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 | %------------------------------------------------------------------------ |
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11 | % [U_smooth,U_tps]=tps_coeff(ctrs,U,rho) |
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12 | %------------------------------------------------------------------------ |
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13 | % OUPUT: |
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14 | % U_smooth: values of the quantity U at the N centres after smoothing |
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15 | % U_tps: tps weights of the centres |
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16 | |
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17 | %INPUT: |
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18 | % ctrs: Nxs matrix representing the postions of the M centers, sources of the tps (s=space dimension) |
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19 | % U: Nx1 column vector representing the initial values of the considered scalar at the centres ctrs |
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20 | % rho: smoothing parameter: the result is smoother for larger rho. |
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21 | |
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22 | |
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23 | function [U_smooth,U_tps]=tps_coeff(ctrs,U,rho) |
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24 | %------------------------------------------------------------------------ |
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25 | %rho smoothing parameter |
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26 | % X=reshape(X,[],1); |
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27 | % Y=reshape(Y,[],1); |
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28 | N=size(ctrs,1); |
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29 | % rhs = reshape(U,[],1); |
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30 | U = [U; zeros(3,1)]; |
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31 | % ctrs = [X Y];% coordinates of measurement sites, radial base functions are located at the measurement sites |
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32 | EM = tps_eval(ctrs,ctrs); |
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33 | RhoMat=rho*eye(N,N);% rho=1/(2*omega) , omega given by fasshauer; |
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34 | RhoMat=[RhoMat zeros(N,3)]; |
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35 | PM=[ones(N,1) ctrs]; |
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36 | IM=[EM+RhoMat; [PM' zeros(3,3)]]; |
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37 | U_tps=(IM\U); |
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38 | U_smooth=EM *U_tps; |
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