when)1735 683 y FH(@)5 b FL(f)1822 646 y FF(>)1878 683 y FL(M)o(f)p 1735 720 272 4 v 1799 808 a FH(@)g FL(f)1886 770 y FF(>)1886 828 y FG(2)2040 739 y FI(=)22 b(0)1037 b FQ(\(C.3\))515 947 y(i.e.)1544 1046 y FI(2)p FL(f)1624 1009 y FF(>)1624 1067 y FG(1)1680 1046 y FL(M)1770 1058 y FG(12)1859 1046 y FI(+)18 b FL(f)1980 1009 y FF(>)1980 1067 y FG(2)2037 1046 y FL(M)2127 1058 y FG(22)2221 1046 y FI(=)k(0)856 b FQ(\(C.4\))515 1178 y(which)19 b(implies)1542 1278 y FL(f)1580 1241 y FF(>)1580 1298 y FG(2)1659 1278 y FI(=)k FA(\000)p FI(2)p FL(f)1891 1241 y FF(>)1891 1298 y FG(1)1947 1278 y FL(M)2037 1290 y FG(12)2108 1278 y FL(M)2198 1241 y FF(\000)p FG(1)2198 1299 y(22)2329 1278 y FH(;)854 b FQ(\(C.5\))515 1410 y(therefore)1248 1509 y FL(f)1287 1472 y FF(>)1343 1509 y FL(M)o(f)33 b FI(=)22 b FL(f)1620 1472 y FF(>)1620 1530 y FG(1)1677 1509 y FI(\()p FL(M)1799 1521 y FG(11)1888 1509 y FA(\000)c FL(M)2062 1521 y FG(12)2132 1509 y FL(M)2223 1473 y FF(\000)p FG(1)2223 1531 y(22)2312 1509 y FL(M)2403 1521 y FG(21)2473 1509 y FI(\))p FL(f)2543 1521 y FG(1)2622 1509 y FH(:)561 b FQ(\(C.6\))515 1641 y(Let)20 b(us)h(de\002ne)1475 1720 y FI(~)1451 1741 y FL(M)1564 1694 y FG(def)1578 1741 y FI(=)36 b FL(M)1770 1753 y FG(11)1858 1741 y FA(\000)18 b FL(M)2032 1753 y FG(12)2102 1741 y FL(M)2193 1704 y FF(\000)p FG(1)2193 1762 y(22)2282 1741 y FL(M)2373 1753 y FG(21)3206 1741 y FQ(\(C.7\))515 1873 y(T)-7 b(o)25 b(minimize)g(this)h(for)f FL(f)1273 1836 y FF(>)1329 1873 y FL(J)q(f)42 b FI(=)33 b FQ(constant)n(,)28 b(let)e FH(\024)g FQ(be)f(a)h(Lagrangian)d(multiplier)i(for)g(the)g(con-)515 1972 y(straint.)58 b(Then)30 b(we)h(must)h(set)f(to)h(zero)e(the)h (deri)n(v)n(ati)n(v)o(e)f(with)h(respect)g(to)g(zero)g(of)g FL(f)3066 1935 y FF(>)3066 1993 y FG(1)3146 1951 y FI(~)3122 1972 y FL(M)o(f)3251 1984 y FG(1)3314 1972 y FA(\000)515 2072 y FH(\024)p FL(f)601 2035 y FF(>)601 2093 y FG(1)657 2072 y FL(Jf)744 2084 y FG(1)782 2072 y FQ(.)25 b(This)c(yields:)1515 2172 y FI(2)p FL(f)1594 2135 y FF(>)1594 2192 y FG(1)1675 2150 y FI(~)1650 2172 y FL(M)i FI(=)g(2)p FH(\024)p FL(f)1979 2135 y FF(>)1979 2192 y FG(1)2035 2172 y FL(J)h FI(=)e(2)p FH(\024)p FL(f)2323 2135 y FF(>)2323 2192 y FG(1)3206 2172 y FQ(\(C.8\))515 2304 y(so)16 b(that)g FH(\024)h FQ(is)g(an)f(eigen)m(v)n(alue)e(of)1459 2282 y FI(~)1435 2304 y FL(M)i FQ(with)g FL(f)1745 2266 y FF(>)1745 2324 y FG(1)1817 2304 y FQ(the)h(corresponding)12 b(eigen)m(v)o(ector)-5 b(.)21 b(The)16 b(eigen)m(v)o(ector)515 2403 y(of)29 b(the)h(best)g(geometric)e(\002t)j(usually)e(corresponds)f(to)i(the)f (smallest)i(eigen)m(v)n(alue.)51 b(It)30 b(shall)h(be)515 2503 y(sho)n(wn)18 b(else)n(where)h(that)h(the)f(matrix)1626 2482 y FI(~)1602 2503 y FL(M)g FQ(is)i(related)e(to)g(the)h(co)o(v)n (ariance)d(matrix)i(of)g(the)g(residuals.)515 2602 y(What)i(form)e(has) 1055 2581 y FI(~)1030 2602 y FL(M)p FQ(?)27 b(In)21 b(the)f(af)n (\002ne)h(case)g(it)g(can)g(be)f(seen)h(that)g FL(M)j FI(=)f FL(Z)2651 2566 y FF(>)2707 2602 y FL(Z)e FQ(leads)g(to)g FL(M)3155 2614 y FG(22)3249 2602 y FI(=)i(1)515 2702 y FQ(thus)1579 2780 y FI(~)1554 2802 y FL(M)g FI(=)f FL(M)1846 2814 y FG(11)1935 2802 y FA(\000)c FL(M)2108 2814 y FG(12)2179 2802 y FL(M)2269 2814 y FG(21)3206 2802 y FQ(\(C.9\))515 2934 y(using)h(the)i(f)o(act)1097 3022 y FC(X)1217 3101 y FI(\()p FH(x)e FA(\000)p 1398 3055 48 4 v 18 w FH(x)q FI(\)\()p FH(y)i FA(\000)p 1655 3055 44 4 v 18 w FH(y)s FI(\))83 b(=)1962 3022 y FC(X)2082 3101 y FI(\()p FH(xy)22 b FA(\000)p 2307 3055 48 4 v 18 w FH(xy)f FA(\000)p 2499 3055 44 4 v 18 w FH(y)s(x)e FI(+)p 2692 3055 48 4 v 18 w FH(x)p 2753 3055 44 4 v 14 w(y)1814 3267 y FI(=)1962 3188 y FC(X)2082 3267 y FI(\()p FH(xy)s FI(\))g FA(\000)2339 3188 y FC(X)2473 3267 y FH(x)2534 3188 y FC(X)2668 3267 y FH(y)515 3431 y FQ(then)g(equation)g(\(C.9\))h(becomes)539 3687 y FI(~)515 3709 y FL(M)82 b FI(=)836 3517 y FC(2)836 3663 y(6)836 3716 y(4)1008 3519 y(P)1109 3581 y FH(z)1152 3551 y FG(2)1148 3602 y FB(i)p FG(1)1367 3519 y FC(P)1454 3581 y FI(\()p FH(z)1525 3593 y FB(i)p FG(1)1586 3581 y FH(z)1625 3593 y FB(i)p FG(2)1685 3581 y FI(\))i FH(:)14 b(:)g(:)933 3618 y FC(P)1020 3680 y FI(\()p FH(z)1091 3692 y FB(i)p FG(2)1152 3680 y FH(z)1191 3692 y FB(i)p FG(1)1251 3680 y FI(\))1442 3618 y FC(P)1543 3680 y FH(z)1586 3650 y FG(2)1582 3702 y FB(i)p FG(2)1801 3680 y FH(:)g(:)g(:)1098 3769 y FQ(.)1098 3802 y(.)1098 3835 y(.)1532 3769 y(.)1532 3802 y(.)1532 3835 y(.)1839 3769 y(.)1839 3802 y(.)1839 3835 y(.)1981 3517 y FC(3)1981 3663 y(7)1981 3716 y(5)2054 3709 y FA(\000)2137 3517 y FC(2)2137 3663 y(6)2137 3716 y(4)2291 3581 y FI(\()2323 3519 y FC(P)2424 3581 y FH(z)2463 3593 y FB(i)p FG(1)2524 3581 y FI(\))2556 3551 y FG(2)2733 3519 y FC(P)2834 3581 y FH(z)2873 3593 y FB(i)p FG(1)2947 3519 y FC(P)3049 3581 y FH(z)3088 3593 y FB(i)p FG(2)3231 3581 y FH(:)g(:)g(:)2234 3618 y FC(P)2336 3680 y FH(z)2375 3692 y FB(i)p FG(2)2449 3618 y FC(P)2550 3680 y FH(z)2589 3692 y FB(i)p FG(1)2789 3680 y FI(\()2821 3618 y FC(P)2923 3680 y FH(z)2962 3692 y FB(i)p FG(2)3022 3680 y FI(\))3054 3650 y FG(2)3231 3680 y FH(:)g(:)g(:)2432 3769 y FQ(.)2432 3802 y(.)2432 3835 y(.)2930 3769 y(.)2930 3802 y(.)2930 3835 y(.)3269 3769 y(.)3269 3802 y(.)3269 3835 y(.)3411 3517 y FC(3)3411 3663 y(7)3411 3716 y(5)688 4096 y FI(=)836 3904 y FC(2)836 4050 y(6)836 4103 y(4)1064 3906 y(P)1166 3968 y FH(z)1209 3938 y FG(2)1205 3989 y FB(i)p FG(1)1284 3968 y FA(\000)k FI(\()1399 3906 y FC(P)1500 3968 y FH(z)1539 3980 y FB(i)p FG(1)1600 3968 y FI(\))1632 3938 y FG(2)1884 3906 y FC(P)1971 3968 y FI(\()p FH(z)2042 3980 y FB(i)p FG(1)2103 3968 y FH(z)2142 3980 y FB(i)p FG(2)2202 3968 y FI(\))h FA(\000)2336 3906 y FC(P)2438 3968 y FH(z)2477 3980 y FB(i)p FG(1)2551 3906 y FC(P)2652 3968 y FH(z)2691 3980 y FB(i)p FG(2)2835 3968 y FH(:)14 b(:)g(:)933 4005 y FC(P)1020 4067 y FI(\()p FH(z)1091 4079 y FB(i)p FG(2)1152 4067 y FH(z)1191 4079 y FB(i)p FG(1)1251 4067 y FI(\))19 b FA(\000)1385 4005 y FC(P)1487 4067 y FH(z)1526 4079 y FB(i)p FG(2)1600 4005 y FC(P)1701 4067 y FH(z)1740 4079 y FB(i)p FG(1)2015 4005 y FC(P)2117 4067 y FH(z)2160 4037 y FG(2)2156 4089 y FB(i)p FG(2)2234 4067 y FA(\000)f FI(\()2349 4005 y FC(P)2451 4067 y FH(z)2490 4079 y FB(i)p FG(2)2551 4067 y FI(\))2583 4037 y FG(2)2835 4067 y FH(:)c(:)g(:)1356 4155 y FQ(.)1356 4189 y(.)1356 4222 y(.)2307 4155 y(.)2307 4189 y(.)2307 4222 y(.)2873 4155 y(.)2873 4189 y(.)2873 4222 