% 'DXYMatrix': calculate the matrix of thin-plate shell derivatives
%
% function DMXY = DXYMatrix(dsites,ctrs)
%
% INPUT:
% dsites: M x s matrix of interpolation site coordinates (s=space dimension)
% ctrs: N x s matrix of centre coordinates (initial data)
%
% OUTPUT:
% DMXY: Mx(N+1+s)xs matrix corresponding to M interpolation sites and
% N centres, with s=space dimension, DMXY(:,:,k) gives the derivatives
% along dimension k (=x, y,z) after multiplication by the N+1+s tps sources.
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;
[Dsites,Ctrs] = ndgrid(dsites(:,2),ctrs(:,2));%d coordinates of interpolation points (Dsites) and initial points (Ctrs)
DY=Dsites-Ctrs;
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) ones(M,1) zeros(M,1)];% effect of mean gradient