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Bounded variation spaces with generalized Orlicz growth related to image denoising

Michela Eleuteri, Petteri Harjulehto, Peter Hästö

TL;DR

This work addresses BV-type variational models with nonstandard, non-doubling generalized Orlicz growth for image denoising. It develops a duality-based BV^φ framework, derives the exact modular formula $\varrho_{V,\varphi}(u)=\varrho_\varphi(|\nabla^a u|)+\int_\Omega \varphi'_{\infty} \, d|D^su|$, and proves Γ-convergence of uniformly convex approximations to the modular. The results unify variable-exponent, double-phase, and Orlicz-type growth in a nonautonomous setting via the $x$-dependent recession function $\varphi'_{\infty}(x)$, providing rigorous foundations for nonstandard-growth regularizers in PDE-based image restoration. The approach delivers stability, approximation, and relaxation properties essential for designing denoising models that avoid stair-casing while adapting to spatial image structure. These contributions advance the mathematical theory of generalized BV and Musielak–Orlicz spaces and have potential for robust, adaptive image processing applications.

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 $Γ$-limit of uniformly convex approximating energies.

Bounded variation spaces with generalized Orlicz growth related to image denoising

TL;DR

This work addresses BV-type variational models with nonstandard, non-doubling generalized Orlicz growth for image denoising. It develops a duality-based BV^φ framework, derives the exact modular formula , and proves Γ-convergence of uniformly convex approximations to the modular. The results unify variable-exponent, double-phase, and Orlicz-type growth in a nonautonomous setting via the -dependent recession function , providing rigorous foundations for nonstandard-growth regularizers in PDE-based image restoration. The approach delivers stability, approximation, and relaxation properties essential for designing denoising models that avoid stair-casing while adapting to spatial image structure. These contributions advance the mathematical theory of generalized BV and Musielak–Orlicz spaces and have potential for robust, adaptive image processing applications.

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 -limit of uniformly convex approximating energies.
Paper Structure (7 sections, 21 theorems, 154 equations)

This paper contains 7 sections, 21 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Auxiliary results
  4. Basic properties of dual norms and modulars
  5. Approximation properties of the dual norm
  6. Explicit expression for the dual modular
  7. Precise approximation and Gamma-convergence

Key Result

Proposition 3.2

Let $\varphi(x,t):=t^{p(x)}$ be a variable exponent energy with $p:\Omega\to[1,\infty)$. Then restricted def:va1 is equivalent to the strong $\log$-Hölder continuity of $\frac{1}{p}$, i.e. $\log$-Hölder continuity with uniformly in $y\in \{p=1\}$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3: Jensen's inequality
  • proof
  • ...and 42 more