[37] | 1 | function [phi,S] = tps(x1,x2,Y) |
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| 2 | % TPS Compute the thin-plate spline basis |
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| 3 | % PHI=TPS(X1,X2,Y) returns the basis PHI of a thin-plate spline |
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| 4 | % (TPS) defined on the domain X1,X2 with control points Y. |
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| 5 | % |
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| 6 | % X1 and X2 are MxN matrices specifying the grid vertices. When |
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| 7 | % warping images, these usually correspond to image pixels. |
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| 8 | % |
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| 9 | % Y is a 2xK matrix specifying the control points, one per |
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| 10 | % column. Ofthen Y is a subset of the domain X1,X2, but this is not |
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| 11 | % required. |
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| 12 | % |
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| 13 | % PHI is a (K+3)xNxM matrix, with one layer per basis element. Each |
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| 14 | % basis element is a function of the domain X1,X2. |
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| 15 | % |
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| 16 | % [PHI,S] = TPS(X1,X2,Y) additionally returns the stiffness matrix S |
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| 17 | % of the TPS. |
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| 18 | % |
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| 19 | % See also WTPS. |
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| 20 | |
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| 21 | % AUTORIGHTS |
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| 22 | % Copyright (C) 2006 Andrea Vedaldi |
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| 23 | % |
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| 24 | % This file is part of VLUtil. |
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| 25 | % |
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| 26 | % VLUtil is free software; you can redistribute it and/or modify |
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| 27 | % it under the terms of the GNU General Public License as published by |
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| 28 | % the Free Software Foundation; either version 2, or (at your option) |
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| 29 | % any later version. |
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| 30 | % |
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| 31 | % This program is distributed in the hope that it will be useful, |
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| 32 | % but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| 33 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| 34 | % GNU General Public License for more details. |
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| 35 | % |
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| 36 | % You should have received a copy of the GNU General Public License |
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| 37 | % along with this program; if not, write to the Free Software Foundation, |
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| 38 | % Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. |
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| 39 | |
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| 40 | X = [x1(:)';x2(:)'] ; |
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| 41 | |
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| 42 | K = size(Y,2) ; |
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| 43 | Q = size(X,2) ; |
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| 44 | U = tpsu(Y,Y) ; |
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| 45 | L = [[ones(1,K); Y], zeros(3) ; U, ones(K,1), Y'] ; |
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| 46 | invL = inv(L) ; |
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| 47 | |
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| 48 | tmp = tpsu(Y,X) ; |
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| 49 | phi = invL * [ ones(1,Q) ; X(1,:) ; X(2,:) ; tmp ] ; |
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| 50 | |
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| 51 | [M,N] = size(x1) ; |
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| 52 | phi = reshape(phi,K+3,M,N) ; |
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| 53 | |
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| 54 | if nargout > 1 |
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| 55 | % See Bookstein; note that here the terms are re-arranged a bit |
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| 56 | invLn = invL(1:K, end-K+1:end) ; |
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| 57 | S = invLn * U * invLn ; |
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| 58 | end |
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