Subspace correction methods for total variation and $\ell_1-$minimization

TL;DR

This paper introduces a new subspace correction method using oblique thresholding for efficient minimization of energy functionals involving total variation and -minimization, with applications in image processing and PDEs.

Contribution

It presents a novel oblique thresholding technique and a subspace correction framework for convex minimization problems, with proven convergence and parallel variants.

Findings

Efficient algorithms for total variation minimization in image processing.

Convergence conditions established for the proposed methods.

Numerical examples demonstrate improved performance in signal and image recovery.

Abstract

This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via \emph{oblique thresholding}, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for singular elliptic PDE's arising in total variation minimization and in accelerated sparse recovery algorithms based on $\ell_1$-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing.