【作者】 李兰兰；

【导师】 吴乐南；

【作者基本信息】 东南大学 ， 信号与信息处理， 2005， 博士

【摘要】 图像处理中采用偏微分方程(PDE:Partial Differential Equation)的思想可以追溯到Gabor和其后Jain的工作,但实质性的创始工作应该归功于Koenderink和Witkin各自独立的工作,他们将图像与高斯核相卷积后的结果与偏微分方程的解联系起来,给出了图像多尺度描述的具体数学表达形式。变分法和偏微分方程具有一定的理论框架、多样的模型形式和快速的数值算法,将它们引入数字图像处理和计算机视觉领域,无疑给该领域提供了有力的解决问题的数学工具。近几年来,基于偏微分方程和几何曲率流驱动扩散的图像分析和处理方法越来越成为图像处理学中的研究热点,PDE是一种比较精细的图像分析和处理方法,可以用于图像降噪、增强和分割等领域。本论文主要围绕基于偏微分方程的图像处理方法而展开的,主要内容涉及图像复原、图像增强、图像的超分辨率处理、图像修描以及图像分割,概括如下:第一章介绍偏微分方程在图像处理中的主要应用及研究价值,概括了基于偏微分方程的图像处理方法的发展历史和研究现状。第二章主要讨论基于变分模型的图像恢复算法。用于图像恢复的变分模型有多种形式,适当选取模型的正则化参数能够提高图像恢复的质量。正则化参数可以是一个常数,也可以是一个变量。根据图像的局部特征来选取自适应变化的正则化参数是提高图像恢复质量的一个重要方法,可以根据像素在迭代过程中的亮度变化量大小来选取正则化参数。全变分正则化函数是Rudin,Osher和Fatemi等人提出的一种变分模型(ROF模型),用于图像恢复时能够很好保持图像边缘,是目前为止保持边缘图像恢复问题中比较成功的方法。全变分最小化图像恢复模型具有几个重要性质,本章根据这些性质对ROF模型进行了改进。提出了一种RGB颜色空间中保持纹理特征的全变分模型,将变分模型推广到CB颜色空间中,可以用来恢复彩色图像,据此提出了一种改进的彩色图像恢复算法。第三章重点研究基于异向扩散滤波的图像平滑。图像预处理是图像分析和处理必要的前期工作,对图像滤波以减弱噪声的影响是预处理中一个重要的环节。对这种初级信号(图像)的处理呈现两个矛盾需求:一方面,我们希望提取信号的总体趋势,而不是局部信息;另一方面,希望较准确地检测信号的突变信息,在图像中的表征形式即为图像的边缘。1990年,Perona和Malik提出了非线性各向异性扩散滤波,在去除图像噪声的同时能够精确定位图像的边缘特征。异向扩散滤波的形式多样,本章分析了这些不同形式,例如复数域内的异向扩散滤波,和具有方向特性的异向扩散滤波。异向扩散滤波模型可以用于平滑压缩图像,去除压缩图像的方块效应,提高压缩图像的恢复质量。在前向各向异性扩散方程的迭代求解过程中,选择合适的尺度(即选择合适的停止时间)对于准确定位图像边缘和去除噪声有十分重要的意义。本章在小波分析的理论基础上,从能量的角度估算扩散迭代过程的停止时间,提出了一中选择异向扩散停止时间的新方法。小波分析能够提取信号的低频分量和高频分量,本章讨论了小波域中的各向异性扩散滤波方程。第四章主要针对基于冲击滤波模型的图像分辨率增强。用于去除图像噪声的反应扩散方程是抛物型的偏微分方程,1990年,Rudin等人首次将非线性双曲型方程用于图像增强中,提出了用冲击滤波模型对图像增强的算法。用冲击滤波模型增强带有噪声的图像会使图像噪声放大,因此,1994年,L. Alvarez等人提出了能够增强带噪声图像的冲击滤波模型,在锐化图像边缘的同时能够平滑噪声。图像的分辨率增强是图像处理的一个重要方面,是一种重要的、代价较低的改善图像质量的方法。在用通常的内插方法对图像进行放大时会引入噪声,并且使图像的边缘变模糊。本章研究了增强图像的冲击滤波模型,并且在异向扩散滤波模型和冲击滤波模型的基础上,提出了一种图像分辨率增强的算法。第五章主要探讨基于全变分模型的图像修描算法。图像修描(inpainting)是图像复原一个重要分支,也是图像处理学研究的热门课题之一。修描图像就是用不显眼的方式来处理图画、照片或影片,包括恢复受损图像,移走或取代照片中被选择的物体等等。在通信领域,由于干扰、噪声和信道拥塞等原因导致信道不可避免会有误码或丢包等现象发生,使接收端重建的视频图像质量大幅度下降。可以利用信号的时空冗余度和视觉生理-心理的掩蔽特性对错误信号进行掩盖处理,用图像修描可以做到这一点。常用的图像修描方法是插值,本章研究了基于全变分模型插值的图像修描方法,结合图像增强模型提出了同时对图像进行局部修描和增强的方法,并提出了修补纹理图像的算法。第六章研究了各向异性扩散滤波在图像分割中的一些应用以及用Snake活动轮廓模型进行图像分割。最后对全文进行了总结,并展望了进一步的研究方向。更多还原

