# Existence Theory for the EED Inpainting Problem

## TL;DR

This paper develops an existence theory for the edge-enhancing diffusion inpainting problem, proving the existence of solutions and their convergence properties in a nonlinear anisotropic diffusion framework.

## Contribution

It introduces a rigorous mathematical existence framework for EED inpainting, applying fixed point theorems and analyzing solution sequences.

## Key findings

- Existence of weak solutions established using Leray-Schauder theorem
- Boundedness of all possible weak solutions shown
- Convergence of iterative solution sequences demonstrated

## Abstract

We establish an existence theory for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoring missing data in an image as steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. We prove existence of a weak solution by applying the Leray-Schauder Fixed point theorem and show that the set of all possible weak solutions is bounded. Moreover, we demonstrate that under certain conditions, the sequences resulting from iterative application of the operator from the existence theory contain convergent subsequences.