Image Inpainting with Fractional Laplacian Regularization: An Lp Norm Approach

Yuan, Hongfang; Su, Weijie; Lian, Xiangkai; Yao, Zheng-An; Hu, Dewen

doi:10.3390/math13142254

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Open AccessArticle

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

by

Hongfang Yuan

^1,†,

Weijie Su

^1,†,

Xiangkai Lian

^2,*

,

Zheng-An Yao

¹ and

Dewen Hu

²

¹

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

²

College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2025, 13(14), 2254; https://doi.org/10.3390/math13142254

Submission received: 14 May 2025 / Revised: 5 July 2025 / Accepted: 8 July 2025 / Published: 11 July 2025

(This article belongs to the Special Issue Numerical and Computational Methods in Engineering)

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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

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MDPI and ACS Style

Yuan, H.; Su, W.; Lian, X.; Yao, Z.-A.; Hu, D. Image Inpainting with Fractional Laplacian Regularization: An L^p Norm Approach. Mathematics 2025, 13, 2254. https://doi.org/10.3390/math13142254

AMA Style

Yuan H, Su W, Lian X, Yao Z-A, Hu D. Image Inpainting with Fractional Laplacian Regularization: An L^p Norm Approach. Mathematics. 2025; 13(14):2254. https://doi.org/10.3390/math13142254

Chicago/Turabian Style

Yuan, Hongfang, Weijie Su, Xiangkai Lian, Zheng-An Yao, and Dewen Hu. 2025. "Image Inpainting with Fractional Laplacian Regularization: An L^p Norm Approach" Mathematics 13, no. 14: 2254. https://doi.org/10.3390/math13142254

APA Style

Yuan, H., Su, W., Lian, X., Yao, Z.-A., & Hu, D. (2025). Image Inpainting with Fractional Laplacian Regularization: An L^p Norm Approach. Mathematics, 13(14), 2254. https://doi.org/10.3390/math13142254

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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