【Abstract】 The idea that Partial Differential Equations (PDE) been used in image processing can be traced back to the work of Gabor and Jain followed. But the substantial and original achievement in this field should own to the individual effort of Koenderink and Witkin who connected the convolution of original image with Gaussian kernel with solution of PDE. Further more, they introduce the definite mathematical form of multiscale representation to image. Variation method and PDE have specific theory frame, various forms and fast numerical algorithm; hence it is undoubted that introducing these methods into digital image field and computer vision field affords a powerful mathematical tool to solve some problem in these fields. In recent years, image analysis and processing methods based on PDE and diffusion driven by geometric curvature flow has been interested in markedly. PDE algorithm is a fine method in image analysis and processing and can be used in many aspects, such as noise reduction, image enhancement and segmentation. This dissertation focuses on the variation problem and PDE in image processing applications. And the study is carried out in image restoration, image enhancement, image super-resolution and image inpainting based on total variation. Following is the outline of the dissertation.In Chapter 1, the major application of PDE in image processing and its value to research are introduced. A review to the history and actuality of image processing based on PDE is presented additionally.In Chapter 2, image restoration algorithm based on variation method is studied. To image restoration, there are various variation models, in which the quality of restored image could be improved by means of choosing regular parameter appropriately. Regular parameter could be constant or variable. It is an important method to improve restoration quality by choosing regular parameter according to local feature of images. For example, the variance of pixel brightness in iterative process is a useful feature. In 1990, Rudin, Osher and Fatemi proposed the Total Variation model (TV/ROF model) for image restoration, which is effective in noise reducing and edge preserving. TV model has many good properties. Here, ROF model has been improved by making use of these properties. A TV model for color image restoration is proposed, by which image texture can be preserved.. Furthermore, TV model has been extended to CB (Chromaticity-Brightness) color space to restore chromatic images. Then, an improved model for color image restoration in CB color space is proposed.Based on anisotropic diffusion filters, image smoothing is interested in Chapter 3. Image pretreatment is the first step in image analysis and processing. It is an important step for noise reduction by using image filter. There are two contrary requirements for this original signal or image processing. One is that we hope to extract not only local information but global features; the other is that we hope to detect sudden variations of signal exactly, which represented as edges in images. In 1990, Perona and Malik proposed the nonlinear anisotropic diffusion, which could locate image edges accurately. There are various anisotropic diffusion models,such as anisotropic diffusion model in complex number field, Gabor model and anti-geometric diffusion model. Anisotropic diffusion filters could be used to smooth compressed images for block removing and quality improving. In the process of forward iterative, for finding solution of anisotropic diffusion equation, it is important to choose accurately appropriate scale, appropriate stopping time, for noise removal and image edge location. Based on wavelet theory, the method for choosing stopping time is proposed from the viewpoint of energy.Chapter 4 focus on image enhancing in resolution based on shock filters. Usually, reaction-diffusion equations for image smoothing are parabolic partial differential equations. In 1990, Rudin etc. applied the shock filter model, a nonlinear hyperbolic equation, to image enhancement firstly. However, noise will beamplified when image is enhanced via this shock filter model. To overcome this problem, in 1994, L. Alvarezs proposed another shock filter model which could enhance the noise-stained images. This model can not only sharp image edges but also reduce noise. Super-resolution is a branch of image processing. It is an important and inexpensive method to improve image quality. Noise and blurred edges will be added to image by using general interpolation method during image enlargement. Here, super-resolution algorithm based on shock filter and anisotropic diffusion is proposed.Chapter 5 has an interest in image inpainting algorithm based on total variation. Image inpainting is a small branch of image restoration and it is a hot topic in digital image processing field. Through a normal and undetectable way, image inpainting aims to produce a revised image, photo or video. And the inpainted region is seamlessly immerged into the original image. Because of interference, noise and channel congestion, bit error and package loss are always inevitable in communications. So, it becomes worse to reconstruct video frames at a receiver. Erring signals can be concealed by making use of space-time redundancy and vision features, and image inpainting can do this work well. General method of inpainting is to interpolate. In this chapter, image inpainting based on total variation is focused on. In addition, an algorithm for local image inpainting and global enhancing is proposed.In Chapter 6, applications of anisotropic diffusion in image segmentation are discussed.更多还原