1 | %'tps_coeff': calculate the thin plate spline (tps) coefficients |
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2 | % (ref fasshauer@iit.edu MATH 590 ? Chapter 19 32) |
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3 | % this interpolation/smoothing minimises a linear combination of the squared curvature |
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4 | % and squared difference form the initial data. |
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5 | % This function calculates the weight coefficients U_tps of the N sites where |
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6 | % data are known. Interpolated data are then obtained as the matrix product |
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7 | % EM*U_tps where the matrix EM is obtained by the function tps_eval. |
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8 | % The spatial derivatives are obtained as EMDX*U_tps and EMDY*U_tps, where |
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9 | % EMDX and EMDY are obtained from the function tps_eval_dxy. |
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10 | % for big data sets, a splitting in subdomains is needed, see functions |
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11 | % set_subdomains and tps_coeff_field. |
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12 | % |
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13 | %------------------------------------------------------------------------ |
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14 | % [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing) |
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15 | %------------------------------------------------------------------------ |
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16 | % OUPUT: |
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17 | % U_smooth: values of the quantity U at the N centres after smoothing |
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18 | % U_tps: tps weights of the centres and columns of the linear |
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19 | |
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20 | %INPUT: |
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21 | % ctrs: NxNbDim matrix representing the positions of the N centers, sources of the tps (NbDim=space dimension) |
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22 | % U: Nx1 column vector representing the values of the considered scalar measured at the centres ctrs |
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23 | % Smoothing: smoothing parameter: the result is smoother for larger Smoothing. |
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24 | % |
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25 | %related functions: |
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26 | % tps_eval, tps_eval_dxy |
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27 | % tps_coeff_field, set_subdomains, filter_tps, calc_field |
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28 | |
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29 | function [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing) |
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30 | %------------------------------------------------------------------------ |
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31 | |
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32 | N=size(ctrs,1);% nbre of source centres |
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33 | NbDim=size(ctrs,2);% space dimension (2 or 3) |
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34 | U = [U; zeros(NbDim+1,1)]; |
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35 | EM = tps_eval(ctrs,ctrs); |
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36 | SmoothingMat=Smoothing*eye(N,N);% Smoothing=1/(2*omega) , omega given by fasshauer; |
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37 | SmoothingMat=[SmoothingMat zeros(N,NbDim+1)]; |
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38 | PM=[ones(N,1) ctrs]; |
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39 | IM=[EM+SmoothingMat; [PM' zeros(NbDim+1,NbDim+1)]]; |
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40 | U_tps=(IM\U); |
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41 | U_smooth=EM *U_tps; |
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