%'tps_coeff': calculate the thin plate spline (tps) coefficients % (ref fasshauer@iit.edu MATH 590 ? Chapter 19 32) % this interpolation/smoothing minimises a linear combination of the squared curvature % and squared difference form the initial data. % This function calculates the weight coefficients U_tps of the N sites where % data are known. Interpolated data are then obtained as the matrix product % EM*U_tps where the matrix EM is obtained by the function tps_eval. % The spatial derivatives are obtained as EMDX*U_tps and EMDY*U_tps, where % EMDX and EMDY are obtained from the function tps_eval_dxy. %------------------------------------------------------------------------ % [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing) %------------------------------------------------------------------------ % OUPUT: % U_smooth: values of the quantity U at the N centres after smoothing % U_tps: tps weights of the centres %INPUT: % ctrs: Nxs matrix representing the postions of the N centers, sources of the tps (s=space dimension) % U: Nx1 column vector representing the values of the considered scalar measured at the centres ctrs % Smoothing: smoothing parameter: the result is smoother for larger Smoothing. function [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing) %------------------------------------------------------------------------ %Smoothing smoothing parameter % X=reshape(X,[],1); % Y=reshape(Y,[],1); N=size(ctrs,1); % rhs = reshape(U,[],1); U = [U; zeros(3,1)]; % ctrs = [X Y];% coordinates of measurement sites, radial base functions are located at the measurement sites EM = tps_eval(ctrs,ctrs); SmoothingMat=Smoothing*eye(N,N);% Smoothing=1/(2*omega) , omega given by fasshauer; SmoothingMat=[SmoothingMat zeros(N,3)]; PM=[ones(N,1) ctrs]; IM=[EM+SmoothingMat; [PM' zeros(3,3)]]; U_tps=(IM\U); U_smooth=EM *U_tps;