Image Restoration And Enhancement Algorithm Based On Partial Differential Equations

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.