# Improved PDE Models for Image Restoration through Backpropagation

## TL;DR

This paper introduces a novel approach to image restoration by optimizing PDE-based models using deep learning, specifically backpropagation, leading to more effective denoising with a solid mathematical basis.

## Contribution

The paper presents a new method that combines PDE models with deep learning to optimize parameters for improved image filtering performance.

## Key findings

- Enhanced denoising quality demonstrated through numerical results.
- Stable learning process with optimized cross-diffusion coefficients.
- Better image restoration compared to traditional PDE methods.

## Abstract

In this paper we focus on learning optimized partial differential equation (PDE) models for image filtering. In this approach, the grey-scaled images are represented by a vector field of two real-valued functions and the image restoration problem is modelled by an evolutionary process such that the restored image at any time satisfies an initial-boundary-value problem of cross-diffusion with reaction type. The coupled evolution of the two components of the image is determined by a nondiagonal matrix that depends on those components. A critical question when designing a good-performing filter lies in the selection of the optimal coefficients and influence functions which define the cross-diffusion matrix. We propose the use of deep learning techniques in order to optimize the parameters of the model. In particular, we use a back propagation technique in order to minimize a cost function related to the quality of the denoising processe, while we ensure stability during the learning procedure. Consequently, we obtain improved image restoration models with solid mathematical foundations. The learning framework and resulting models are presented along with related numerical results and image comparisons.