The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising

TL;DR

This paper introduces a new technique for analyzing the jump set properties of solutions to geometric regularization problems, avoiding reliance on the co-area formula, and applies it to first-order regularizers like TV and Huber-regularized TV.

Contribution

It develops a novel Lipschitz transformation method to establish jump set containment for a broad class of regularizers, extending results beyond traditional total variation.

Findings

Proves jump set containment for TV and Huber-regularized TV.

Introduces a technique avoiding the co-area formula.

Lays groundwork for higher-order regularizer analysis in Part 2.

Abstract

Let $u \in \mbox{BV}(\Omega)$ solve the total variation denoising problem with $L^2$-squared fidelity and data $f$. Caselles et al. [Multiscale Model. Simul. 6 (2008), 879--894] have shown the containment $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ of the jump set $J_u$ of $u$ in that of $f$. Their proof unfortunately depends heavily on the co-area formula, as do many results in this area, and as such is not directly extensible to higher-order, curvature-based, and other advanced geometric regularisers, such as total generalised variation (TGV) and Euler's elastica. These have received increased attention in recent times due to their better practical regularisation properties compared to conventional total variation or wavelets. We prove analogous jump set containment properties for a general class of regularisers. We do this with novel Lipschitz transformation techniques, and do not require the co-area formula. In the present Part 1 we demonstrate the general technique on first-order regularisers, while in Part 2 we will extend it to higher-order regularisers. In particular, we concentrate in this part on TV and, as a novelty, Huber-regularised TV. We also demonstrate that the technique would apply to non-convex TV models as well as the Perona-Malik anisotropic diffusion, if these approaches were well-posed to begin with.