A remark on the denoising of greyscale images using energy densities with varying growth rates

TL;DR

This paper proves the solvability of a class of variational problems with variable growth energy densities, extending the TV-model for greyscale image denoising to more general functional forms.

Contribution

It introduces a generalized framework for denoising models with energy densities of varying growth rates, broadening the applicability of TV-based methods.

Findings

Proves existence of solutions in Sobolev spaces for the generalized models.

Extends the TV-model to include energy densities with (1, p)-growth.

Provides a mathematical foundation for more flexible denoising functionals.

Abstract

We prove the solvability in Sobolev spaces for a class of variational problems related to the TV-model proposed by Rudin, Osher and Fatemi in [1] for the denoising of greyscale images. In contrast to their approach we discuss energy densities with variable growth rates depending on $|\nabla u|$ in a rather general form including functionals of $(1, p)$-growth.