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iCAPs/packages/gspbox/prox/gsp_proj_filterbank.m
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gsp_proj_filterbank.m
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function [sol, info] = gsp_proj_filterbank(x, ~ , G, W, y, param)
%GSP_PROJ_FILTERBANK Projection onto the synthesis coefficients
% Usage: sol = gsp_proj_filterbank(x, 0, G, W, y, param);
% sol = gsp_proj_filterbank(x, 0, G, W, y);
% [sol, info] = gsp_proj_filterbank(...)
%
% Input parameters:
% x : Input signal
% G : Graph structure
% W : Filterbank (cell array of functions)
% y : Measurements
% param : Structure of optional parameters
% Output parameters
% sol : Solution.
% info : Structure summarizing informations at convergence
%
% GSP_PROJ_FILTERBANK(x, gamma, G, W, param) can solves:
%
% sol = argmin_{z} 0.5*||x - z||_2^2 such that W^* x =y
%
% Where W is the linear analysis operator associated with the
% filterbank.
%
% The function can use different techniques
%
% 'exact' : if the Fourier basis is computed, go for this one
% 'cheby' : use the pseudo-inverse filters of the filterbank with
% chebyshev approximation. It works well for well-conditionned
% filterbanks.
% 'lanczos' : use the pseudo-inverse filters of the filterbank with
% lanczos approximation. It works well for well-conditionned
% filterbanks.
% 'proj_b2': scallable and robust way to do it. However, the
% convergence maybe slow and might require a lot of filtering
% operations.
%
% param is a Matlab structure containing the following fields:
%
% param.verbose : 0 no log, 1 a summary at convergence, 2 print main
% steps (default: 1)
% param.eps : tolerance for the pseudo inverse method
% param.proj_method*: selected method
%
% info is a Matlab structure containing the following fields:
%
% info.algo : Algorithm used
% info.iter : Number of iteration
% info.time : Time of exectution of the function in sec.
% info.final_eval : Final evaluation of the function
% info.crit : Stopping critterion used
%
%
% See also: gsp_solve_l1 gsp_proj_b2_filterbank
%
%
% Url: http://lts2research.epfl.ch/gsp/doc/prox/gsp_proj_filterbank.php
% Copyright (C) 2013-2016 Nathanael Perraudin, Johan Paratte, David I Shuman.
% This file is part of GSPbox version 0.7.0
%
% 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
% Author: Nathanael Perraudin
% Date: 25 March 2014
% Testing: test_gsp_proj_filerbank
if nargin < 5
error('GSP_PROJ_FILTERBANK: Not enought input arguments');
end
if nargin < 6, param=struct; end
if ~isfield(param, 'eps'), param.eps = 1e-8; end
if ~isfield(param, 'proj_method'),
if gsp_check_fourier(G)
param.proj_method = 'exact';
else
[A,B] = gsp_filterbank_bounds(G,W);
if B/A > 20
warning('Filterbank ill-conditioned, going for a slow method');
param.proj_method = 'primal_dual';
else
param.proj_method = 'cheby';
end
end
end
t = tic;
if size(y,2)==1 && size(x,2)>1
y = repmat(y,1,size(x,2));
end
if isfield(param,'method')
param2 = rmfield(param,'method');
else
param2 = param;
end
if strcmp(param.proj_method, 'proj_b2') || strcmp(param.proj_method, 'primal_dual')
[~,B] = gsp_filterbank_bounds(G,W);
if ~isfield(param,'paramproj'), param.paramproj = struct; end
paramproj = param.paramproj;
paramproj.A = @(x) gsp_filter_synthesis(G,W,x,param2);
paramproj.At = @(x) gsp_filter_analysis(G,W,x,param2);
paramproj.nu = B^2;
paramproj.epsilon = sqrt(G.N)*param.eps;
paramproj.y = y;
paramproj.method = param.proj_method;
[sol,info] = proj_linear_eq(x,0,paramproj);
else
if strcmp(param.proj_method, 'cheby') || strcmp(param.proj_method, 'lanczos')
[A,B] = gsp_filterbank_bounds(G,W);
if B/A > 20
warning('Filterbank ill-conditioned, check your solution!');
end
end
Wd = gsp_design_can_dual(W,param.eps);
sol = x - gsp_filter_analysis(G,Wd,(gsp_filter_synthesis(G,W,x,param2)-y),param2);
info.iter = 1;
info.final_eval = 0;
info.crit = 'Direct computation';
end
info.time = toc(t);
info.algo = param.proj_method;
end
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