y(.)3014 3904 y FC(3)3014 4050 y(7)3014 4103 y(5)515 4390 y FQ(from)19 b(which)h(it)g(can)g(be)h(seen)f(that)1561 4369 y FI(~)1536 4390 y FL(M)h FQ(is)g(the)f(centred)f(moment)g (matrix.)639 4490 y(A)33 b(formal)f(proof)f(is)i(not)g(presented,)h (rather)d(it)j(is)f(noted)f(that)h(if)g(a)g(set)g(of)f(points)g FL(x)3241 4502 y FB(i)3314 4490 y FA(\032)515 4590 y(P)579 4554 y FG(3)643 4590 y FQ(are)26 b(visible)f(to)h(a)g(pair)g(of)f (camera)g(with)h(transform)e(matrices)i FL(P)g FQ(and)f FL(P)2822 4560 y FF(0)2846 4590 y FQ(,)i(and)f(if)g FL(G)g FQ(is)h(an)515 4689 y(arbitrary)18 b(non-singular)f FI(4)f FA(\002)g FI(4)k FQ(matrix,)f(then)g(replacing)f FL(x)2263 4701 y FB(i)2311 4689 y FQ(by)h FL(G)2489 4653 y FF(\000)p FG(1)2578 4689 y FL(x)2629 4701 y FB(i)2656 4689 y FQ(,)h FL(P)h FQ(by)e FL(P)q(G)h FQ(and)f FL(P)3252 4659 y FF(0)3296 4689 y FQ(by)515 4789 y FL(P)580 4759 y FF(0)603 4789 y FL(G)j FQ(preserv)o(es)e(the)i(object-point)d(to)j(image-space)d (correspondences.)26 b(As)c(may)f(be)g(seen,)h(the)515 4889 y(internal)d(parameters)g(of)h(one)g(of)g(the)g(cameras)g(may)g (be)g(changed)e(arbitrarily)-5 b(.)24 b(Thus)c(points)f(that)515 4988 y(are)g(consistent)g(with)g(Equation)f(\(4.10\))g(can)h(be)g(said) h(to)f(mo)o(v)o(e)f(rigidly)g(modulo)g(a)i(collineation.)p eop %%Page: 73 75 73 74 bop 515 1146 a FE(A)l(ppendix)42 b(D)515 1561 y FN(V)-19 b(ariance)51 b(of)h(r)l(esiduals)515 1993 y FQ(In)20 b(Section)g(4.7)f(it)i(w)o(as)g(sho)n(wn)f(ho)n(w)f(each)h (constraint)f(may)h(be)g(re)n(weighted)f(by)h(the)g(v)n(ariance)f(of) 515 2092 y(its)j(corresponding)c(residual,)j(in)h(order)e(to)i(pro)o (vide)e(a)i(more)e(statistically)j(sound)d(minimization)515 2192 y(measure.)40 b(In)26 b(this)g(appendix)e(an)i(e)o(xpression)e (for)h(the)h(v)n(ariance)e(of)h(each)h(residual)f(is)i(deri)n(v)o(ed,) 515 2291 y(also)20 b(bias)h(in)f(linear)g(estimation)g(is)h(discussed.) 639 2391 y(Consider)30 b(the)h(problem)e(of)i(\002tting)f(a)h (fundamental)e(matrix)h(to)h(the)f(data)h(points)f FL(z)3179 2403 y FB(i)3206 2391 y FH(;)14 b(i)42 b FI(=)515 2491 y(1)14 b FH(:)g(:)g(:)f(n)p FQ(,)19 b(de\002ned)e(in)i(\(4.20\).)k(Let) 18 b FL(f)p 1494 2504 39 4 v 29 w FQ(be)g(the)h(e)o(xact)f(fundamental) e(matrix,)i(written)h(in)f(v)o(ector)g(form)515 2590 y(and)h(let)i FL(f)30 b FQ(be)20 b(the)g(estimate.)26 b(If)20 b FL(f)30 b FQ(is)21 b(computed)d(by)i(the)g(least)h(squares)f (optimization:)1570 2837 y FL(f)32 b FI(=)23 b FQ(min)1848 2866 y FL(f)1944 2733 y FB(n)1904 2758 y FC(X)1910 2935 y FB(i)p FG(=1)2048 2781 y FI(\()p FL(f)2118 2743 y FF(>)2174 2781 y FL(z)2217 2793 y FB(i)2244 2781 y FI(\))2276 2750 y FG(2)p 2048 2818 267 4 v 2137 2894 a FH(w)2196 2906 y FB(i)3201 2837 y FQ(\(D.1\))515 3091 y(where)17 b FH(w)795 3103 y FB(i)841 3091 y FQ(is)i(the)e(optimal)g(weight)g(\(being)g(the)g (v)n(ariance)g(of)g(the)h(residual\).)23 b(Let)17 b FL(Z)i FQ(be)f(the)f(matrix)515 3191 y(whose)e(ro)n(ws)g(are)h FL(z)1078 3161 y FF(>)1078 3212 y FB(i)1134 3191 y FH(=w)1235 3203 y FB(i)1279 3191 y FQ(and)e FL(M)23 b FI(=)g FL(Z)1674 3154 y FF(>)1730 3191 y FL(Z)16 b FQ(be)g(the)f FH(p)p FQ(-dimensional)f(symmetric)g(moment)g(matrix)1685 3441 y FL(M)23 b FI(=)1926 3337 y FB(n)1886 3362 y FC(X)1893 3539 y FB(i)p FG(=1)2030 3385 y FL(z)2073 3397 y FB(i)2100 3385 y FL(z)2143 3355 y FF(>)2143 3406 y FB(i)p 2030 3422 169 4 v 2065 3498 a FH(w)2126 3469 y FG(2)2124 3521 y FB(i)3201 3441 y FQ(\(D.2\))515 3696 y(If)28 b FL(M)h FQ(has)g(eigen)m(v)n(alues,)f(in)h(increasing)e(order)m(,)i FH(\025)2033 3708 y FG(1)2084 3696 y FH(:)14 b(:)g(:)g(\025)2243 3708 y FB(p)2311 3696 y FQ(and)28 b(corresponding)d(eigen)m(v)o(ectors) 515 3796 y FL(u)568 3808 y FG(1)619 3796 y FH(:)14 b(:)g(:)g FL(u)783 3808 y FB(p)842 3796 y FQ(then)515 3978 y FM(Theor)o(em)20 b(7)g(\(W)-5 b(eng)20 b Ft(et)g(al)h FM([61)n(]\))41 b Ft(If)21 b(co)o(variance)d(matrix)j(for)g FL(f)32 b FI(=)22 b FL(u)2561 3990 y FG(1)2619 3978 y Ft(is)1637 4161 y FI(\000)1689 4173 y FB(f)1755 4161 y FI(=)h FA(E)7 b(f)p FH(\016)s FL(u)2029 4173 y FG(1)2066 4161 y FH(\016)s FL(u)2159 4127 y FF(>)2159 4182 y FG(1)2215 4161 y FA(g)944 b FQ(\(D.3\))515 4344 y Ft(the)20 b(optimal)f(weighting)h(is)1713 4443 y FH(w)1772 4455 y FB(i)1823 4443 y FI(=)j FL(f)1949 4406 y FF(>)2005 4443 y FI(\000)2057 4455 y FB(z)2089 4463 y Fl(i)2119 4443 y FL(f)9 b FH(:)1021 b FQ(\(D.4\))515 4593 y Ft(and)19 b(that)h(the)g(co)o(variance)f(matrix)h(for)h FL(f)30 b Ft(is)1603 4842 y FI(\000)1655 4854 y FB(f)1721 4842 y FI(=)23 b FH(\033)1859 4808 y FG(2)1953 4735 y FB(p)1911 4763 y FC(X)1910 4942 y FB(k)q FF(6)p FG(=1)2055 4786 y FL(u)2108 4798 y FB(k)2149 4786 y FL(u)2202 4756 y FF(>)2202 4810 y FB(k)p 2055 4823 203 4 v 2112 4899 a FH(\025)2160 4911 y FB(k)2268 4842 y FH(:)910 b FQ(\(D.5\))1905 5255 y(73)p eop %%Page: 74 76 74 75 bop 515 232 a FJ(APPENDIX)20 b(D.)41 b(V)-11 b(ARIANCE)21 b(OF)f(RESIDU)m(ALS)1209 b FQ(74)515 523 y Ft(Pr)l(oof:)30 b FQ(F)o(ollo)n(wing)21 b(the)i(analysis)g(of)f(W)-7 b(eng)23 b Ft(et)g(al)g FQ([61)o(].)33 b(Let)23 b(the)f(homogeneous)e (image)i(points)515 623 y FL(x)565 635 y FB(i)620 623 y FI(=)27 b(\()p FH(x)791 635 y FB(i)820 623 y FH(;)14 b(y)898 635 y FB(i)925 623 y FH(;)g(\020)6 b FI(\))24 b FQ(in)f(the)f(\002rst)h(image)g(be)f(matched)f(to)i FL(x)2195 593 y FF(0)2195 644 y FB(i)2250 623 y FI(=)k(\()p FH(x)2421 593 y FF(0)2421 644 y FB(i)2449 623 y FH(;)14 b(y)2530 593 y FF(0)2527 644 y FB(i)2554 623 y FH(;)g(\020)6 b FI(\))24 b FQ(in)f(the)g(second)e(image.)515 722 y(The)j(noise)h(in)g (the)f(data)h(matrix)f FL(Z)h FQ(is)h(due)e(to)h(noise)g(in)g(the)f (image)g(coordinates,)g(arising)h(from)515 822 y(spatial)19 b(quantisation,)e(feature)h(detection)f(errors,)h(point)g(mismatching)f (and)h(camera)g(distortion.)515 922 y(W)-7 b(e)31 b(shall)f(assume)g (that)f(the)h(noise)g(in)g(the)f(image)h(coordinates)e(has)i(zero)f (mean)g(and)g(kno)n(wn)515 1021 y(v)n(ariance.)h(If)22 b(the)g(noise)h(arises)g(from)e(man)o(y)g(sources)h(and)g(is)h (in\003uenced)e(by)h(the)g(sum)h(of)f(man)o(y)515 1121 y(f)o(actors,)c(its)h(distrib)n(ution)f(is)h(roughly)d(Gaussian)j(by)f (the)g(central)g(limit)h(theorem.)k(\(In)18 b(f)o(act,)h(as)g(we)515 1220 y(shall)24 b(see,)i(steps)e(can)g(be)g(tak)o(en)g(to)g(ensure)g (that)g(the)g(methods)f(we)h(use)h(are)f(rob)n(ust)f(not)h(only)f(to) 515 1320 y(outliers,)16 b(b)n(ut)h(also)g(to)f(departure)f(from)h(the)g (Gaussian)h(distrib)n(utions.