Parameter learning and fractional differential operators: application in image regularization and decomposition

TL;DR

This paper develops a bilevel optimization framework for learning parameters in PDE-based image regularization using fractional Laplacian operators, demonstrating improved efficiency and explainability in denoising and decomposition tasks.

Contribution

It introduces a novel parameter learning approach for PDE-based image regularization with fractional operators, including explicit solutions and stability analysis.

Findings

Reduced computational time compared to ROF model

Stable and explainable parameter optimization

Effective image denoising and decomposition results

Abstract

In this paper, we focus on learning optimal parameters for PDE-based image regularization and decomposition. First we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to other machine learning approaches. The numerical experiments correlate with our theoretical model setting and show a reduction of computing time in contrast to the ROF model. Second we introduce a new image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.