Iterative TV minimization on the graph

TL;DR

This paper introduces a new iterative algorithm for TV denoising on graphs, leveraging a novel characterization of the dual space of functions of bounded variation, and demonstrates its competitive performance in image denoising tasks.

Contribution

The paper develops a new graph-based TV denoising algorithm with a provable convergence, based on a novel dual space characterization, and compares it favorably with existing methods.

Findings

The algorithm converges reliably and efficiently.

It achieves high-quality denoising comparable to state-of-the-art methods.

Experimental results confirm its competitive convergence rate and visual quality.

Abstract

We define the space of functions of bounded variation ($BV$) on the graph. Using the notion of divergence of flows on graphs, we show that the unit ball of the dual space to $BV$ in the graph setting can be described as the image of the unit ball of the space $\ell^{\infty}$ by the divergence operator. Based on this result, we propose a new iterative algorithm to find the exact minimizer for the total variation (TV) denoising problem on the graph. The proposed algorithm is provable convergent and its performance on image denoising examples is compared with the Split Bregman and Primal-Dual algorithms as benchmarks for iterative methods and with BM3D as a benchmark for other state-of-the-art denoising methods. The experimental results show highly competitive empirical convergence rate and visual quality for the proposed algorithm.