1 | %'tps_eval_dxy': calculate the derivatives of thin plate spline (tps) interpolation at a set of points (limited to the 2D case) |
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2 | %------------------------------------------------------------------------ |
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3 | % function [DMX,DMY] = tps_eval_dxy(dsites,ctrs) |
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4 | %------------------------------------------------------------------------ |
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5 | % OUTPUT: |
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6 | % DMX: Mx(N+3) matrix representing the contributions to the X |
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7 | % derivatives at the M sites from unit sources located at each of the N |
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8 | % centers, + 3 columns representing the contribution of the linear gradient part. |
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9 | % DMY: idem for Y derivatives |
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10 | % |
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11 | % INPUT: |
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12 | % dsites: M x s matrix of interpolation site coordinates (s=space dimension=2 here) |
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13 | % ctrs: N x s matrix of centre coordinates (initial data) |
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14 | % |
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15 | % related functions: |
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16 | % tps_coeff, tps_eval |
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17 | |
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18 | function [DMX,DMY] = tps_eval_dxy(dsites,ctrs) |
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19 | %% matrix declarations |
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20 | [M,s] = size(dsites); [N,s] = size(ctrs); |
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21 | Dsites=zeros(M,N); |
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22 | DM = zeros(M,N); |
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23 | % DMXY = zeros(M,N+1+s); |
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24 | |
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25 | %% Accumulate sum of squares of coordinate differences |
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26 | % The ndgrid command produces two MxN matrices: |
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27 | % Dsites, consisting of N identical columns (each containing |
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28 | % the d-th coordinate of the M interpolation sites) |
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29 | % Ctrs, consisting of M identical rows (each containing |
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30 | % the d-th coordinate of the N centers) |
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31 | |
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32 | [Dsites,Ctrs] = ndgrid(dsites(:,1),ctrs(:,1));%d coordinates of interpolation points (Dsites) and initial points (Ctrs) |
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33 | DX=Dsites-Ctrs; |
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34 | [Dsites,Ctrs] = ndgrid(dsites(:,2),ctrs(:,2));%d coordinates of interpolation points (Dsites) and initial points (Ctrs) |
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35 | DY=Dsites-Ctrs; |
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36 | DM = DX.*DX + DY.*DY;% add d component squared |
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37 | |
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38 | %% calculate matrix of tps derivatives |
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39 | DM(DM~=0) = log(DM(DM~=0))+1; %=2 log(r)+1 derivative of the tps r^2 log(r) |
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40 | |
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41 | DMX=[DX.*DM zeros(M,1) ones(M,1) zeros(M,1)];% effect of mean gradient |
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42 | DMY=[DY.*DM zeros(M,1) ones(M,1) zeros(M,1)];% effect of mean gradient |
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43 | |
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