\))22 b(W)-7 b(e)18 b(shall)f(further)e (assume)515 1420 y(that)24 b(the)f(noise)h(is)g(uncorrelated)d(between) i(dif)n(ferent)f(image)h(points)h(and)f(that)h(the)f(noise)h(in)g(the) 515 1519 y(tw)o(o)e(components)e(of)h(the)h(image)g(coordinates)e(is)j (also)f(uncorrelated.)28 b(The)21 b(co)o(v)n(ariance)f(matrix)515 1619 y(of)d FL(f)28 b FQ(is)19 b(deri)n(v)o(ed)d(on)h(the)h(basis)g(of) g(\002rst)g(order)f(perturbations.)22 b(Consider)17 b(tw)o(o)h(image)f (coordinates)515 1719 y FH(x)k FQ(and)f FH(y)j FQ(with)d(small)h (errors:)1682 1883 y FH(x)83 b FI(=)g FH(x)p 1960 1896 48 4 v 19 w FI(+)18 b FH(\016)s(x)1686 2007 y(y)85 b FI(=)e FH(y)p 1960 2037 44 4 v 21 w FI(+)18 b FH(\016)s(y)s(:)515 2171 y FQ(T)-7 b(o)20 b(\002rst)h(order)1556 2271 y FH(xy)26 b FA(\031)d FH(x)p 1758 2284 48 4 v(y)p 1805 2300 44 4 v 22 w FI(+)18 b FH(\016)s(xy)p 2038 2300 V 21 w FI(+)g FH(\016)s(y)s(x)p 2267 2284 48 4 v(:)864 b FQ(\(D.6\))515 2409 y(The)20 b(perturbed)d(data)k(matrix)e FL(Z)k FI(=)g FL(Z)p 1569 2422 59 4 v 19 w FI(+)18 b FH(\016)s FL(Z)j FQ(gi)n(v)o(es)f(rise)g(to)h(a)f(perturbed)e(moment)h(matrix)h FL(M)1156 2573 y(M)p 1156 2586 91 4 v 18 w FI(+)e FH(\016)s FL(M)83 b FI(=)f(\()p FL(Z)p 1741 2586 59 4 v 19 w FI(+)18 b FH(\016)s FL(Z)q FI(\))2032 2539 y FF(>)2088 2573 y FI(\()p FL(Z)p 2120 2586 V 19 w FI(+)g FH(\016)s FL(Z)q FI(\))1562 2697 y(=)82 b FL(Z)p 1709 2710 V 1768 2661 a FF(>)1824 2697 y FL(Z)p 1824 2710 V 18 w FI(+)18 b FH(\016)s FL(Z)2082 2661 y FF(>)2138 2697 y FL(Z)p 2138 2710 V 19 w FI(+)g FL(Z)p 2298 2710 V -36 x FF(>)2412 2697 y FH(\016)s FL(Z)h FI(+)f FH(O)2677 2663 y FG(2)2715 2697 y FH(:)515 2861 y FQ(Thus)i(to)g(\002rst)h(order)1551 2961 y FH(\016)s FL(M)i FI(=)g FH(\016)s FL(Z)1891 2925 y FF(>)1947 2961 y FL(Z)p 1947 2974 V 19 w FI(+)18 b FL(Z)p 2107 2974 V -36 x FF(>)2221 2961 y FH(\016)s FL(Z)q FH(;)858 b FQ(\(D.7\))515 3099 y(where)19 b(the)i(ro)n(ws)f(of)g FH(\016)s FL(Z)h FQ(are)3720 3263 y FH(\016)s FL(z)3803 3275 y FB(i)3853 3216 y FG(def)3867 3263 y FI(=)515 3312 y FC(\020)606 3405 y FH(\016)s(x)693 3417 y FB(i)721 3405 y FH(x)p 721 3418 48 4 v -56 x Fn(0)768 3426 y FB(i)814 3405 y FI(+)e FH(\016)s(x)985 3349 y Fn(0)985 3426 y FB(i)1013 3405 y FH(x)p 1013 3418 V 20 x FB(i)1171 3405 y FH(\016)s(y)1252 3417 y FB(i)1279 3405 y FH(x)p 1279 3418 V 1327 3349 a Fn(0)1327 3426 y FB(i)1373 3405 y FI(+)f FH(\016)s(y)1537 3417 y FB(i)1564 3405 y FH(x)p 1564 3418 V 1612 3349 a Fn(0)1612 3426 y FB(i)1722 3405 y FH(\016)s(x)1809 3349 y Fn(0)1809 3426 y FB(i)1837 3405 y FH(\020)90 b(\016)s(y)2047 3349 y Fn(0)2044 3426 y FB(i)2073 3405 y FH(x)p 2073 3418 V 2121 3425 a FB(i)2167 3405 y FI(+)18 b FH(\016)s(x)2337 3417 y FB(i)2365 3405 y FH(y)p 2365 3434 44 4 v 2409 3349 a Fn(0)2409 3441 y FB(i)2519 3405 y FH(\016)s(y)2603 3349 y Fn(0)2600 3426 y FB(i)2630 3405 y FH(y)p 2630 3434 V 2673 3441 a FB(i)2719 3405 y FI(+)g FH(\016)s(y)2883 3417 y FB(i)2911 3405 y FH(y)p 2911 3434 V 2954 3349 a Fn(0)2954 3441 y FB(i)3065 3405 y FH(\016)s(y)3149 3349 y Fn(0)3146 3426 y FB(i)3175 3405 y FH(\020)90 b(\016)s(x)3388 3417 y FB(i)3416 3405 y FH(\020)f(\016)s(y)3622 3417 y FB(i)3650 3405 y FH(\020)g FI(0)3858 3312 y FC(\021)3922 3405 y FH(:)515 3693 y FQ(Under)28 b(the)h(assumption)f(that)h(there)g(is)h (no)e(cross)i(correlation)d(of)i(error)f(between)g(correspon-)515 3793 y(dences)19 b(we)i(de\002ne)f(the)g(co)o(v)n(ariance)e(matrix)h (of)h(an)o(y)g(tw)o(o)g(ro)n(ws)g(of)g FL(Z)h FQ(to)g(be)1408 4004 y FA(E)8 b FI(\()p FH(\016)s FL(z)1574 4016 y FB(i)1602 4004 y FH(\016)s FL(z)1684 3969 y FF(>)1684 4024 y FB(j)1740 4004 y FI(\))24 b(=)1883 3887 y FC(\032)1987 3953 y FL(0)149 b FQ(if)21 b FH(i)h FA(6)p FI(=)h FH(j)1987 4053 y FI(\000)2039 4065 y FB(z)2071 4073 y Fl(i)2184 4053 y FQ(if)e FH(i)h FI(=)h FH(j)3201 4004 y FQ(\(D.8\))515 4212 y(where)878 4469 y FI(\000)930 4481 y FB(z)962 4489 y Fl(i)1015 4469 y FI(=)g FH(\033)1153 4434 y FG(2)1205 4302 y FC(2)1205 4451 y(4)1301 4364 y FL(x)p 1301 4377 51 4 v 1352 4309 a Fn(0)1352 4386 y FB(i)1379 4364 y FL(x)p 1379 4377 V 1430 4309 a Fn(0)1430 4386 y FB(i)1598 4364 y FI(0)197 b(0)1359 4472 y(0)139 b FL(x)p 1540 4485 V 1591 4417 a Fn(0)1591 4494 y FB(i)1619 4472 y FL(x)p 1619 4485 V -55 x Fn(0)1669 4494 y FB(i)1837 4472 y FI(0)1359 4580 y(0)197 b(0)140 b FL(x)p 1780 4593 V -55 x Fn(0)1830 4602 y FB(i)1858 4580 y FL(x)p 1858 4593 V -55 x Fn(0)1908 4602 y FB(i)1977 4302 y FC(3)1977 4451 y(5)2051 4469 y FI(+)18 b FH(\033)2184 4434 y FG(2)2235 4302 y FC(2)2235 4451 y(4)2336 4364 y FH(x)p 2336 4377 48 4 v -30 x FG(2)2420 4364 y FL(J)87 b FH(x)p 2556 4377 V(y)p 2603 4393 44 4 v 3 w FL(J)c FH(x)p 2779 4377 48 4 v 1 w(\020)6 b FL(J)2332 4466 y FH(x)p 2332 4479 V 1 w(y)p 2380 4496 44 4 v 2 w FL(J)89 b FH(y)p 2561 4496 V 2605 4436 a FG(2)2642 4466 y FL(J)h FH(y)p 2781 4496 V 3 w(\020)6 b FL(J)2333 4569 y FH(x)p 2333 4582 48 4 v(\020)g FL(J)87 b FH(y)p 2558 4598 44 4 v 3 w(\020)6 b FL(J)91 b FH(\020)2826 4539 y FG(2)2864 4569 y FL(J)2960 4302 y FC(3)2960 4451 y(5)3201 4469 y FQ(\(D.9\))515 4739 y(and)19 b(with)1606 4919 y FL(J)24 b FI(=)1766 4752 y FC(2)1766 4902 y(4)1863 4819 y FI(1)83 b(0)f(0)1863 4918 y(0)h(1)f(0)1863 5018 y(0)h(0)f(0)2195 4752 y FC(3)2195 4902 y(5)2264 4919 y FH(:)873 b FQ(\(D.10\))p eop %%Page: 75 77 75 76 bop 515 232 a FJ(APPENDIX)20 b(D.)41 b(V)-11 b(ARIANCE)21 b(OF)f(RESIDU)m(ALS)1209 b FQ(75)515 523 y(Gi)n(v)o(en)18 b FI(\000)786 535 y FB(z)818 543 y Fl(i)868 523 y FQ(we)i(can)f(deri)n (v)o(e)f(the)h(estimated)g(v)n(ariance)f(of)h(each)g(residual.)24 b(If)19 b FL(z)p 2747 536 43 4 v 21 x FB(i)2837 523 y FQ(is)h(perturbed)d(into)515 623 y FL(z)557 635 y FB(i)608 623 y FI(=)23 b FL(z)p 696 636 V 20 x FB(i)784 623 y FI(+)18 b FH(\016)s FL(z)950 635 y FB(i)977 623 y FQ(,)j(then)e FH(r)1219 635 y FB(i)1271 623 y FI(=)j FL(f)p 1358 636 39 4 v 1396 586 a FF(>)1452 623 y FL(z)1495 593 y FF(0)1539 623 y FQ(can)e(be)g(written)g(to)g(a)h(\002rst)g(approximation)c(as) 1757 805 y FH(r)1794 817 y FB(i)1845 805 y FI(=)23 b FL(f)p 1933 818 V 1971 768 a FF(>)2027 805 y FH(\016)s FL(z)2109 817 y FB(i)3160 805 y FQ(\(D.11\))515 988 y(assuming)g(that)g FL(f)p 1001 1001 V 34 w FQ(is)i(kno)n(wn.)34 b(As)24 b FL(f)p 1541 1001 V 33 w FQ(is)h(unkno)n(wn)c(we)j(must)g(use)g(its)h (estimation)e FL(f)33 b FQ(in)24 b(its)h(place,)515 1088 y(it)h(is)g(hoped)d(that)j FL(f)35 b FQ(will)26 b(con)m(v)o(er)o(ge)c (to)j FL(f)p 1680 1101 V 35 