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

Jian Wang, Ziwei Han, Junseok Kim

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number44
JournalZeitschrift fur Angewandte Mathematik und Physik
Issue number2
Publication statusPublished - 2024 Apr

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.


  • Allen–Cahn equation
  • Image inpainting
  • Phase separation

ASJC Scopus subject areas

  • General Mathematics
  • General Physics and Astronomy
  • Applied Mathematics


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