Theoretical Foundation of the Weighted Laplace Inpainting Problem

TL;DR

This paper provides a rigorous theoretical analysis of the weighted Laplace inpainting problem, establishing conditions for the existence and uniqueness of solutions using advanced mathematical tools like weighted Sobolev spaces.

Contribution

It offers the first detailed theoretical foundation for the weighted Laplace inpainting model, extending previous discrete results to continuous settings.

Findings

Solutions require specific regularity and growth conditions on weights

Existence and uniqueness depend on weighted Sobolev space properties

The analysis complements prior discrete case studies

Abstract

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.