w FQ(as)h(the)f(algorithm)f(progresses.)39 b(The)25 b(residual)f FH(r)3207 1100 y FB(i)3261 1088 y FQ(is)i(a)515 1187 y(random)18 b(v)n(ariable)h(with)i(mean)e(zero)h (and)f(v)n(ariance)g(gi)n(v)o(en)g(by)1257 1370 y FH(\033)1307 1336 y FG(2)1304 1390 y FB(i)1368 1370 y FI(=)k FA(E)7 b FI(\()p FH(r)1578 1336 y FG(2)1576 1390 y FB(i)1616 1370 y FI(\))24 b(=)e FL(f)1797 1333 y FF(>)1853 1370 y FA(E)8 b FI(\()p FH(\016)s FL(z)2019 1382 y FB(i)2047 1370 y FH(\016)s FL(z)2130 1336 y FF(>)2130 1390 y FB(i)2186 1370 y FI(\))p FL(f)32 b FI(=)23 b FL(f)2405 1333 y FF(>)2461 1370 y FI(\000)2513 1382 y FB(z)2545 1390 y Fl(i)2575 1370 y FL(f)9 b FH(:)524 b FQ(\(D.12\))515 1553 y(Thus)24 b(we)i(set)g FH(w)1006 1565 y FB(i)1066 1553 y FI(=)32 b FH(\033)1213 1522 y FG(2)1210 1574 y FB(i)1277 1553 y FQ(in)25 b(equation)e(\(D.1\),)j(pro)o(viding)c(the)j(maximum)f(lik)o (elihood)f(estimate)515 1652 y(of)c FL(f)30 b FQ(\(if)19 b(the)h FH(r)920 1664 y FB(i)968 1652 y FQ(follo)n(w)f(a)h(Gaussian)g (distrib)n(ution\).)j(By)d(equating)e(coef)n(\002cients)h(it)h(can)g (be)f(sho)n(wn)515 1752 y(that)1606 1851 y FL(f)1645 1814 y FF(>)1701 1851 y FI(\000)1753 1863 y FB(z)1785 1871 y Fl(i)1815 1851 y FL(f)32 b FI(=)23 b FH(\033)2014 1817 y FG(2)2051 1851 y FI(\()p FA(r)p FH(r)2189 1863 y FB(i)2218 1851 y FI(\))2250 1817 y FG(2)3160 1851 y FQ(\(D.13\))515 2001 y(where)c(the)i(gradient,)d FA(r)p FH(r)r FQ(,)k(is)f(easily)f(computed:)1387 2193 y FA(r)p FH(r)86 b FI(=)d(\()p FH(r)1798 2159 y FG(2)1796 2214 y FB(x)1857 2193 y FI(+)18 b FH(r)1979 2159 y FG(2)1977 2214 y FB(y)2036 2193 y FI(+)g FH(r)2158 2159 y FG(2)2156 2214 y FB(x)2194 2197 y Fn(0)2239 2193 y FI(+)g FH(r)2361 2159 y FG(2)2359 2214 y FB(y)2395 2197 y Fn(0)2422 2193 y FI(\))2464 2137 y Fq(1)p 2464 2146 29 4 v 2464 2179 a(2)1417 2318 y FH(r)1454 2330 y FB(x)1579 2318 y FI(=)83 b FH(f)1768 2330 y FG(1)1805 2318 y FH(x)1852 2284 y FF(0)1894 2318 y FI(+)18 b FH(f)2018 2330 y FG(4)2055 2318 y FH(y)2099 2284 y FF(0)2140 2318 y FI(+)g FH(f)2264 2330 y FG(7)2301 2318 y FH(\020)1419 2443 y(r)1456 2455 y FB(y)1579 2443 y FI(=)83 b FH(f)1768 2455 y FG(2)1805 2443 y FH(x)1852 2408 y FF(0)1894 2443 y FI(+)18 b FH(f)2018 2455 y FG(5)2055 2443 y FH(y)2099 2408 y FF(0)2140 2443 y FI(+)g FH(f)2264 2455 y FG(8)2301 2443 y FH(\020)1395 2567 y(r)1432 2579 y FB(x)1470 2563 y Fn(0)1579 2567 y FI(=)83 b FH(f)1768 2579 y FG(1)1805 2567 y FH(x)19 b FI(+)f FH(f)1995 2579 y FG(2)2032 2567 y FH(y)j FI(+)d FH(f)2218 2579 y FG(3)2255 2567 y FH(\020)1396 2692 y(r)1433 2704 y FB(y)1469 2687 y Fn(0)1579 2692 y FI(=)83 b FH(f)1768 2704 y FG(4)1805 2692 y FH(x)19 b FI(+)f FH(f)1995 2704 y FG(5)2032 2692 y FH(y)j FI(+)d FH(f)2218 2704 y FG(6)2255 2692 y FH(\020)6 b(;)515 2874 y FQ(where)16 b FH(r)772 2886 y FB(x)832 2874 y FQ(denotes)g(the)h(partial)g(deri)n(v)n(ati)n(v) o(e)e(of)i FH(r)k FQ(\(gi)n(v)o(en)15 b(in)i(Equation)f(\(4.23\)\))f (with)i(respect)g(to)g FH(x)p FQ(.)515 2974 y(The)24 b(optimal)f(weights)h(in)m(v)n(olv)o(e)f(the)h(unperturbed)d(points)j FL(z)p 2278 2987 43 4 v FQ(,)i(so)e(we)h(must)f(approximate)e(them)515 3074 y(by)e(their)f(sample)h(v)n(alues.)25 b(This)20 b(forms)g(the)g(basis)h(of)f(Sampson')-5 b(s)19 b(method.)639 3273 y(Ne)o(xt)26 b(we)h(shall)g(deri)n(v)o(e)e(the)h(co)o(v)n(ariance) e(matrix)i(for)g FL(f)43 b FI(=)34 b FL(u)2468 3285 y FG(1)2506 3273 y FQ(.)43 b(If)27 b(the)f(moment)f(matrix)h(is)515 3372 y(perturbed)18 b(by)h FH(\016)s FL(M)i FQ(then)f(the)g(solution)f (is)i(perturbed)d(by)1526 3623 y FH(\016)s FL(u)1619 3635 y FG(1)1679 3623 y FI(=)1810 3515 y FB(p)1767 3544 y FC(X)1767 3723 y FB(k)q FF(6)p FG(=1)1911 3567 y FL(u)1964 3536 y FF(>)1964 3590 y FB(k)2020 3567 y FH(\016)s FL(Mu)2204 3579 y FG(1)p 1911 3604 331 4 v 2032 3680 a FH(\025)2080 3692 y FB(k)2251 3623 y FL(u)2304 3635 y FB(k)2345 3623 y FH(:)792 b FQ(\(D.14\))515 3890 y(Noting)19 b(that)1434 4053 y FL(u)1487 4019 y FF(>)1487 4074 y FB(k)1543 4053 y FH(\016)s FL(Mu)1727 4065 y FG(1)1787 4053 y FI(=)1914 3950 y FB(n)1875 3975 y FC(X)1881 4151 y FB(i)p FG(=1)2019 3997 y FL(u)2072 3967 y FF(>)2072 4021 y FB(k)2128 3997 y FL(z)2170 4009 y FB(i)2198 3997 y FH(\016)s FL(z)2280 3967 y FF(>)2280 4019 y FB(i)2336 3997 y FL(u)2389 4009 y FG(1)p 2019 4034 408 4 v 2179 4110 a FH(w)2238 4122 y FB(i)2437 4053 y FH(;)700 b FQ(\(D.15\))515 4274 y(and)19 b(de\002ning)g(v)o(ectors)1634 4442 y FL(s)1671 4454 y FB(i)1722 4442 y FI(=)1853 4334 y FB(p)1810 4363 y FC(X)1810 4542 y FB(k)q FF(6)p FG(=1)1954 4386 y FL(u)2007 4355 y FF(>)2007 4409 y FB(k)2063 4386 y FL(z)2106 4398 y FB(i)p 1954 4423 180 4 v 1999 4499 a FH(\025)2047 4511 y FB(k)2143 4442 y FL(u)2196 4454 y FB(k)2237 4442 y FH(;)900 b FQ(\(D.16\))515 4676 y(it)21 b(can)f(be)g(seen)g(that)1591 4822 y FH(\016)s FL(u)1684 4834 y FG(1)1744 4822 y FI(=)1871 4719 y FB(n)1832 4743 y FC(X)1838 4920 y FB(i)p FG(=1)1976 4766 y FH(\016)s FL(z)2058 4736 y FF(>)2058 4788 y FB(i)2114 4766 y FL(u)2167 4778 y FG(1)2204 4766 y FL(s)2242 4778 y FB(i)p 1976 4803 295 4 v 2079 4879 a FH(w)2138 4891 y FB(i)2280 4822 y FH(:)857 b FQ(\(D.17\))p eop %%Page: 76 78 76 77 bop 515 232 a FJ(APPENDIX)20 b(D.)41 b(V)-11 b(ARIANCE)21 b(OF)f(RESIDU)m(ALS)1209 b FQ(76)515 523 y(Hence)20 b(the)g(co)o(v)n (ariance)e(matrix)h FI(\000)1536 535 y FB(f)1600 523 y FQ(is)i(gi)n(v)o(en)e(by)1232 775 y FI(\000)1284 787 y FB(f)1410 775 y FI(=)1616 671 y FB(n)1577 696 y FC(X)1558 873 y FB(i;j)s FG(=1)1729 775 y FA(E)1795 633 y FC( )1870 709 y FI(\()p FH(\016)s FL(z)1985 679 y FF(>)1985 731 y FB(i)2041 709 y FL(u)2094 721 y FG(1)2131 709 y FH(\016)s FL(z)2214 679 y FF(>)2214 731 y FB(j)2270 709 y FL(u)2323 721 y FG(1)2360 709 y FI(\))p FL(s)2430 721 y FB(i)2458 709 y FL(s)2495 679 y FF(>)2495 731 y FB(j)p 1870 756 682 4 v 2120 832 a FH(w)2179 844 y FB(i)2207 832 y FH(w)2266 844 y FB(j)2561 633 y FC(!)1410 1071 y FI(=)1616 967 y FB(n)1577 992 y FC(X)1558 1169 y FB(i;j)s FG(=1)1729 1071 y FA(E)1795 929 y FC( )1870 1005 y FI(\()p FL(u)1956 975 y FF(>)1956 1026 y FG(1)2012 1005 y FH(\016)s FL(z)2094 975 y FF(>)2094 1027 y FB(i)2150 1005 y FH(\016)s FL(z)2233 1017 y FB(j)2268 1005 y FI(\))p FL(u)2353 1017 y FG(1)2390 1005 y FL(s)2428 1017 y FB(i)2455 1005 y FL(s)2493 975 y FF(>)2493 1027 y FB(j)p 1870 1052 679 4 v 2119 1128 a FH(w)2178 1140 y FB(i)2206 1128 y FH(w)2265 1140 y FB(j)2559 929 y FC(!)2639 1071 y FH(:)515 1328 y FQ(Using)h(\(D.8\))f (we)i(see)f(that)1393 1565 