Association Journal of CSIAM
Supervised by Ministry of Education of PRC
Sponsored by Xi'an Jiaotong University
ISSN 1005-3085  CN 61-1269/O1
Chinese Journal of Engineering Mathematics

Chinese Journal of Engineering Mathematics ›› 2018, Vol. 35 ›› Issue (6): 722-732.doi: 10.3969/j.issn.1005-3085.2018.06.011

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An Operator Splitting Method for Image Inpainting Based on the Allen-Cahn Equation

QIAO Yuan-yang1,   ZHAI Shu-ying1,2,   FENG Xin-long1   

  1. 1- College of Mathematics and System Sciences, Xinjiang  University, Urumqi 830046
    2- School of Mathematics Science, Huaqiao University, Quanzhou 362021
  • Received:2016-11-08 Accepted:2017-03-24 Online:2018-12-15 Published:2019-02-15
  • Supported by:
    The National Natural Science Foundation of China (11526094); the China Postdoctoral Science Foundation (2015M582739); the Graduate Student Research Innovation Program of Xinjiang Municipality (XJGRI2016006); the Natural Science Foundation of Fujian Province (2016J05007); the Excellent Doctor Innovation Program of Xinjiang University (XJUBSCX-2016007).

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Abstract: In this paper, we propose an operator splitting method for image inpainting, based on the Allen-Cahn (AC) equation. The core idea is using an operator splitting method to decompose the original problem into a linear equation and a nonlinear equation. The linear equation and the nonlinear equation are solved by the finite difference Crank-Nicolson scheme and analytical method, respectively. So both time and space accuracy can achieve the second order. The method is only applied in the inpainting domain, while the pixel values of the rest region are kept as those in the original input image, which can improve the computational efficiency greatly. Accuracy and validity of the proposed method is illustrated through numerical experiments on synthetic and actual images.

Key words: image inpainting, Allen-Cahn equation, operator splitting method, finite difference Crank-Nicolson scheme

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