Image Inpainting with Fractional Laplacian Regularization: An L p Norm Approach

Hongfang Yuan, Weijie Su, Xiangkai Lian (), Zheng-An Yao and Dewen Hu
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Hongfang Yuan: School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
Weijie Su: School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
Xiangkai Lian: College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China
Zheng-An Yao: School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
Dewen Hu: College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China

Mathematics, 2025, vol. 13, issue 14, 1-81

Abstract: This study presents an image inpainting model based on an energy functional that incorporates the L p norm of the fractional Laplacian operator as a regularization term and the H − 1 norm as a fidelity term. Using the properties of the fractional Laplacian operator, the L p norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional L 2 norm with the H − 1 norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks.

Keywords: image inpainting; fractional Laplacian; well-posedness (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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