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An efficient and explicit local image inpainting method using the Allen–Cahn equation

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Abstract

Image inpainting is the process of restoring damaged areas in an image using information available from neighboring regions. In this paper, we present a novel, efficient, and simple local image inpainting algorithm based on the Allen–Cahn (AC) equation with a fidelity term. We utilize the phase separation property of the AC equation and introduce a new phase-dependent fidelity parameter to preserve the original values in the neighboring regions of an inpainting region. The governing partial differential equation is solved using the finite difference method, with the values of the neighboring cells serving as the Dirichlet boundary condition. The proposed algorithm is both local and explicit, making it is fast and easy to implement. We demonstrate the performance of the proposed model through several numerical experiments. Furthermore, comparing this method to other image inpainting methods demonstrates its superiority in image inpainting.

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Acknowledgements

The first author Jian Wang expresses thanks for the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant Nos. 22KJB110020). The corresponding author (J.S. Kim) was supported by the National Research Foundation(NRF), Korea, under project BK21 FOUR. We would like to extend our deepest gratitude to the reviewers for their invaluable comments and suggestions that greatly improved the quality of this manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Jian Wang & Ziwei Han

  2. Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Jian Wang

  3. Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Jian Wang

  4. Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

    Junseok Kim

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  1. Jian Wang
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  2. Ziwei Han
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Correspondence to Junseok Kim.

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Wang, J., Han, Z. & Kim, J. An efficient and explicit local image inpainting method using the Allen–Cahn equation. Z. Angew. Math. Phys. 75, 44 (2024). https://doi.org/10.1007/s00033-023-02184-6

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  • DOI: https://doi.org/10.1007/s00033-023-02184-6

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