Generalized Intersection Algorithms with Fixpoints for Image Decomposition Learning
This work addresses the lack of theoretical guarantees for learned image decomposition algorithms, offering a foundational framework for researchers in image processing and machine learning.
The authors formalized a general class of intersection point problems for image decomposition models, providing an existence result for fixpoints, and demonstrated that novel non-learned choices within this class achieve comparable results in denoising and texture removal tasks.
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.