基于Allen-Cahn方程图像修复的算子分裂方法（英文）

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

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CrankNicolson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.

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.