Modular proximal optimization for multidimensional total-variation regularization

TL;DR

This paper introduces efficient algorithms for computing proximal operators for multidimensional total-variation regularization, especially for the -norm, using a novel geometric approach linked to taut-string methods, enabling improved image and video processing techniques.

Contribution

The paper presents a new geometric analysis for -norm TV proximal operators, leading to more efficient solvers and a modular framework for higher-dimensional TV problems.

Findings

Our 1D-TV solvers outperform existing methods in speed and accuracy.

The modular approach improves performance in image and video denoising tasks.

Open-source library implementation facilitates reproducibility and practical use.

Abstract

We study \emph{TV regularization}, a widely used technique for eliciting structured sparsity. In particular, we propose efficient algorithms for computing prox-operators for $\ell_p$-norm TV. The most important among these is $\ell_1$-norm TV, for whose prox-operator we present a new geometric analysis which unveils a hitherto unknown connection to taut-string methods. This connection turns out to be remarkably useful as it shows how our geometry guided implementation results in efficient weighted and unweighted 1D-TV solvers, surpassing state-of-the-art methods. Our 1D-TV solvers provide the backbone for building more complex (two or higher-dimensional) TV solvers within a modular proximal optimization approach. We review the literature for an array of methods exploiting this strategy, and illustrate the benefits of our modular design through extensive suite of experiments on (i) image denoising, (ii) image deconvolution, (iii) four variants of fused-lasso, and (iv) video denoising. To underscore our claims and permit easy reproducibility, we provide all the reviewed and our new TV solvers in an easy to use multi-threaded C++, Matlab and Python library.