Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications

TL;DR

This paper introduces a novel variational model using non-Muckenhoupt weighted Sobolev spaces for image processing, addressing non-standard weights and proposing a finite element scheme with improved denoising results.

Contribution

It develops a new variational framework in weighted Sobolev spaces with non-Muckenhoupt weights, including a finite element method and weight identification algorithm for image denoising.

Findings

Better denoising results compared to total variation methods

Successfully identifies unknown weights in the model

Applicable to a range of test problems

Abstract

We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted $L^2$ space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.