y FI(\000)1445 1577 y FB(f)1571 1565 y FI(=)1777 1461 y FB(n)1737 1486 y FC(X)1718 1663 y FB(i;j)s FG(=1)1900 1500 y FH(\016)1937 1512 y FB(ij)1996 1500 y FL(u)2049 1470 y FF(>)2049 1521 y FG(1)2105 1500 y FI(\000)2157 1512 y FB(z)2189 1520 y Fl(i)2219 1500 y FL(u)2272 1512 y FG(1)2309 1500 y FL(s)2347 1512 y FB(i)2374 1500 y FL(s)2412 1470 y FF(>)2412 1522 y FB(j)p 1900 1546 568 4 v 2093 1622 a FH(w)2152 1634 y FB(i)2180 1622 y FH(w)2239 1634 y FB(j)2478 1565 y FH(;)515 1823 y FQ(where)f(the)i(Kroneck)o(er)d(delta)i FH(\016)1450 1835 y FB(j)s(k)1545 1823 y FI(=)i(0)e FQ(if)h FH(j)28 b FA(6)p FI(=)23 b FH(k)s FQ(,)d FH(\016)2040 1835 y FB(j)s(k)2135 1823 y FI(=)i(1)e FQ(if)h FH(j)28 b FI(=)23 b FH(k)s FQ(.)i(From)20 b(\(D.4\))f(we)h(see)h(that)1648 2067 y FI(=)1854 1963 y FB(n)1815 1988 y FC(X)1796 2165 y FB(i;j)s FG(=1)1977 2010 y FH(\016)2014 2022 y FB(ij)2073 2010 y FH(w)2132 2022 y FB(i)2160 2010 y FL(s)2198 2022 y FB(i)2225 2010 y FL(s)2263 1980 y FF(>)2263 2032 y FB(i)p 1977 2047 342 4 v 2057 2124 a FH(w)2116 2136 y FB(i)2144 2124 y FH(w)2203 2136 y FB(j)1648 2354 y FI(=)1835 2251 y FB(n)1796 2276 y FC(X)1802 2452 y FB(i)p FG(=1)1939 2298 y FL(s)1977 2310 y FB(i)2005 2298 y FL(s)2042 2268 y FF(>)2042 2320 y FB(i)p 1939 2335 159 4 v 1975 2411 a FH(w)2034 2423 y FB(i)2108 2354 y FH(:)515 2600 y FQ(W)-7 b(e)21 b(are)f(no)n(w)g(in)g(a)h(position)e(to)h(e)o(xpand)f(the)h FL(s)g FQ(v)o(ectors)g(gi)n(ving)1328 2842 y FI(\000)1380 2854 y FB(f)1446 2842 y FI(=)1573 2738 y FB(n)1534 2763 y FC(X)1540 2940 y FB(i)p FG(=1)1700 2785 y FI(1)p 1677 2822 88 4 v 1677 2899 a FH(w)1736 2911 y FB(i)1852 2734 y(p)1809 2763 y FC(X)1788 2941 y FB(k)q(;l)p FF(6)p FG(=1)1974 2785 y FL(u)2027 2755 y FF(>)2027 2809 y FB(k)2083 2785 y FL(z)2125 2797 y FB(i)2153 2785 y FL(z)2195 2755 y FF(>)2195 2807 y FB(i)2251 2785 y FL(u)2304 2797 y FB(l)p 1974 2822 357 4 v 2070 2899 a FH(\025)2118 2911 y FB(k)2159 2899 y FH(\025)2207 2911 y FB(l)2340 2842 y FL(u)2393 2854 y FB(k)2434 2842 y FL(u)2487 2807 y FF(>)2487 2862 y FB(k)2543 2842 y FH(:)594 b FQ(\(D.18\))515 3101 y(Since)20 b FL(u)773 3113 y FB(k)835 3101 y FQ(and)f FL(u)1028 3113 y FB(l)1075 3101 y FQ(are)h(eigen)m(v)o(ectors)d(of)j FL(M)h FQ(we)f(ha)n(v)o(e:)1438 3338 y FL(u)1491 3304 y FF(>)1491 3359 y FB(k)1547 3338 y FL(M)o(u)1690 3350 y FB(l)1799 3338 y FI(=)1986 3235 y FB(n)1946 3259 y FC(X)1953 3436 y FB(i)p FG(=1)2090 3282 y FL(u)2143 3252 y FF(>)2143 3306 y FB(k)2199 3282 y FL(z)2242 3294 y FB(i)2269 3282 y FL(z)2312 3252 y FF(>)2312 3304 y FB(i)2368 3282 y FL(u)2421 3294 y FB(l)p 2090 3319 V 2225 3395 a FH(w)2284 3407 y FB(i)1799 3535 y FI(=)82 b FH(\025)1994 3547 y FB(k)2036 3535 y FH(\016)2073 3547 y FB(k)q(l)2135 3535 y FH(:)515 3709 y FQ(and)19 b(thus)i(we)f(obtain)1603 3860 y FI(\000)1655 3872 y FB(f)1721 3860 y FI(=)j FH(\033)1859 3826 y FG(2)1953 3752 y FB(p)1911 3781 y FC(X)1910 3960 y FB(k)q FF(6)p FG(=1)2055 3804 y FL(u)2108 3816 y FB(k)2149 3804 y FL(u)2202 3774 y FF(>)2202 3827 y FB(k)p 2055 3841 203 4 v 2112 3917 a FH(\025)2160 3929 y FB(k)2268 3860 y FH(:)869 b FQ(\(D.19\))515 4199 y FD(D)n(.1)119 b(Bias)29 b(in)i(Linear)f(Estimation)515 4385 y FQ(An)18 b(estimate)g(is)g(statistically)h(biased)e(if)i(the)e(e)o(xpectation)f (of)i(the)g(error)e(is)j(zero,)f(and)f(statistically)515 4484 y(biased)j(otherwise.)k(F)o(ollo)n(wing)19 b([28)n(])i(it)g(can)f (be)g(seen)g(that)g(the)h(true)e(perturbation)f(of)i FL(M)g FQ(is)1386 4658 y FH(\016)s FL(M)i FI(=)h FH(\016)s FL(Z)1725 4622 y FF(>)1781 4658 y FL(Z)p 1781 4671 59 4 v 19 w FI(+)18 b FL(Z)p 1941 4671 V -36 x FF(>)2056 4658 y FH(\016)s FL(Z)g FI(+)g FH(\016)s FL(Z)2354 4622 y FF(>)2410 4658 y FH(\016)s FL(Z)652 b FQ(\(D.20\))515 4832 y(rather)19 b(than)h(that)g(gi)n(v)o(en)f(in)h(Equation)f (\(D.7\),)g(and)h(its)h(e)o(xpectation)d(is)j(not)f(zero:)1610 5006 y FA(E)7 b FI(\()p FH(\016)s FL(M)p FI(\))23 b(=)g FH(\016)s FL(Z)2065 4970 y FF(>)2121 5006 y FH(\016)s FL(Z)42 b FH(:)876 b FQ(\(D.21\))p eop %%Page: 77 79 77 78 bop 515 232 a FJ(APPENDIX)20 b(D.)41 b(V)-11 b(ARIANCE)21 b(OF)f(RESIDU)m(ALS)1209 b FQ(77)639 523 y(Kanatani)18 b(suggests)g(\(in)g(the)g(case)g(of)g(conics)g([28)n(]\))g(that)h(the)f (bias)g(may)g(be)g(remo)o(v)o(ed)d(by)j(sub-)515 623 y(tracting)e(the)h(bias)g(term)f FA(E)8 b FI(\()p FH(\016)s FL(M)p FI(\))17 b FQ(from)f(the)h(moment)f(matrix)g(prior)g(to)h (\002nding)e(the)i(eigen)m(v)o(ectors.)515 722 y(In)26 b(other)f(w)o(ords)g(taking)h(as)g(our)g(solution)f(the)h(eigen)m(v)o (ector)d(corresponding)g(to)j(the)g(minimum)515 822 y(eigen)m(v)n(alue) 18 b(of)1695 922 y FL(M)g FA(\000)g(E)8 b FI(\()p FH(\016)s FL(M)p FI(\))41 b FH(:)962 b FQ(\(D.22\))p eop %%Page: 78 80 78 79 bop 515 1187 a FN(Bibliograph)m(y)556 1619 y FQ([1])41 b(P)-9 b(.)21 b(A.)g(Beardsle)o(y)-5 b(.)32 b Ft(Applications)19 b(of)i(pr)l(ojective)g(g)o(eometry)g(to)h(r)l(obot)f(vision)p FQ(.)32 b(PhD)21 b(thesis,)695 1719 y(Oxford)d(Uni)n(v)o(ersity)-5 b(,)18 b(1992.)556 1885 y([2])41 b(P)-9 b(.)18 b(A.)g(Beardsle)o(y)-5 b(,)16 b(A.)i(Zisserman,)g(and)f(D.)h(W)-8 b(.)19 b(Murray)-5 b(.)21 b(Sequential)c(update)g(of)g(projecti)n(v)o(e)695 1984 y(and)j(af)n(\002ne)g(structure)g(from)f(motion.)30 b(T)-6 b(echnical)19 b(Report)h(OUEL)h(2012/94,)d(Dept)j(of)f(Eng)695 2084 y(Science,)f(Uni)n(v)o(ersity)g(of)h(Oxford,)e(1994.)556 2250 y([3])41 b(P)-9 b(.)24 b(A.)h(Beardsle)o(y)-5 b(,)24 b(A.)h(Zisserman,)f(and)g(D.)h(W)-8 b(.)25 b(Murray)-5 b(.)41 b(Sequential)23 b(update)h(of)g(projec-)695 2350 y(ti)n(v)o(e)d(and)f(af)n(\002ne)h(structure)g(from)f(motion.)32 b Ft(International)19 b(J)n(ournal)h(of)h(Computer)g(V)-6 b(ision)p FQ(,)695 2449 y(23\(3\):235\226259,)15 b(1997.)556 2615 y([4])41 b(D.A.)24 b(Belsle)o(y)-5 b(.)45 b Ft(Conditioning)23 b(Dia)o(gnostics:)33 b(Collinearity)25 b(and)f(weak)h(data)g(in)g(r)m (e)m(gr)m(es-)695 2715 y(sion)p FQ(.)k(W)m(ile)o(y)-5 b(,)19 b(1991.)556 2881 y([5])41 b(S.)24 b(Birch\002eld)g(and)f(C.)i(T) -7 b(omasi.)42 b(A)24 b(pix)o(el)g(dissimilarity)f(measure)h(that)g(is) h(insensiti)n(v)o(e)e(to)695 2980 y(image)c(sampling.)28 b Ft(IEEE)20 b(P)-7 b(AMI)p FQ(,)19 b(v)n(ol.20\(4\):401\226405,)14 b(1998.)556 3147 y([6])41 b(F)-7 b(.)21 b(Bookstein.)30 b(Fitting)21 b(conic)f(sections)h(to)g(scattered)f(data.)31 b Ft(Computer)20 b(V)-6 b(ision)20 b(Gr)o(aphics)695 3246 y(and)f(Ima)o(g)o(e)g(Pr)l(ocessing)p FQ(,)h(9:56\22671,)e(1979.) 