一种基于微小区域的TV双调和型偏微分方程图像修复方法  

A TV harmonic type PDE approach of image inpainting based on observation on a meager domain

作　　者：孙俊岭[1,2] 杨杰[1] SUN Junling;YANG Jie(School of Information Engineering,Wuhan University of Technology,Wuhan 430070,Hubei,China;School of Mathematics and Information Science,Henan University of Polytechnic,Jiaozuo 454000,Henan,China)

机构地区：[1]武汉理工大学信息工程学院,湖北武汉430070 [2]河南理工大学数学与信息科学学院,河南焦作454003

出　　处：《河南理工大学学报（自然科学版）》2018年第3期150-156,共7页Journal of Henan Polytechnic University（Natural Science）

基　　金：国家自然科学基金-河南省人才培养联合基金资助项目(U1404103)

摘　　要：针对具有微小缺失或破损的图像修复问题,基于变分和偏微分方程理论基础提出一种TV双调和偏微分方程图像修复方法。该模型在Sobolev对偶分布空间H^(-1)(Ω)上考虑一个变分问题,它的Euler-Lagrange方程是一个与多孔介质方程相关的非线性双调和椭圆扩散方程。首先,将方程归结为一个最小化问题,再利用双调和椭圆边值问题得到Euler-Lagrange最优条件下的广义解,最后采用尺度空间的稳态解和最小二乘法进行数值模拟,从峰值信噪比、平均结构相似度和视觉效果等3方面对模拟实验的图像进行评价。该模型允许观测图像f可以不是标准L^1函数,而是H^(-1)分布或测度,因此,适用于修复一组任意小的带条或者像素有限点集。该算法充分利用现有像素的邻近信息,对3种非纹理类型的具有较小或较细缺损的图像修复效果有所改进,数值实验也说明这种方法在修复过程中,明显保护了边缘特征、有效避免了阶梯效应,提高了视觉质量。We present a new TV-PDE approach of image inpainting based on an incomplete observation of the blurred image. The method is proposed starts from a related variational problem as well as considered in the Sobolev distribution space H^-1（ Ω）. Its Euler-Lagrange equation is a nonlinear bi-harmonic elliptic diffusion equation related to porous media equation. The equation is belong to a minimization problem. The biharmonic elliptic boundary value problem is used to obtain the generalized solution under the Euler-Lagrange optimal condition. The advantage of the approach is that the observation f can be taken as a distribution with support in a finite number of pointsk{（ X,yh） }l,h = 1^n,m. This means that it can be applied to the image inpainting for a subdomain with zero Lebesgue measure. Steady scale space solutions and the least squares method are used for numerical simulation. We evaluate analog images from three aspects： peak signal to noise ratio（ PSNR）,structural similarity index and visual intuition. The simulation experiments of nontexture image show that the proposed algorithm can make full use of the existing information adjacent pixels and has better performance of inpainting. The algorithm is suitable for small defect cases. In the process of restoration,this method protects the edge feature,avoids the step effect and improves the visual quality significantly.

关 键 词：图像修复 变分偏微分方程 平均结构相似度 欧拉-拉格朗日方程 

分 类 号：N03[自然科学总论—科学技术哲学]