%'tps_eval_dxy': calculate the derivatives of thin plate spline (tps) interpolation at a set of points (limited to the 2D case) %------------------------------------------------------------------------ % function [DMX,DMY] = tps_eval_dxy(dsites,ctrs) %------------------------------------------------------------------------ % OUTPUT: % DMX: Mx(N+3) matrix representing the contributions to the X % derivatives at the M sites from unit sources located at each of the N % centers, + 3 columns representing the contribution of the linear gradient part. % DMY: idem for Y derivatives % % INPUT: % dsites: M x s matrix of interpolation site coordinates (s=space dimension=2 here) % ctrs: N x s matrix of centre coordinates (initial data) % % RELATED FUNCTIONS: % tps_coeff, tps_eval % tps_coeff_field, set_subdomains, filter_tps, calc_field %======================================================================= % Copyright 2008-2017, LEGI UMR 5519 / CNRS UGA G-INP, Grenoble, France % http://www.legi.grenoble-inp.fr % Joel.Sommeria - Joel.Sommeria (A) legi.cnrs.fr % % This file is part of the toolbox UVMAT. % % UVMAT is free software; you can redistribute it and/or modify % it under the terms of the GNU General Public License as published % by the Free Software Foundation; either version 2 of the license, % or (at your option) any later version. % % UVMAT is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License (see LICENSE.txt) for more details. %======================================================================= function [DMX,DMY] = tps_eval_dxy(dsites,ctrs) %% matrix declarations [M,s] = size(dsites); [N,s] = size(ctrs); Dsites=zeros(M,N); DM = zeros(M,N); % DMXY = zeros(M,N+1+s); %% Accumulate sum of squares of coordinate differences % The ndgrid command produces two MxN matrices: % Dsites, consisting of N identical columns (each containing % the d-th coordinate of the M interpolation sites) % Ctrs, consisting of M identical rows (each containing % the d-th coordinate of the N centers) [Dsites,Ctrs] = ndgrid(dsites(:,1),ctrs(:,1));%d coordinates of interpolation points (Dsites) and initial points (Ctrs) DX=Dsites-Ctrs;% set of x wise distances between sites and centres [Dsites,Ctrs] = ndgrid(dsites(:,2),ctrs(:,2));%d coordinates of interpolation points (Dsites) and initial points (Ctrs) DY=Dsites-Ctrs;% set of y wise distances between sites and centres DM = DX.*DX + DY.*DY;% add d component squared %% calculate matrix of tps derivatives DM(DM~=0) = log(DM(DM~=0))+1; %=2 log(r)+1 derivative of the tps r^2 log(r) DMX=[DX.*DM zeros(M,1) ones(M,1) zeros(M,1)];% effect of mean gradient DMY=[DY.*DM zeros(M,1) zeros(M,1) ones(M,1)];% effect of mean gradient