source: trunk/src/tps_coeff.m @ 1109

Last change on this file since 1109 was 1107, checked in by g7moreau, 2 years ago

Update Copyright to 2022

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[356]1%'tps_coeff': calculate the thin plate spline (tps) coefficients
2% (ref fasshauer@iit.edu MATH 590 ? Chapter 19 32)
3% this interpolation/smoothing minimises a linear combination of the squared curvature
4%  and squared difference form the initial data.
5% This function calculates the weight coefficients U_tps of the N sites where
6% data are known. Interpolated data are then obtained as the matrix product
7% EM*U_tps where the matrix EM is obtained by the function tps_eval.
8% The spatial derivatives are obtained as EMDX*U_tps and EMDY*U_tps, where
9% EMDX and EMDY are obtained from the function tps_eval_dxy.
[581]10% for big data sets, a splitting in subdomains is needed, see functions
11% set_subdomains and tps_coeff_field.
12%
[246]13%------------------------------------------------------------------------
[434]14% [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing)
[246]15%------------------------------------------------------------------------
[811]16% OUTPUT:
17%  U_smooth: values of the quantity U at the N centres after smoothing
18%  U_tps: tps weights of the centres and columns of the linear
[356]19
[811]20% INPUT:
21%  ctrs: NxNbDim matrix  representing the positions of the N centers, sources of the tps (NbDim=space dimension)
22%  U: Nx1 column vector representing the values of the considered scalar measured at the centres ctrs
23%  Smoothing: smoothing parameter: the result is smoother for larger Smoothing.
[581]24%
[811]25% RELATED FUNCTIONS:
26%  tps_eval, tps_eval_dxy
27%  tps_coeff_field, set_subdomains, filter_tps, calc_field
[356]28
[809]29%=======================================================================
[1107]30% Copyright 2008-2022, LEGI UMR 5519 / CNRS UGA G-INP, Grenoble, France
[809]31%   http://www.legi.grenoble-inp.fr
32%   Joel.Sommeria - Joel.Sommeria (A) legi.cnrs.fr
33%
34%     This file is part of the toolbox UVMAT.
35%
36%     UVMAT is free software; you can redistribute it and/or modify
37%     it under the terms of the GNU General Public License as published
38%     by the Free Software Foundation; either version 2 of the license,
39%     or (at your option) any later version.
40%
41%     UVMAT is distributed in the hope that it will be useful,
42%     but WITHOUT ANY WARRANTY; without even the implied warranty of
43%     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
44%     GNU General Public License (see LICENSE.txt) for more details.
45%=======================================================================
46
[434]47function [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing)
[356]48%------------------------------------------------------------------------
[586]49warning off
[581]50N=size(ctrs,1);% nbre of source centres
51NbDim=size(ctrs,2);% space dimension (2 or 3)
52U = [U; zeros(NbDim+1,1)];
[246]53EM = tps_eval(ctrs,ctrs);
[434]54SmoothingMat=Smoothing*eye(N,N);%  Smoothing=1/(2*omega) , omega given by fasshauer;
[581]55SmoothingMat=[SmoothingMat zeros(N,NbDim+1)];
[246]56PM=[ones(N,1) ctrs];
[581]57IM=[EM+SmoothingMat; [PM' zeros(NbDim+1,NbDim+1)]];
[356]58U_tps=(IM\U);
[809]59U_smooth=EM *U_tps;
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