Bounded variation spaces with generalized Orlicz growth related to image denoising

TL;DR

This paper introduces a new class of bounded variation spaces with generalized Orlicz growth to improve image denoising, addressing issues like stair-casing in total variation methods.

Contribution

It develops a novel mathematical framework that generalizes existing models, including variable exponent and double phase spaces, and analyzes their properties and limits.

Findings

Derived a formula for the modular in terms of Lebesgue decomposition.

Showed the modular as a $ ext{Gamma}$-limit of convex energies.

Extended the theory to cover earlier variable exponent and double phase models.

Abstract

Motivated by the image denoising problem and the undesirable stair-casing effect of the total variation method, we introduce bounded variation spaces with generalized Orlicz growth. Our setup covers earlier variable exponent and double phase models. We study the norm and modular of the new space and derive a formula for the modular in terms of the Lebesgue decomposition of the derivative measure and a location dependent recession function. We also show that the modular can be obtained as the $\Gamma$-limit of uniformly convex approximating energies.