A General Decomposition Method for a Convex Problem Related to Total Variation Minimization

TL;DR

This paper introduces a general decomposition method for convex total variation minimization problems, providing convergence analysis, rate comparisons, and demonstrating effectiveness in various image processing applications.

Contribution

It develops a unified decomposition framework with convergence guarantees for total variation minimization, including approximate solutions and practical algorithms.

Findings

Convergence of the methods is proven in a Hilbert space setting.

The proposed algorithms perform well in image inpainting and optical flow estimation.

The convergence rate matches or exceeds existing results in the literature.

Abstract

We consider sequential and parallel decomposition methods for a dual problem of a general total variation minimization problem with applications in several image processing tasks, like image inpainting, estimation of optical flow and reconstruction of missing wavelet coefficients. The convergence of these methods to a solution of the global problem is analysed in a Hilbert space setting and a convergence rate is provided. Thereby, these convergence result hold not only for exact local minimization but also if the subproblems are just solved approximately. As a concrete example of an approximate local solution process a surrogate technique is presented and analysed. Further, the obtained convergence rate is compared with related results in the literature and shown to be in agreement with or even improve upon them. Numerical experiments are presented to support the theoretical findings and to show the performance of the proposed decomposition algorithms in image inpainting, optical flow estimation and wavelet inpainting tasks.