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

arXiv: https://arxiv.org/abs/2001.03394 · HTML: https://mlsanity.com/paper/2001.03394/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2020.

## Built on by (most cited first, 1 of 1)

- [Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory](https://mlsanity.com/paper/2110.02273/) (2021)

## Abstract (arXiv)

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.