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Scripts_Sharing/ISFC/Utils_Graphs/gspbox/sgwt_require/sgwt_kernel_abspline5.m
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sgwt_kernel_abspline5.m
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% sgwt_kernel_abspline5 : Monic polynomial / quintic spline / power law decay kernel
%
% function r = sgwt_kernel_abspline5(x,alpha,beta,t1,t2)
%
% Defines function g(x) with g(x) = c1*x^alpha for 0<x<x1
% g(x) = c3/x^beta for x>t2
% quintic spline for t1<x<t2,
% Satisfying g(t1)=g(t2)=1
% g'(t1)=g'(t2)
% g''(t1)=g''(t2)
%
% Inputs :
% x : array of independent variable values
% alpha : exponent for region near origin
% beta : exponent decay
% t1, t2 : determine transition region
%
% Outputs :
% r - result (same size as x)
%
% Url: http://lts2research.epfl.ch/gsp/doc/sgwt_require/sgwt_kernel_abspline5.php
% Copyright (C) 2013-2016 Nathanael Perraudin, Johan Paratte, David I Shuman.
% This file is part of GSPbox version 0.5.2
%
% This program 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 3 of the License, or
% (at your option) any later version.
%
% This program 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 for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% If you use this toolbox please kindly cite
% N. Perraudin, J. Paratte, D. Shuman, V. Kalofolias, P. Vandergheynst,
% and D. K. Hammond. GSPBOX: A toolbox for signal processing on graphs.
% ArXiv e-prints, Aug. 2014.
% http://arxiv.org/abs/1408.5781
% This file is part of the SGWT toolbox (Spectral Graph Wavelet Transform toolbox)
% Copyright (C) 2010, David K. Hammond.
%
% The SGWT toolbox 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 3 of the License, or
% (at your option) any later version.
%
% The SGWT toolbox 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 for more details.
%
% You should have received a copy of the GNU General Public License
% along with the SGWT toolbox. If not, see <http://www.gnu.org/licenses/>.
function r = sgwt_kernel_abspline5(x,alpha,beta,t1,t2)
r=zeros(size(x));
% compute spline coefficients
% M a = v
M=[[1 t1 t1^2 t1^3 t1^4 t1^5];...
[1 t2 t2^2 t2^3 t2^4 t2^5];...
[0 1 2*t1 3*t1^2 4*t1^3 5*t1^4];...
[0 1 2*t2 3*t2^2 4*t2^3 5*t2^4];
[0 0 2 6*t1 12*t1^2 20*t1^3];...
[0 0 2 6*t2 12*t2^2 20*t2^3]...
];
%v=[t1^alpha ; t2^(-beta) ; alpha*t1^(alpha-1) ; -beta*t2^(-beta-1)];
v=[1 ; 1 ; ...
t1^(-alpha)*alpha*t1^(alpha-1) ; -beta*t2^(-beta-1)*t2^beta; ...
t1^(-alpha)*alpha*(alpha-1)*t1^(alpha-2);-beta*(-beta-1)*t2^(-beta-2)*t2^beta
];
a=M\v;
r1=find(x<t1);
%r1=find(x>=0 & x<t1);
r2=find(x>=t1 & x<t2);
r3=find(x>=t2);
r(r1)=x(r1).^alpha*t1^(-alpha);
r(r3)=x(r3).^(-beta)*t2^(beta);
x2=x(r2);
r(r2)=a(1)+a(2)*x2+a(3)*x2.^2+a(4)*x2.^3+a(5)*x2.^4+a(6)*x2.^5;
% tmp=polyval(flipud(a),x2);
% keyboard
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