A Convergent Overlapping Domain Decomposition Method for Total Variation Minimization

TL;DR

This paper introduces a novel convergent overlapping domain decomposition method for total variation minimization, effectively addressing nonlinear, nonsmooth problems with promising applications in image restoration and compressed sensing.

Contribution

It presents the first successful domain decomposition strategy for total variation minimization, combining convergence analysis with practical algorithms.

Findings

Effective restoration of 1D signals and 2D images

Successful application to medical image recovery

Demonstrated convergence of the proposed method

Abstract

This paper is concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.