# source:trunk/src/tps_coeff.m@977

Last change on this file since 977 was 977, checked in by g7moreau, 7 years ago
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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.
10% for big data sets, a splitting in subdomains is needed, see functions
11% set_subdomains and tps_coeff_field.
12%
13%------------------------------------------------------------------------
14% [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing)
15%------------------------------------------------------------------------
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
19
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.
24%
25% RELATED FUNCTIONS:
26%  tps_eval, tps_eval_dxy
27%  tps_coeff_field, set_subdomains, filter_tps, calc_field
28
29%=======================================================================
30% Copyright 2008-2017, LEGI UMR 5519 / CNRS UGA G-INP, Grenoble, France
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
47function [U_smooth,U_tps]=tps_coeff(ctrs,U,Smoothing)
48%------------------------------------------------------------------------
49warning off
50N=size(ctrs,1);% nbre of source centres
51NbDim=size(ctrs,2);% space dimension (2 or 3)
52U = [U; zeros(NbDim+1,1)];
53EM = tps_eval(ctrs,ctrs);
54SmoothingMat=Smoothing*eye(N,N);%  Smoothing=1/(2*omega) , omega given by fasshauer;
55SmoothingMat=[SmoothingMat zeros(N,NbDim+1)];
56PM=[ones(N,1) ctrs];
57IM=[EM+SmoothingMat; [PM' zeros(NbDim+1,NbDim+1)]];
58U_tps=(IM\U);
59U_smooth=EM *U_tps;
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