556 3412 y([7])41 b(M.)25 b(Brooks,)h(L.)g(De)g(Agapito,)g(D.)f(Huynh,) h(and)f(L.)g(Baumela.)47 b(T)-7 b(o)n(w)o(ards)25 b(rob)n(ust)g(metric) 695 3512 y(reconstruction)d(via)k(a)f(dynamic)f(uncalibrated)f(stereo)i (head.)45 b Ft(Ima)o(g)o(e)24 b(and)h(V)-6 b(ision)25 b(Com-)695 3611 y(puting)p FQ(,)18 b(16\(14\):989\2261002,)c(1998.)556 3778 y([8])41 b(S.)30 b(Chaterjee)g(and)g(A.)g(S.)h(Hadi.)61 b Ft(Sensitivity)30 b(Analysis)g(in)h(Linear)f(Re)m(gr)m(ession)p FQ(.)60 b(John)695 3877 y(W)m(ile)o(y)-5 b(,)19 b(Ne)n(w)h(Y)-9 b(ork,)19 b(1988.)556 4043 y([9])41 b(W)-8 b(.)25 b(Chojnacki,)f(M.)h (J.)g(Brooks,)f(A.)h(v)n(an)f(den)f(Hengel,)i(and)f(D.)g(Ga)o(wle)o(y) -5 b(.)42 b(On)24 b(the)h(\002tting)695 4143 y(of)c(surf)o(aces)h(to)f (data)h(with)g(co)o(v)n(ariances.)32 b Ft(IEEE)21 b(T)-5 b(r)o(ans.)22 b(P)-7 b(attern)22 b(Analysis)g(and)f(Mac)o(hine)695 4242 y(Intellig)o(ence)p FQ(,)d(22\(2\):1294\2261303,)c(2000.)515 4408 y([10])40 b(A.)18 b(P)-9 b(.)18 b(Dempster)m(,)g(N.)g(M.)g(Laird,) g(and)g(D.)g(B.)h(Rubin.)24 b(Maximum)17 b(lik)o(elihood)g(from)g (incom-)695 4508 y(plete)j(data)g(via)g(the)g(em)g(algorithm.)28 b Ft(J)n(.)20 b(R.)g(Statist.)h(Soc.)p FQ(,)e(39)h(B:1\22638,)f(1977.) 515 4674 y([11])40 b(R.)30 b(Deriche,)g(Z.)g(Zhang,)g(Q.)f(T)-6 b(.)30 b(Luong,)f(and)g(O.)h(F)o(augeras.)57 b(Rob)n(ust)30 b(reco)o(v)o(ery)c(of)k(the)695 4774 y(epipolar)19 b(geometry)g(for)i (an)g(uncalibrated)e(stereo)i(rig.)31 b(In)21 b(J.)h(O.)f(Ecklundh,)e (editor)m(,)h Ft(Pr)l(oc.)695 4873 y(3r)m(d)f(Eur)l(opean)f(Confer)m (ence)h(on)g(Computer)g(V)-6 b(ision,)19 b(LNCS)i(800/801,)c(Stoc)n (kholm)p FQ(,)h(pages)695 4973 y(567\226576.)f(Springer)n(-V)-9 b(erlag,)17 b(1994.)1905 5255 y(78)p eop %%Page: 79 81 79 80 bop 515 232 a FJ(BIBLIOGRAPHY)2163 b FQ(79)515 523 y([12])40 b(O.D.)18 b(F)o(augeras.)25 b(What)19 b(can)g(be)g(seen)g (in)g(three)f(dimensions)f(with)i(an)g(uncalibrated)e(stereo)695 623 y(rig?)43 b(In)22 b(G.)g(Sandini,)g(editor)m(,)f Ft(Pr)l(oc.)h(2nd)f(Eur)l(opean)g(Confer)m(ence)g(on)g(Computer)h(V)-6 b(ision,)695 722 y(LNCS)21 b(588,)e(Santa)g(Mar)m(gherita)g(Ligur)m(e)p FQ(,)h(pages)g(563\226578.)d(Springer)n(\226V)-9 b(erlag,)17 b(1992.)515 888 y([13])40 b(O.D.)28 b(F)o(augeras.)55 b Ft(Thr)m(ee-Dimensional)26 b(Computer)j(V)-6 b(ision:)41 b(A)28 b(Geometric)h(V)-6 b(ie)o(wpoint)p FQ(.)695 988 y(The)19 b(MIT)h(Press,)h(1993.)515 1154 y([14])40 b(M.)28 b(Fischler)h(and)f(R.)h(Bolles.)57 b(Random)28 b(sample)g(consensus:)42 b(a)29 b(paradigm)d(for)i(model)695 1254 y(\002tting)23 b(with)h(application)f(to)h(image)f(analysis)h(and)f(automated)f (cartography)-5 b(.)37 b Ft(Commun.)695 1353 y(Assoc.)20 b(Comp.)g(Mac)o(h.)p FQ(,)f(v)n(ol.)h(24:381\22695,)c(1981.)515 1519 y([15])40 b(A.)27 b(W)-8 b(.)29 b(Fitzgibbon,)f(M.)f(Pilu,)j(and)d (R.)h(B.)h(Fisher)-5 b(.)53 b(Direct)28 b(least-squares)f(\002tting)g (of)g(el-)695 1619 y(lipses.)44 b Ft(IEEE)24 b(T)-5 b(r)o(ans.)25 b(P)-7 b(attern)25 b(Analysis)f(and)g(Mac)o(hine)f(Intellig)o(ence)p FQ(,)i(21\(5\):476\226480,)695 1719 y(May)19 b(1999.)515 1885 y([16])40 b(G.H.)20 b(Golub)g(and)f(R.)j(Underw)o(ood.)27 b(Stationary)19 b(v)n(alues)h(of)h(the)f(ratio)g(of)g(quadratic)f (forms)695 1984 y(subject)h(to)g(linear)g(constraints.)28 b Ft(Z.)21 b(Ang)o(e)o(w)-6 b(.)19 b(Math.)h(Phys.)p FQ(,)f(21:318\226326,)e(1970.)515 2150 y([17])40 b(G.H.)24 b(Golub)f(and)g(C.F)-7 b(.)25 b(V)-9 b(an)24 b(Loan.)41 b Ft(Matrix)25 b(Computations)p FQ(.)40 b(The)23 b(John)h(Hopkins)f (Uni-)695 2250 y(v)o(ersity)c(Press,)i(1989.)515 2416 y([18])40 b(C.)20 b(Harris)f(and)g(M.)h(Stephens.)26 b(A)20 b(combined)e(corner)g(and)h(edge)g(detector)-5 b(.)26 b(In)20 b Ft(Pr)l(oc.)f(Alve)n(y)695 2516 y(Conf)o(.)p FQ(,)g(pages)h(189\226192,)d(1987.)515 2682 y([19])40 b(R.)26 b(I.)f(Hartle)o(y)-5 b(.)46 b(Estimation)25 b(of)g(relati)n(v)o (e)g(camera)g(positions)g(for)g(uncalibrated)e(cameras.)695 2781 y(In)17 b(G.)h(Sandini,)f(editor)m(,)g Ft(Pr)l(oc.)h(2nd)f(Eur)l (opean)e(Confer)m(ence)i(on)h(Computer)f(V)-6 b(ision,)18 b(LNCS)695 2881 y(588,)h(Santa)f(Mar)m(gherita)i(Ligur)m(e)p FQ(,)g(pages)g(579\22687.)d(Springer)n(\226V)-9 b(erlag,)17 b(1992.)515 3047 y([20])40 b(R.)c(I.)g(Hartle)o(y)-5 b(.)79 b(Estimation)35 b(of)h(relati)n(v)o(e)f(camera)g(positions)h (for)f(uncalibrated)f(cam-)695 3147 y(eras.)39 b(In)23 b Ft(Pr)l(oc.)g(2nd)f(Eur)l(opean)g(Confer)m(ence)g(on)h(Computer)g(V) -6 b(ision,)24 b(LNCS)g(588,)f(Santa)695 3246 y(Mar)m(gherita)c(Ligur)m (e)p FQ(,)h(pages)g(579\226587.)d(Springer)n(-V)-9 b(erlag,)17 b(1992.)515 3412 y([21])40 b(R.)20 b(I.)f(Hartle)o(y)-5 b(.)27 b(Cheirality)20 b(in)m(v)n(ariants.)26 b(In)19 b Ft(Pr)l(oc.)h(D)m(ARP)-7 b(A)19 b(Ima)o(g)o(e)g(Under)o(standing)e(W) -8 b(ork-)695 3512 y(shop)p FQ(,)19 b(pages)h(745\226753,)d(1993.)515 3678 y([22])40 b(R.)28 b(I.)g(Hartle)o(y)-5 b(.)54 b(In)28 b(defence)e(of)i(the)g(8-point)f(algorithm.)53 b(In)28 b Ft(Pr)l(oc.)g(5th)g(Int'l)f(Conf)o(.)h(on)695 3778 y(Computer)19 b(V)-6 b(ision,)20 b(Boston)p FQ(,)f(pages)h (1064\2261075,)c(1995.)515 3944 y([23])40 b(R.)18 b(I.)g(Hartle)o(y)f (and)h(P)-9 b(.)18 b(Sturm.)24 b(T)m(riangulation.)d(In)d Ft(D)m(ARP)-7 b(A)17 b(Ima)o(g)o(e)g(Under)o(standing)f(W)-8 b(ork-)695 4043 y(shop,)19 b(Monter)m(e)n(y)-5 b(,)19 b(CA)p FQ(,)i(pages)f(957\226966,)d(1994.)515 4209 y([24])40 b(R.)29 b(I.)f(Hartle)o(y)g(and)g(P)-9 b(.)29 b(Sturm.)55 b(T)m(riangulation.)f(In)28 b Ft(American)g(Ima)o(g)o(e)g(Under)o (standing)695 4309 y(W)-8 b(orkshop)p FQ(,)19 b(pages)h(957\226966,)d (1994.)515 4475 y([25])40 b(R.)29 b(I.)g(Hartle)o(y)g(and)f(A.)h (Zisserman.)57 b Ft(Multiple)29 b(V)-6 b(ie)o(w)29 b(Geometry)g(in)h (Computer)e(V)-6 b(ision)p FQ(.)695 4575 y(Cambridge)18 b(Uni)n(v)o(ersity)h(Press,)i(ISBN:)g(0521623049,)16 b(2000.)515 4741 y([26])40 b(P)-9 b(.)20 b(J.)h(Huber)-5 b(.)28 b(Projection)19 b(pursuit.)29 b Ft(Annals)19 b(of)h(Statistics)p FQ(,)h(13:433\226475,)16 b(1985.)515 4907 y([27])40 b(K.)29 b(Kanatani.)58 b Ft(Geometric)30 b(Computation)d(for)j(Mac)o(hine)f(V) -6 b(ision)p FQ(.)58 b(Oxford)28 b(Uni)n(v)o(ersity)695 5006 y(Press,)20 b(Oxford,)f(1992.)p eop %%Page: 80 82 80 81 bop 515 232 a FJ(BIBLIOGRAPHY)2163 b FQ(80)515 523 y([28])40 b(K.)17 b(Kanatani.)23 b(Renormalization)15 b(for)i(unbiased)f(estimation.)23 b(In)17 b Ft(Pr)l(oc.)h(4th)f(Int'l)g (Conf)o(.)g(on)695 623 y(Computer)24 b(V)-6 b(ision,)25 b(Berlin)p FQ(,)g(pages)f(599\226606,)f(Los)i(Alamitos,)g(CA,)g(1993.)e (IEEE)h(Com-)695 722 y(puter)19 b(Society)h(Press.)515 888 y([29])40 b(K.)28 b(Kanatani.)52 b(Automatic)27 b(singularity)g (test)h(for)f(motion)g(analysis)h(by)f(an)h(information)695 988 y(criterion.)48 b(In)27 b Ft(Pr)l(oc.)g(4th)f(Eur)l(opean)f(Confer) m(ence)h(on)g(Computer)h(V)-6 b(ision,)27 b(LNCS)h(1064,)695 1088 y(Cambridg)o(e)p FQ(,)19 b(pages)h(697\226708,)d(Springer)n(\226V) -9 b(erlag,)17 b(1996.)h(Buxton,)h(B.)i(and)f(Cipolla)g(R.)515 1254 y([30])40 b(M.)21 b(G.)g(K)n(endall)f(and)g(A.)h(Stuart.)32 b Ft(The)21 b(Advanced)e(Theory)h(of)h(Statistics)p FQ(.)32 b(Charles)21 b(Grif)n(\002n)695 1353 y(and)e(Compan)o(y)-5 b(,)18 b(London,)g(1983.)515 1519 y([31])40 b(H.C.)28 b(Longuet-Higgins.)53 b(A)29 b(computer)d(algorithm)h(for)h (reconstructing)e(a)j(scene)g(from)695 1619 y(tw)o(o)20 b(projections.)28 b Ft(Natur)m(e)p FQ(,)20 b(v)n(ol.293:133\226135,)15 b(1981.)515 1785 y([32])40 b(Q.)24 b(T)-6 b(.)24 b(Luong,)g(R.)h (Deriche,)f(O.)h(D.)f(F)o(augeras,)g(and)g(T)-6 b(.)24 b(P)o(apadopoulo.)39 b(On)25 b(determining)695 1885 y(the)f (fundamental)f(matrix:)33 b(analysis)25 b(of)f(dif)n(ferent)f(methods)h (and)g(e)o(xperimental)e(results.)695 1984 y(T)-6 b(echnical)19 b(Report)h(1894,)e(INRIA)i(\(Sophia)f(Antipolis\),)h(1993.)515 2150 y([33])40 b(S.J.)24 b(Maybank.)41 b Ft(Theory)24 b(of)h(Reconstruction)d(F)-5 b(r)l(om)25 b(Ima)o(g)o(e)f(Motion)p FQ(.)42 b(Springer)n(-V)-9 b(erlag,)695 2250 y(Berlin,)20 b(1993.)515 2416 y([34])40 b(G.I.)23 b(McLachlan)f(and)h(K.)g(Basford.) 39 b Ft(Mixtur)m(e)24 b(models:)31 b(infer)m(ence)23 b(and)g(applications)e(to)695 2516 y(clustering)p FQ(.)28 b(Marcel)20 b(Dekk)o(er)-5 b(.)20 b(Ne)n(w)g(Y)-9 b(ork,)19 b(1988.)515 2682 y([35])40 b(J.)28 b(Mundy)f(and)h(A.)g(Zisserman.)55 b Ft(Geometric)28 b(In)m(variance)e(in)i(Computer)g(V)-6 b(ision)p FQ(.)55 b(MIT)695 2781 y(Press,)20 b(1992.)515 2947 y([36])40 b(S.)31 b(I.)g(Olsen.)64 b(Epipolar)29 b(line)i(estimation.)63 b(In)31 b(G.)g(Sandini,)i(editor)m(,)f Ft(Pr)l(oc.)f(2nd)f(Eur)l(o-)695 3047 y(pean)14 b(Confer)m(ence)g(on)h (Computer)g(V)-6 b(ision,)15 b(LNCS)i(588,)e(Santa)f(Mar)m(gherita)g (Ligur)m(e)p FQ(,)i(pages)695 3147 y(307\226311.)h(Springer)n(\226V)-9 b(erlag,)17 b(1992.)515 3313 y([37])40 b(K.)16 b(Pearson.)k(On)d(lines) f(and)g(planes)g(of)g(closest)h(\002t)g(to)g(systems)g(of)f(points)g (in)g(space.)21 b Ft(Philos.)695 3412 y(Ma)o(g)o(.)e(Ser)-9 b(.)20 b(6)p FQ(,)g(2:559,)f(1901.)515 3578 y([38])40 b(V)-11 b(.)26 b(Pratt.)47 b(Direct)25 b(least)i(squares)e(\002tting)g (of)h(algebraic)e(surf)o(aces.)46 b Ft(Computer)25 b(Gr)o(aphics)p FQ(,)695 3678 y(21\(4\):145\226152,)15 b(1987.)515 3844 y([39])40 b(J.O.)20 b(Ra)o(wlings.)28 b Ft(Applied)19 b(Re)m(gr)m(ession)g(Analysis)p FQ(.)28 b(W)-7 b(adsw)o(orth)20 b(and)f(Brooks,)h(California,)695 3944 y(1988.)515 4110 y([40])40 b(P)-9 b(.)27 b(J.)h(Rousseeuw)-5 b(.)51 b Ft(Rob)n(ust)27 b(Re)m(gr)m(ession)g(and)f(Outlier)i(Detection)p FQ(.)51 b(W)m(ile)o(y)-5 b(,)28 b(Ne)n(w)g(Y)-9 b(ork,)695 4209 y(1987.)515 4375 y([41])40 b(P)-9 b(.D.)28 b(Sampson.)57 b(Fitting)29 b(conic)f(sections)h(to)g(`v)o(ery)f(scattered')g(data:)43 b(An)29 b(iterati)n(v)o(e)f(re-)695 4475 y(\002nement)i(of)h(the)g (Bookstein)g(algorithm.)63 b Ft(Computer)30 b(V)-6 b(ision,)34 b(Gr)o(aphics,)f(and)d(Ima)o(g)o(e)695 4575 y(Pr)l(ocessing)p FQ(,)19 b(18:97\226108,)e(1982.)515 4741 y([42])40 b(D.)26 b(Scharstein)g(and)g(R.)i(Szeliski.)50 b(A)27 b(taxonomy)d(and)i(e)n(v) n(aluation)f(of)h(dense)g(tw)o(o-frame)695 4840 y(stereo)32 b(correspondence)d(algorithms.)67 b Ft(IJCV)p FQ(,)33 b(47\(1\):7\22642,)e(2002.)68 b(Ev)n(aluation)30 b(page)695 4940 y(http://www)-5 b(.middleb)n(ury)g(.edu/stere)o(o/e)n(v)m(al/.)p eop %%Page: 81 83 81 82 bop 515 232 a FJ(BIBLIOGRAPHY)2163 b FQ(81)515 523 y([43])40 b(J.)22 b(G.)h(Semple)e(and)h(G.)g(T)-6 b(.)22 b(Kneebone.)33 b Ft(Alg)o(ebr)o(aic)21 b(Pr)l(ojective)h (Geometry)p FQ(.)35 b(Oxford)21 b(Uni-)695 623 y(v)o(ersity)e(Press,)i (Great)f(Britain,)g(1952.)515 789 y([44])40 b(L.)21 b(S.)h(Shapiro.)32 b Ft(Af)o(\002ne)20 b(Analysis)i(of)f(Ima)o(g)o(e)g(Sequences)p FQ(.)32 b(Cambridge)20 b(Uni)n(v)o(ersity)g(Press,)695 888 y(Cambridge,)e(England,)g(1995.)515 1054 y([45])40 b(L.S.)25 b(Shapiro)g(and)g(J.M.)h(Brady)-5 b(.)46 b(Rejecting)25 b(outliers)h(and)f(estimating)g(errors)g(in)h(an)g(or)n(-)695 1154 y(thogonal)18 b(re)o(gression)h(frame)n(w)o(ork.)27 b(Oxford)18 b(T)-6 b(ech)20 b(Report)g(OUEL)g(1974/93,)d(1993.)515 1320 y([46])40 b(C.)26 b(V)-11 b(.)27 b(Ste)n(w)o(art.)47 b(Bias)27 b(in)f(rob)n(ust)g(estimation)f(caused)h(by)f (discontinuities)g(and)g(multiple)695 1420 y(structures.)d Ft(IEEE)16 b(T)-5 b(r)o(ans.)18 b(on)f(P)-7 b(attern)17 b(Analysis)g(and)g(Mac)o(hine)f(Intellig)o(ence)p FQ(,)g(v)n(ol.P)-8 b(AMI-)695 1519 y(19,no.8:818\226833)o(,)15 b(1997.)515 1685 y([47])40 b(P)-9 b(.)22 b(Sturm.)36 b(On)22 b(focal)g(length)f (calibration)g(from)g(tw)o(o)i(vie)n(ws.)36 b(pages)22 b(145\226150,)e(booktitle)695 1785 y(=.)515 1951 y([48])40 b(R.)19 b(A.)g(Thisted.)26 b Ft(Elements)18 b(of)h(Statistical)g (Computing)p FQ(.)25 b(Chapman)17 b(and)h(Hall,)i(Ne)n(w)f(Y)-9 b(ork,)695 2051 y(1988.)515 2217 y([49])40 b(P)-9 b(.)23 b(H.)g(S.)h(T)-7 b(orr)i(.)39 b Ft(Outlier)23 b(Detection)g(and)f (Motion)g(Se)m(gmentation)p FQ(.)36 b(PhD)24 b(thesis,)g(Dept.)f(of)695 2316 y(Engineering)17 b(Science,)j(Uni)n(v)o(ersity)f(of)h(Oxford,)e (1995.)515 2482 y([50])40 b(P)-9 b(.)19 b(H.)h(S.)g(T)-7 b(orr)i(.)27 b(Bayesian)19 b(model)f(estimation)h(and)g(selection)g (for)g(epipolar)f(geometry)f(and)695 2582 y(generic)i(manifold)f (\002tting.)29 b Ft(IJCV)p FQ(,)20 b(?:?,)h(2002.)515 2748 y([51])40 b(P)-9 b(.)28 b(H.)h(S.)g(T)-7 b(orr)m(,)29 b(P)-9 b(.)29 b(A.)f(Beardsle)o(y)-5 b(,)29 b(and)f(D.)h(W)-8 b(.)29 b(Murray)-5 b(.)54 b(Rob)n(ust)28 b(vision.)55 b(In)28 b(J.)h(Illing-)695 2848 y(w)o(orth,)17 b(editor)m(,)g Ft(Pr)l(oc.)g(5th)h(British)g(Mac)o(hine)e(V)-6 b(ision)18 b(Confer)m(ence)o(,)f(Y)-8 b(ork)p FQ(,)18 b(pages)g(145\226155.)695 2947 y(BMV)-11 b(A)20 b(Press,)h(1994.)515 3113 y([52])40 b(P)-9 b(.)24 b(H.)g(S.)h(T)-7 b(orr)23 b(and)h(D.)g(W)-8 b(.)25 b(Murray)-5 b(.)40 b(Outlier)24 b(detection)f(and)g(motion)g(se) o(gmentation.)40 b(In)695 3213 y(P)-9 b(.)32 b(S.)g(Schenk)o(er)m(,)h (editor)m(,)g Ft(Sensor)f(Fusion)f(VI)p FQ(,)g(pages)h(432\226443.)c (SPIE)k(v)n(olume)f(2059,)695 3313 y(1993.)c(Boston.)515 