数字图像修复的变分方法与实现过程

VARIATIONAL METHOD FOR IMAGE INPAINTING AND ITS IMPLEMENTATION

图像修复是数字图像处理过程的一个很重要的方面.图像修复目的是将图像中污损或破损的部分运用相关的方法将其恢复.本文主要讨论数字图像恢复的变分方法及其实现过程,重点讨论变分方法之中的偏微分方程模型建立的基本过程和求解方法.图像恢复的变分方法的核心思想是将恢复过程归结为求解一个含约束条件的泛函极小值问题.为得到此泛函极小值问题的解,先根据拉格朗日乘子法,将含约束条件的泛函极小值问题化为无约束的泛函极小值问题.由于无约束的泛函极小值问题的解满足一偏微分方程,于是可构造一梯度流并通过它找出该偏微分方程的解.最终用偏微分方程数值方法-有限差分法来离散得到此梯度流的稳态解的近似,并将此近似解作为图像修复之后的结果表示.

Image inpainting is one of the most important part in digital image processing.It aims to fill in missing parts of damaged or degraded images based on the information around.In this paper,we discuss how to do image inpainting with variational methods and how to implement it in computers.We mainly investigate the general way on how to build partial differential equations based on variational methods and how to solve subsequent models.The key idea of variational methods for image inpainting is turning the inpainting problem into a functional minimization problem with constraints.To solve this minimization problem,we first apply the Lagrangian multiplier and solve a free functional minimization problem instead.The solution of the latter one satisfies Euler-Lagrangian equation.We next construct a continuous gradient flow and discretize it with finite difference method in order to find the steady state solution of the gradient flow,which is in fact the solution of related Euler-Lagrangian equation.We finally use the numerical solution as the approximation of inpainted image.