数字图像修复技术综述

图像修复是图像复原研究中的一个重要内容,它的目的是根据图像现有的信息来自动恢复丢失的信息,其可以用于旧照片中丢失信息的恢复、视频文字去除以及视频错误隐藏等。为了使人们对该技术有个概略了解,在对目前有关数字图像修复技术的文献进行理解和综合的基础上,首先通过对数字图像修复问题的描述,揭示了数字图像修复的数学背景;接着分别介绍了以下两类图像修复技术:一类是基于几何图像模型的图像修补(inpainting)技术,该技术特别适用于修补图像中的小尺度缺损;另一类是基于纹理合成的图像补全(comp letion)技术,该技术对于填充图像中大的丢失块有较好的效果;然后给出了这两类方法的应用实例;最后基于对数字图像修复问题的理解,提出了对数字图像修复技术的一些展望。  

Dual Norms and Image Decomposition Models

Following a recent work by Y. Meyer, decomposition models into a geometrical component and a textured component have recently been proposed in image processing. In such approaches, negative Sobolev norms have seemed to be useful to modelize oscillating patterns. In this paper, we compare the properties of various norms that are dual of Sobolev or Besov norms. We then propose a decomposition model which splits an image into three components: a first one containing the structure of the image, a second one the texture of the image, and a third one the noise. Our decomposition model relies on the use of three different semi-norms: the total variation for the geometrical component, a negative Sobolev norm for the texture, and a negative Besov norm for the noise. We illustrate our study with numerical examples.  

整体变分算法在图像修补中的应用研究

对如何将整体变分模型用于图像修补进行了讨论．主要研究整体变分算法在待修补区为长条形空白区域的图像修补，并根据待修补区为长条形的特点，通过引入权值，对整体变分模型的离散化算法作了改进，使该算法利用邻域信息仅对待修补空白区域进行填充，而不改变待修补区邻域的像素值．实验表明，文中算法对窄长条状或线状空白区域的图像修补是有效的。  

Image Decomposition into a Bounded Variation Component and an Oscillating Component

We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.Jean-François Aujol graduated from 1 Ecole Normale Supérieure de Cachan in 2001. He was a PHD student in Mathematics at the University of Nice-Sophia-Antipolis (France). He was a member of the J.A. Dieudonné Laboratory at Nice, and also a member of the Ariana research group (CNRS/INRIA/UNSA) at Sophia-Antipolis (France). His research interests are calculus of variations, nonlinear partial differential equations, numerical analysis and mathematical image processing (and in particular classification, texture, decomposition model, restoration). He is Assistant Researcher at UCLA (Math Department).Gilles Aubert received the These dEtat es-sciences Mathematiques from the University of Paris 6, France, in 1986. He is currently professor of mathematics at the University of Nice-Sophia Antipolis and member of the J.A. Dieudonne Laboratory at Nice, France. His research interests are calculus of variations, nonlinear partial differential equations and numerical analysis; fields of applications including image processing and, in particular, restoration, segmentation, optical flow and reconstruction in medical imaging.Laure Blanc-Féraud received the Ph.D. degree in image restoration in 1989 and the Habilitation á Diriger des Recherches on inverse problems in image processing in 2000, from the University of Nice-Sophia Antipolis, France. She is currently director of research at CNRS in Sophia Antipolis. Her research interests are inverse problems in image processing by deterministic approach using calculus of variation and PDEs. She is also interested in stochastic models for parameter estimation and their relationship with the deterministic approach. She is currently working in the Ariana research group (I3S/INRIA) which is focussed on Earth observation.Antonin Chambolle studied mathematics and physics at the Ecole normale Supérieure in Paris and received the Ph.D. degree in applied mathematics from the Université de Paris-Dauphine in 1993. Since then he has been a CNRS researcher at the CEREMADE, Université de Paris-Dauphine, and, for a short period, a researcher at the SISSA, Trieste, Italy. His research interest include calculus of variations, with applications to shape optimization, mechanics and image processing.  

一种基于整体变分的图象修补算法

图象修补是图象恢复研究中的一个重要内容，它的目的是根据图象现有的信息来自动恢复丢失的信息，可以用于旧照片中丢失信息的恢复。由于图象中的边缘代表了图象的重要信息，所以在设计修补算法时，必须着重考虑边缘的恢复，采用整体变分模型设计了一个图象修补算法，整体变分模型能够模拟人的低层视层，在修补图象时可以恢复图象中的边缘，数值实验表明，该模型能够较好地恢复待修补区域的信息，但是受修补区域大小的影响，同时又采用了一种向前传播操作来缩小修补区域。  

数字破损图像的非线性各向异性扩散修补算法

首先从局部坐标角度分析整体变分（TV）模型与p-Laplace算子的物理意义,从本质上说明p-Laplace算子的扩散性能优于TV模型,进而提出一种基于p-Laplace算子的图像修补算法.该算法利用p-Laplace算子的非线性各向异性扩散的性能来填充受损区域.与TV修补算法相比,文中算法能快速收敛,并达到更好的修补效果,其综合性能优于TV修补算法.  

On the Incorporation of Time-delay Regularization into Curvature-based Diffusion

A new anisotropic nonlinear diffusion model incorporating time-delay regularization into curvature-based diffusion is proposed for image restoration and edge detection. A detailed mathematical analysis of the proposed model in the form of the proof of existence, uniqueness and stability of the viscosity solution of the model is presented. Furthermore, implementation issues and computational methods for the proposed model are also discussed in detail. The results obtained from testing our denoising and edge detection algorithm on several synthetic and real images showed the effectiveness of the proposed model in prserving sharp edges and fine structures while removing noise.  

一种新的运动模糊图像恢复方法

通过对运动模糊产生原因的分析，提出了一种去运动模糊的新方法。首先应用Hough变换和自相关函数估计出运动模糊的方向和长度，然后应用迭代步长自适应的整体变分模型进行图像恢复。实验结果表明，这样的空间域处理方法,不但可以避免传统的频率域去模糊方法产生的震铃效应，而且该方法具有良好的抗噪性和对运动模糊参数估计误差的低敏感性。  

基于图的数字全变差模型及其带噪图像任意精度放大

分析了利用Sobolev空间Tikhonov正则化模型对带噪图像进行放大的不足，基于图像的修复模型，提出带噪图像放大的数字全变差模型．利用有向图构造出兼顾噪声去除和图像放大的数字TV滤波器，并利用该滤波器提出一种新颖的图像放大算法．作为算法对比，利用Sobolev空间Tikhonor正则化模型，提出相对应的数字Tikhonov放大算法．结果表明：数字TV放大算法明显优于数字Tikhonov放大算法，不仅较好地抑制了噪声的影响，而且使得任意精度放大的图像边缘清晰、过渡自然，特别适合于目标边缘明显的一类非纹理的医学图像的放大。  

基于总变差的图像放大和增强方法

利用小波的多分辨分析和总变差极小原理，提出了一种实现图像放大和增强的新方法。该方法把原图像作为放大图像的小波子带，对放大图像强加了一种小波系数型的限制，放大图像利用图像总变差极小模型进行正则化c经求变分产生带限制的非线性扩散方程作为总变差极小的必要条件，求解偏微分方程得到增强的放大图像。对人工合成图像、医学MRI心脏切片和人物图像进行了实验。实验结果说明该方法同时实现图像放大和增强的有效性。