## Theoretical foundation of the weighted Laplace inpainting problem.(English)Zbl 07088741

Summary: 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 must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.

### MSC:

 35J15 Second-order elliptic equations 35J70 Degenerate elliptic equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

### Software:

MATLAB-Python-inpainting-codes; SIPI Image Database
Full Text:

### References:

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