Variational Coupling Revisited: Simpler Models, Theoretical Connections, and Novel Applications

TL;DR

This paper revisits variational coupling models in image analysis, revealing their ability to achieve edge-preserving denoising with quadratic data terms, establishing connections to Mumford-Shah segmentation, and proposing a convex, convergent algorithm.

Contribution

It demonstrates the effectiveness of first-order coupling models for denoising, introduces a novel $L^1$ coupling for edge detection, and links TV regularization with Mumford-Shah segmentation through a convex approach.

Findings

Quadratic data terms with $L^1$ coupling preserve edges.

A convex split Bregman algorithm guarantees convergence.

First-order coupling models outperform classical TV regularization.

Abstract

Variational models with coupling terms are becoming increasingly popular in image analysis. They involve auxiliary variables, such that their energy minimisation splits into multiple fractional steps that can be solved easier and more efficiently. In our paper we show that coupling models offer a number of interesting properties that go far beyond their obvious numerical benefits. We demonstrate that discontinuity-preserving denoising can be achieved even with quadratic data and smoothness terms, provided that the coupling term involves the $L^1$ norm. We show that such an $L^1$ coupling term provides additional information as a powerful edge detector that has remained unexplored so far. While coupling models in the literature approximate higher order regularisation, we argue that already first order coupling models can be useful. As a specific example, we present a first order coupling model that outperforms classical TV regularisation. It also establishes a theoretical connection between TV regularisation and the Mumford-Shah segmentation approach. Unlike other Mumford-Shah algorithms, it is a strictly convex approximation, for which we can guarantee convergence of a split Bregman algorithm.