Functions of bounded Musielak-Orlicz-type deformation and anisotropic Total Generalized Variation for image-denoising problems
Giacomo Bertazzoni, Elisa Davoli, Samuele Riccò, Elvira Zappale
TL;DR
This work develops a Musielak–Orlicz framework for anisotropic, higher-order regularization in image denoising by introducing the space $BD^φ$ of bounded deformation fields with generalized growth and the anisotropic Total Generalized Variation $TGV^{φ,2}_α$. It delivers a dual modular decomposition $\varrho_{\widetilde{V}^{n\times n}_φ}(Eu)=\varrho_φ(|\mathcal{E}u|)+\int_Ω φ^∞ \, d|E^su|$ and proves well-posedness of the reconstruction problem, including existence, stability, and Gamma-convergence for variational denoising problems. The variable-exponent case $φ(x,t)=\frac{1}{p(x)}t^{p(x)}$ is treated, showing a clear split between absolutely continuous and singular parts with $BD^φ(Ω)$ aligning with classical BD/LD spaces depending on the asymptotic growth. Collectively, the results provide an adaptive, anisotropic regularization mechanism that can better preserve textures and suppress staircasing in images, while accommodating spatially varying material growth through a robust Musielak–Orlicz theory.
Abstract
In the first part of this paper we introduce the space of bounded deformation fields with generalized Orlicz growth. We establish their main properties, provide a modular representation, and characterize a decomposition of the modular into an absolutely continuous part and a singular part weighted via a recession function. A further analysis in the variable exponent case is also provided. The second part of the paper contains a notion of Musielak-Orlicz anisotropic Total Generalized Variation. We establish a duality representation, and show well-posedness of the corresponding image reconstruction problem.