Generalized Intersection Algorithms with Fixpoints for Image Decomposition Learning

TL;DR

This paper introduces a broad theoretical framework for image decomposition algorithms using fixed points, extending classical models and demonstrating their effectiveness in denoising and texture removal.

Contribution

It formalizes a general class of intersection point problems for image decomposition, proving existence of solutions and connecting classical models with learned algorithms.

Findings

Existence of fixpoints for a wide class of image decomposition algorithms.

Generalization of classical variational models like TV-l2 and TV-Hilbert.

Comparable denoising and texture removal results with novel algorithm choices.

Abstract

In image processing, classical methods minimize a suitable functional that balances between computational feasibility (convexity of the functional is ideal) and suitable penalties reflecting the desired image decomposition. The fact that algorithms derived from such minimization problems can be used to construct (deep) learning architectures has spurred the development of algorithms that can be trained for a specifically desired image decomposition, e.g. into cartoon and texture. While many such methods are very successful, theoretical guarantees are only scarcely available. To this end, in this contribution, we formalize a general class of intersection point problems encompassing a wide range of (learned) image decomposition models, and we give an existence result for a large subclass of such problems, i.e. giving the existence of a fixpoint of the corresponding algorithm. This class generalizes classical model-based variational problems, such as the TV-l2 -model or the more general TV-Hilbert model. To illustrate the potential for learned algorithms, novel (non learned) choices within our class show comparable results in denoising and texture removal.