function [sol, info] = solve_bpdn(y, epsilon, A, At, Psi, Psit, param)
%SOLVE_BPDN Solve BPDN (basis pursuit denoising) problem
% Usage: sol = solve_bpdn(y, epsilon, A, At, Psi, Psit, param)
% sol = solve_bpdn(y, epsilon, A, At, Psi, Psit)
% [sol, info] = solve_bpdn(...)
%
% Input parameters:
% y : Measurements
% epsilon: Radius of the L2 ball
% A : Operator
% At : Adjoint of A
% Psi : Operator
% Psit : Adjoint of Psi
% param : Optional parameter
% Output parameters:
% sol : Solution
% info : Structure summarizing informations at convergence
%
% sol = solve_BPDN(y, A, At, Psi, Psit, param) solves:
%
% sol arg min ||Psi x||_1 s.t. ||y-A x||_2 < epsilon
%
% Y contains the measurements. A is the forward measurement operator and
% At the associated adjoint operator. Psit is a sparfying transform and Psi
% its adjoint. PARAM a Matlab structure containing the following fields:
%
% General parameters:
%
% param.verbose : 0 no log, 1 print main steps, 2 print all steps.
%
% param.maxit : max. nb. of iterations (default: 200).
%
% param.tol : is stop criterion for the loop. The algorithm stops if
%
% ( n(t) - n(t-1) ) / n(t) < tol,
%
% where n(t) = Psi(x)|| is the objective function at iteration t*
% by default, tol=10e-4.
%
% param.gamma : control the converge speed (default: 1).
%
%
% Projection onto the L2-ball :
%
% param.tight_b2 : 1 if A is a tight frame or 0 if not (default = 1)
%
% nu_b2 : bound on the norm of the operator A, i.e.
%
% ` ||A x||^2 <= nu * ||x||^2
%
% tol_b2 : tolerance for the projection onto the L2 ball (default: 1e-3):
%
% epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
%
% maxit_b2 : max. nb. of iterations for the projection onto the L2
% ball (default 200).
%
%
% Proximal L1 operator:
%
% tol_l1 : Used as stopping criterion for the proximal L1
% operator. Min. relative change of the objective value between two
% successive estimates.
%
% maxit_l1 : Used as stopping criterion for the proximal L1
% operator. Maximum number of iterations.
%
% param.nu_l1 : bound on the norm^2 of the operator Psi, i.e.
%
% ` ||Psi x||^2 <= nu * ||x||^2
%
% param.tight_l1 : 1 if Psit is a tight frame or 0 if not (default = 1)
%
% param.weights : weights (default = 1) for a weighted L1-norm defined
% as:
%
% sum_i{weights_i.*abs(x_i)}
%
% The problem is solved thanks to a Douglas-Rachford splitting
% algorithm.
%
% Demos: demo_weighted_l1
%
% References:
% P. Combettes and J. Pesquet. A douglas--rachford splitting approach to
% nonsmooth convex variational signal recovery. Selected Topics in Signal
% Processing, IEEE Journal of, 1(4):564--574, 2007.
%
%
% Url: https://epfl-lts2.github.io/unlocbox-html/doc/solver/solve_bpdn.html
% Copyright (C) 2012-2016 Nathanael Perraudin.
% This file is part of UNLOCBOX version 1.7.4
%
% 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/>.
% Author: Gilles Puy, Nathanael Perraudin
% Date: Nov. 1, 2012
%
% Optional input arguments
if nargin<7, param=struct; end
if nargin < 5; Psi = @(x) x; end
if nargin < 6; Psit = Psi; end
% Optional input arguments
if ~isfield(param, 'verbose'), param.verbose = 1; end
if ~isfield(param, 'tol'), param.tol = 1e-4; end
if ~isfield(param, 'maxit'), param.maxit = 200; end
if ~isfield(param, 'pos_l1'), param.pos_l1 = 0; end
% Input arguments for projection onto the L2 ball
param_b2.A = A; param_b2.At = At;
param_b2.y = y; param_b2.epsilon = epsilon;
param_b2.verbose = param.verbose-1;
if isfield(param, 'nu_b2'), param_b2.nu = param.nu_b2; end
if isfield(param, 'tol_b2'), param_b2.tol = param.tol_b2; end
if isfield(param, 'tight_b2'), param_b2.tight = param.tight_b2; end
if isfield(param, 'maxit_b2')
param_b2.maxit = param.maxit_b2;
end
% Input arguments for prox L1
param_l1.A = Psi; param_l1.At = Psit; param_l1.pos = param.pos_l1;
param_l1.verbose = param.verbose-1; param_l1.tol = param.tol;
if isfield(param, 'nu_l1')
param_l1.nu = param.nu_l1;
end
if isfield(param, 'tight_l1')
param_l1.tight = param.tight_l1;
end
if isfield(param, 'maxit_l1')
param_l1.maxit = param.maxit_l1;
end
if isfield(param, 'tol_l1')
param_l1.tol = param.tol_l1;
end
if isfield(param, 'weights')
param_l1.weights = param.weights;
f1.eval = @(x) norm(param_l1.weights(:).*x(:),1);
else
param_l1.weights = 1;
f1.eval = @(x) norm(x(:),1);
end
f1.prox = @(x,T) prox_l1(x,T,param_l1);
f2.prox = @(x,T) proj_b2(x,T,param_b2);
f2.eval = @(x) eps;
[sol, info] = douglas_rachford(At(y),f2, f1, param);
end