SOLVE_TVDN - Solve TVDN problem

Usage

sol = solve_tvdn(y, epsilon, A, At, param)
sol = solve_tvdn(y, epsilon, A, At)
[sol, info] = solve_tvdn(...)

Input parameters

Output parameters

Description

sol = solve_tvdn(Y, epsilon, A, At, PARAM) solves:

Y contains the measurements. A is the forward measurement operator and At the associated adjoint operator. 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.useGPU : Use GPU to compute the TV prox operator. Please prior call init_gpu and free_gpu to launch and release the GPU library (default: 0).

• param.tol : is stop criterion for the loop. The algorithm stops if

  \begin{equation*} \frac{ n(t) - n(t-1) }{ n(t)} < tol, \end{equation*}

  where \(n(t) = ||(x)||_{TV}\) is the objective function at iteration t by default, tol=10e-4.

• param.gamma : control the converge speed (default: 1e-1).

Projection onto the L2-ball :

• param.tight_b2 : 1 if A is a tight frame or 0 if not (default = 1)

• param.nu_b2 : bound on the norm of the operator A, i.e.

  \begin{equation*} \|A x\|^2 \leq \nu \|x\|^2 \end{equation*}

• param.tol_b2 : tolerance for the projection onto the L2 ball (default: 1e-3):

  \begin{equation*} \frac{\epsilon}{1-tol} \leq \|y - A z\|_2 \leq \frac{\epsilon}{1+tol} \end{equation*}

• param.maxit_b2 : max. nb. of iterations for the projection onto the L2 ball (default 200).

Proximal TV operator:

• param.maxit_tv : Used as stopping criterion for the proximal TV operator. Maximum number of iterations.

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 objectivs functions
• info.crit : Stopping critterion used
• info.rel_norm : Relative norm at convergence
• info.residue : Final residue

The problem is solved thanks to a Douglas-Rachford splitting algorithm.