Nested Bregman Iterations for Decomposition Problems

TL;DR

This paper introduces Nested Bregman iterations, a novel approach for image decomposition that optimizes regularization weights through a structured iterative process, improving the robustness and quality of image reconstruction.

Contribution

The paper proposes a new Nested Bregman iteration method that automates the selection of regularization weights for image decomposition tasks, enhancing existing variational approaches.

Findings

Numerical experiments demonstrate improved image decomposition quality.

The method converges under certain conditions, ensuring reliable performance.

It effectively automates regularization weight selection, reducing manual tuning.

Abstract

We consider the task of image reconstruction while simultaneously decomposing the reconstructed image into components with different features. A commonly used tool for this is a variational approach with an infimal convolution of appropriate functions as a regularizer. Especially for noise corrupted observations, incorporating these functionals into the classical method of Bregman iterations provides a robust method for obtaining an overall good approximation of the true image, by stopping early the iteration according to a discrepancy principle. However, crucially, the quality of the separate components depends further on the proper choice of the regularization weights associated to the infimally convoluted functionals. Here, we propose the method of Nested Bregman iterations to improve a decomposition in a structured way. This allows to transform the task of choosing the weights into the problem of stopping the iteration according to a meaningful criterion based on normalized cross-correlation. We discuss the well-definedness and the convergence behavior of the proposed method, and illustrate its strength numerically with various image decomposition tasks employing infimal convolution functionals.