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sopt_mltb_solve_TVDN

PURPOSE ^

sopt_mltb_solve_TVDN - Solve TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param)

DESCRIPTION ^

 sopt_mltb_solve_TVDN - Solve TVDN problem

 Solve the total variation denoising (TVDN) problem

   min ||x||_TV   s.t.  ||y - A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator. The structure param should
 contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of TV norm when solving for
       TV proximal operator) (default = 1e-1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - max_iter_TV: Used as stopping criterion for the proximal TV
       operator. Maximun number of iterations (default = 200).

 References:
 P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
 to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
 Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param)
0002 % sopt_mltb_solve_TVDN - Solve TVDN problem
0003 %
0004 % Solve the total variation denoising (TVDN) problem
0005 %
0006 %   min ||x||_TV   s.t.  ||y - A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator. The structure param should
0010 % contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of TV norm when solving for
0025 %       TV proximal operator) (default = 1e-1).
0026 %
0027 %   - initsol: Initial solution for a warmstart.
0028 %
0029 %   Projection onto the L2-ball:
0030 %
0031 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0032 %
0033 %   - nu_B2: Bound on the norm of the operator A, i.e.
0034 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0035 %
0036 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0037 %       stops if
0038 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0039 %       (default = 1e-3).
0040 %
0041 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0042 %       L2 ball (default = 200).
0043 %
0044 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0045 %       default = 0).
0046 %
0047 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   Proximal L1 operator:
0051 %
0052 %   - max_iter_TV: Used as stopping criterion for the proximal TV
0053 %       operator. Maximun number of iterations (default = 200).
0054 %
0055 % References:
0056 % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
0057 % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
0058 % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0059 
0060 % Optional input arguments
0061 if ~isfield(param, 'verbose'), param.verbose = 1; end
0062 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0063 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0064 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0065 
0066 % Input arguments for projection onto the L2 ball
0067 param_B2.A = A; param_B2.At = At;
0068 param_B2.y = y; param_B2.epsilon = epsilon;
0069 param_B2.verbose = param.verbose;
0070 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0071 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0072 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0073 if isfield(param, 'max_iter_B2')
0074     param_B2.max_iter = param.max_iter_B2;
0075 end
0076 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0077 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0078 
0079 % Input arguments for prox TV
0080 param_TV.verbose = param.verbose; param_TV.rel_obj = param.rel_obj;
0081 if isfield(param, 'max_iter_TV')
0082     param_TV.max_iter= param.max_iter_TV;
0083 end
0084 
0085 % Initialization
0086 if isfield(param,'initsol')
0087     xhat = param.initsol;
0088 else
0089     xhat = At(y); 
0090 end 
0091 
0092 iter = 1; prev_norm = 0;
0093 
0094 % Main loop
0095 while 1
0096     
0097     if param.verbose>=1
0098         fprintf('Iteration %i:\n', iter);
0099     end
0100     
0101     % Projection onto the L2-ball
0102     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0103     
0104     % Global stopping criterion
0105     curr_norm = sopt_mltb_TV_norm(sol, 0);
0106     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0107     if param.verbose >= 1
0108         fprintf('  ||x||_TV = %e, rel_norm = %e\n', ...
0109             curr_norm, rel_norm);
0110     end
0111     if (rel_norm < param.rel_obj)
0112         crit_BPDN = 'REL_NORM';
0113         break;
0114     elseif iter >= param.max_iter
0115         crit_BPDN = 'MAX_IT';
0116         break;
0117     end
0118     
0119     % Proximal L1 operator
0120     xhat = 2*sol - xhat;
0121     temp = sopt_mltb_prox_TV(xhat, param.gamma, param_TV);
0122     xhat = temp + sol - xhat;
0123     
0124     % Update variables
0125     iter = iter + 1;
0126     prev_norm = curr_norm;
0127     
0128 end
0129 
0130 % Log
0131 if param.verbose >= 1
0132   
0133     % L1 norm
0134     fprintf('\n Solution found:\n');
0135     fprintf(' Final TV norm: %e\n', curr_norm);
0136     
0137     % Residual
0138     temp = A(sol);
0139     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ...
0140         norm(y(:)-temp(:)));
0141     
0142     % Stopping criterion
0143     fprintf(' %i iterations\n', iter);
0144     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0145     
0146 end
0147 
0148 end

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