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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2018, Vol. 35 ›› Issue (6): 722-732.doi: 10.3969/j.issn.1005-3085.2018.06.011

• • 上一篇    

基于Allen-Cahn方程图像修复的算子分裂方法（英）

乔远阳1,   翟术英1,2,   冯新龙1   

  1. 1- 新疆大学数学与系统科学学院，乌鲁木齐  830046
    2- 华侨大学数学科学学院，泉州  362021
  • 收稿日期:2016-11-08 接受日期:2017-03-24 出版日期:2018-12-15 发布日期:2019-02-15
  • 基金资助:
    国家自然科学基金(11526094)；国家博士后科学基金(2015M582739)；新疆自治区研究生科研创新计划(XJGRI2016006)；福建省自然科学基金(2016J05007)；新疆大学优秀博士创新项目(XJUBSCX-2016007).

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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摘要： 本文提出了一种基于Allen-Cahn方程图像修复的算子分裂方法．其核心思想是利用算子分裂方法将原问题分解为一个线性方程和一个非线性方程，线性方程使用有限差分Crank-Nicolson格式进行离散，非线性方程利用解析方法进行求解，因此时间和空间都能达到二阶精度．由于该方法只作用于图像需要修复的区域，而其余区域的像素值与原始图像的保持一样，可以大大提高计算效率．合成图像和真实图像的数值实验验证了该算法的正确性和有效性．

关键词: 图像修复, Allen-Cahn方程, 算子分裂方法, 有限差分Crank-Nicolson格式

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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