3479 y([53])40 b(P)-9 b(.)31 b(H.)g(S.)h(T)-7 b(orr)30 b(and)h(D.)g(W)-8 b(.)33 b(Murray)-5 b(.)62 b(The)30 b(de)n(v)o(elopment)f(and)h(comparison)f(of)i(rob)n(ust)695 3578 y(methods)19 b(for)g(estimating)h(the)g(fundamental)e(matrix.)28 b Ft(IJCV)p FQ(,)20 b(24\(3\):271\226300,)15 b(1997.)515 3744 y([54])40 b(P)-9 b(.)31 b(H.)g(S.)h(T)-7 b(orr)30 b(and)h(D.)g(W)-8 b(.)33 b(Murray)-5 b(.)62 b(The)30 b(de)n(v)o(elopment)f(and)h(comparison)f(of)i(rob)n(ust)695 3844 y(methods)20 b(for)h(estimating)g(the)h(fundamental)d(matrix.)33 b Ft(Int)22 b(J)n(ournal)e(of)i(Computer)f(V)-6 b(ision)p FQ(,)695 3944 y(24\(3\):271\226300,)15 b(1997.)515 4110 y([55])40 b(P)-9 b(.)16 b(H.)h(S.)g(T)-7 b(orr)16 b(and)g(A.)h (Zisserman.)j(Rob)n(ust)d(computation)d(and)i(parametrization)e(of)i (multi-)695 4209 y(ple)j(vie)n(w)g(relations.)26 b(In)19 b(U)g(Desai,)h(editor)m(,)e Ft(ICCV6)p FQ(,)h(pages)g(727\226732.)d (Narosa)j(Publishing)695 4309 y(House,)g(1998.)515 4475 y([56])40 b(P)-9 b(.)30 b(H.)h(S.)g(T)-7 b(orr)m(,)31 b(A)g(Zisserman,)h(and)e(S.)h(Maybank.)60 b(Rob)n(ust)30 b(detection)g(of)g(de)o(generate)695 4575 y(con\002gurations)17 b(for)j(the)g(fundamental)e(matrix.)28 b Ft(CVIU)p FQ(,)21 b(71\(3\):312\226333,)15 b(1998.)515 4741 y([57])40 b(P)-9 b(.H.S.)18 b(T)-7 b(orr)i(.)27 b(An)19 b(assessment)g(of)g(information) e(criteria)h(for)h(motion)f(model)g(selection.)26 b(In)695 4840 y Ft(CVPR97)p FQ(,)19 b(pages)g(47\22653,)g(1997.)p eop %%Page: 82 84 82 83 bop 515 232 a FJ(BIBLIOGRAPHY)2163 b FQ(82)515 523 y([58])40 b(R.Y)-11 b(.)23 b(Tsai)g(and)f(T)-6 b(.S.)22 b(Huang.)36 b(Uniqueness)21 b(and)h(estimation)g(of)h (three-dimensional)c(mo-)695 623 y(tion)g(parameters)g(of)h(rigid)g (objects)g(with)g(curv)o(ed)f(surf)o(aces.)28 b(6:13\22627,)18 b(1984.)515 789 y([59])40 b(S.)29 b(A.)h(T)-6 b(euk)o(olsk)o(y)28 b(W)-8 b(.)30 b(H.)g(Press,)i(B.)e(P)-9 b(.)30 b(Flannery)e(and)h(W)-8 b(.)31 b(T)-6 b(.)29 b(V)-9 b(etterling.)57 b Ft(Numerical)695 888 y(Recipes)21 b(in)h(C,)h(the)f(art)g(of)g(scienti\002c)f(computing) p FQ(.)33 b(Cambridge)20 b(Uni)n(v)o(ersity)h(Press,)i(Cam-)695 988 y(bridge,)18 b(1988.)515 1154 y([60])40 b(J.)17 b(W)-7 b(eng,)18 b(N.)f(Ahuja,)g(and)f(T)-6 b(.)17 b(Huang.)k(Optimal)c (motion)f(and)g(structure)g(estimation.)22 b Ft(IEEE)695 1254 y(P)-7 b(AMI)p FQ(,)19 b(v)n(ol.15\(9\):864\226884,)14 b(1993.)515 1420 y([61])40 b(J.)24 b(W)-7 b(eng,)24 b(T)-6 b(.S.)23 b(Huang,)g(and)g(N.)g(Ahuja.)39 b(Motion)23 b(and)g(structure)f(from)h(tw)o(o)g(perspecti)n(v)o(e)695 1519 y(vie)n(ws:)i(Algorithms,)19 b(error)g(analysis,)h(and)f(error)h (estimation.)28 b(11:451\226476,)16 b(1989.)515 1685 y([62])40 b(M.)17 b(W)-7 b(erman)18 b(and)f(D.)h(K)n(eren.)k(A)c (bayesian)f(method)f(for)h(\002tting)h(parametric)e(and)h(nonpara-)695 1785 y(metric)i(model)h(to)g(noisy)g(data.)29 b Ft(IEEE)19 b(P)-7 b(AMI)p FQ(,)20 b(v)n(ol.23\(5\):528\226534)o(,)15 b(2001.)515 1951 y([63])40 b(G.)21 b(Xu)h(and)f(Z.)g(Zhang.)32 b Ft(Epipolar)20 b(Geometry)i(in)f(Ster)m(eo,)g(Motion)g(and)g(Object)g (Reco)o(gni-)695 2051 y(tion)p FQ(.)28 b(1996.)515 2217 y([64])40 b(Z.)15 b(Zhang.)i(Determining)d(the)h(epipolar)f(geometry)g (and)h(its)h(uncertainty:)21 b(A)16 b(re)n(vie)n(w)-5 b(.)18 b Ft(IJCV)p FQ(,)695 2316 y(27\(2\):161\226195,)d(1997.)p eop %%Page: 83 85 83 84 bop 515 994 a FN(Index)515 1412 y FO(torr)p 720 1412 25 4 v 29 w(self)p 949 1412 V 29 w(calib)p 1228 1412 V 29 w(f,)49 b(54)515 1512 y(torr)p 720 1512 V 29 w(test)p 949 1512 V 29 w(calib)p 1228 1512 V 29 w(sc,)g(56)515 1611 y(torr)p 720 1611 V 29 w(mapsac)p 1049 1611 V 29 w(F,)g(43)515 1711 y(torr)p 720 1711 V 29 w(F)p 799 1711 V 30 w(constrained)p 1379 1711 V 27 w(fit,)g(42)515 1811 y(torr)p 720 1811 V 29 w(PfromF,)f(50)515 1910 y(torr)p 720 1910 V 29 w(add)p 899 1910 V 29 w(manual)p 1228 1910 V 29 w(matches,)g(65)515 2010 y(torr)p 720 2010 V 29 w(correctx4F,)g(52)515 2109 y(torr)p 720 2109 V 29 w(display)p 1099 2109 V 28 w(epipoles,)h(16)515 2209 y(torr)p 720 2209 V 29 w(display)p 1099 2209 V 28 w(matches,)g(13)515 2309 y(torr)p 720 2309 V 29 w(display)p 1099 2309 V 28 w(structure,)f(59)515 2408 y(torr)p 720 2408 V 29 w(errf2,)h(33)515 2508 y(torr)p 720 2508 V 29 w(errg)p 949 2508 V 29 w(sse,)g(56)515 2608 y(torr)p 720 2608 V 29 w(evalFsc,)f(36)515 2707 y(torr)p 720 2707 V 29 w(g2F,)h(56)515 2807 y(torr)p 720 2807 V 29 w(gen)p 899 2807 V 29 w(2view)p 1178 2807 V 29 w(matches,)f(61)515 2906 y(torr)p 720 2906 V 29 w(get)p 899 2906 V 29 w(right)p 1178 2906 V 29 w(epipole,)g(22)515 3006 y(torr)p 720 3006 V 29 w(linear)p 1049 3006 V 29 w(EtoPX,)g(55)515 3106 y(torr)p 720 3106 V 29 w(nonlinG,)g(56)515 3205 y(torr)p 720 3205 V 29 w(nonlinf)p 1099 3205 V 28 w(mincon2x2,)g(36)515 3305 y(torr)p 720 3305 V 29 w(skew)p 949 3305 V 29 w(sym,)h(21)515 3405 y(torr)p 720 3405 V 29 w(sphere2unit,)e(56)515 3504 y(torr)p 720 3504 V 29 w(test)p 949 3504 V 29 w(F,)i(16)515 3604 y(torr)p 720 3604 V 29 w(test)p 949 3604 V 29 w(SFMsc,)g(59)515 3703 y(torr)p 720 3703 V 29 w(test)p 949 3703 V 29 w(correct)p 1328 3703 V 29 w(sc,)g(52)515 3803 y(torr)p 720 3803 V 29 w(test)p 949 3803 V 29 w(mat,)g(62)515 3903 y(torr)p 720 3903 V 29 w(triangulate,)e(51)515 4002 y(torr)p 720 4002 V 29 w(unit2sphere,)g(56)515 4185 y(,)i(33)515 4285 y(torr)p 720 4285 V 29 w(estf)p 949 4285 V 29 w(bookstein)p 1428 4285 V 28 w(sampson,)847 4384 y(33)515 4484 y(torr)p 720 4484 V 29 w(estimateF,)f(14)515 4583 y(torr)p 720 4583 V 29 w(matcher)p 1099 4583 V 28 w(script,)h(12)515 4766 y(birch)p 770 4766 V 29 w(match,)f(12)515 4866 y(Birchfield)g(and) h(Tomasi)g(Cor-)847 4965 y(relation,)f(12)2092 1412 y(correlation,)g (11)2092 1595 y(display)p 2447 1595 V 29 w(corners)p 2826 1595 V 28 w(in)p 2954 1595 V 30 w(figure,)g(8)2092 1777 y(feature)h(matching,)f(10)2092 1877 y(Fundamental)g(Matrix,)g(14) 2092 2060 y(Harris)h(corners,)f(7)2092 2242 y(patch)p 2347 2242 V 29 w(match,)h(11)2092 2425 y(sub)g(pixel)g(features,)f(8) 2092 2608 y(torr)p 2297 2608 V 29 w(charris,)h(8)2092 2707 y(torr)p 2297 2707 V 29 w(compare)p 2676 2707 V 29 w(epipoles,)f(17)2092 2807 y(torr)p 2297 2807 V 29 w(cor)p 2476 2807 V 30 w(script,)g(9)2092 2906 y(torr)p 2297 2906 V 29 w(corn)p 2526 2906 V 29 w(matcher,)h(11)2092 3006 y(torr)p 2297 3006 V 29 w(estf,)g(26)2092 3106 y(torr)p 2297 3106 V 29 w(ls,)h(26)2092 3205 y(Torr)p 2297 3205 V 29 w(tool,)f(63)1905 5255 y FQ(83)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF