Convergence of level sets in fractional Laplacian regularization

TL;DR

This paper investigates the convergence of level sets in fractional Laplacian regularization for image denoising and inverse problems, demonstrating boundary convergence and rates under certain conditions.

Contribution

It provides the first rigorous analysis of level set convergence for fractional Laplacian regularization, including convergence rates and technical tools involving fractional Allen-Cahn barriers.

Findings

Level set boundaries converge in Hausdorff distance.

Convergence rates are established for denoising with indicatrix data.

Fractional Laplacian behaves less aggressively than $H^1$ norms in discontinuous functions.

Abstract

The use of the fractional Laplacian in image denoising and regularization of inverse problems has enjoyed a recent surge in popularity, since for discontinuous functions it can behave less aggressively than methods based on $H^1$ norms, while being linear and computable with fast spectral numerical methods. In this work, we examine denoising and linear inverse problems regularized with fractional Laplacian in the vanishing noise and regularization parameter regime. The clean data is assumed piecewise constant in the first case, and continuous and satisfying a source condition in the second. In these settings, we prove results of convergence of level set boundaries with respect to Hausdorff distance, and additionally convergence rates in the case of denoising and indicatrix clean data. The main technical tool for this purpose is a family of barriers constructed by Savin and Valdinoci for studying the fractional Allen-Cahn equation. To help put these fractional methods in context, comparisons with the total variation and classical Laplacian are provided throughout.