Researches On Digital Image Processing Based On Partial Differential Equation

Low level image processing always restricts the development of computer vision. In recent years, partial differential equation based image processing method has gained much attention in the international academic community, and hopefully is a solution to shortcomings of classical heuristic image processing method. Illustrated by existing research results, The results from the present investigation suggest that, partial differential equation based image processing method can be used in the following areas: image restoration, image segmentation, super-resolution, stereo vision, image inpainting, image classification, video sequence analysis, etc. and related achievements can be promptly transferred to many occasions, such as medical image analysis, remote sensing image processing, robot vision, video surveillance, video compression and so on. Actually, partial differential equation based image processing belongs to basic research, and crossed by multiple subjects, such as partial differential equation, variation calculus, differential geometry, numerical analysis, classical mechanics and image processing. Because partial differential equation based image processing covers broad and deep theories, currently researches on this aspect is still in the early stage in our country.Based on widely investigating the latest developments of theory and practice, this paper systematically and detailedly presented basic principles, design ideas, numerical solutions and experimental results of some famous partial differential equation based image processing models, and involve following research contents:In this dissertation, the framework and mathematical background of image processing technique is first introduced, then, the derivation of partial differential equation in image processing, and the development course as well as advantages of partial differential equation based image processing methods are also introduced. Some important international journals and conferences, monographs, and leading research institutes in this field are given at the end of the first section.At the beginning of the second part, the explaination of digital image was first given from mathematical viewpoint, and then we detailedly introduced the mathematical theoretical basis of partial differential equation based image processing, mainly including the following aspects: partial differential equation, variation calculus and gradient descend flow, planar differential geometry, numerical analysis. Especially in the differential geometry section, the most famous level set theory proposed by Osher-Sethian is covered.The chapter three and four are central parts of the dissertation. Here two key problems in image processing: image restoration and image segmentation, are detailedly investigated separately, especially, in this paper, we focus on image processing method based on the evolution flow of partial differential equation. In the third section, some essential concepts and theories, such as Tikhonov regularization, anisotropic diffusion and scale space are first introduced. There is a focus on the analysis of the Perona-Malik anisotropic diffusion and its estimation in Robust Statistics, and also our improvement on Rudin-Osher-Fatemi Total Variation model. At the end of the chapter three, Osher-Rudin's shock filter and its improvement are briefly introduced.In chapter four, three kinds of image segmentation methods: edge-based, region-based and knowledge-based active contour models are mainly analyzed. In the edge-based image segmentation algorithm, we begin with geodesic active contour proposed by Caselles et al, and drawbacks of related algorithm are also analyzed. As to numerical implement, level set evolution without re-initialization is especially introduced. In the section about the region-based segmentation method, Mumford-Shah model and Chan-Vese simplified model are research emphases; here we proposed an improvement idea and speedup convergence of these two models. In the knowledge-based method, a kind of "smart" active contour model, i.e. active shape model proposed by Cootes-Taylor is briefly surveyed, and then a new snakes-searching based active shape model is proposed.In the five part of this paper, a kind of numerical scheme-finite difference method is discussed for solving three kinds of partial differential equations; here the emphasis and also the difficulty lie in how to numerically solve hyperbolic partial differential equation, and upwind scheme, CFL stable condition and several numerical discrete schemes for the Hamilton-Jacobi equation are introduced in turn.In the end we conclude the dissertation, briefly speaking, the main contributions of this dissertation include:1. After widely investigating latest research production, we systematically anddetailedly expatiate on fundamental, designment and approximation of partialdifferential equation based image processing method, and we believe this is thefirst attempt in our country. In addition, we give valuable matlab experimental results of almost all significant partial differential equation image processing models in image restoration and segmentation, and we hope our efforts can bring reference utility for our national scholars to research in this area.2. Among partial differential equation based image restoration methods, we first analyze the main drawback of Rudin-Osher-Fatemi Total Variation image restoration model is hard to initialization, i.e. mean and deviation restricted conditions must be satisfied simultaneously at the beginning, after this we can calculate the Lagrangian parameter. Here we propose a new Lagrangian parameter selection method implemented by separated iteration, thus improve the Total Variation model, speed up its convergence and enhance the quality of restored image.3. Enlightened by Li Chunming's new geodesic active contour model based on level set evolution without re-initialization, here we improve Chan-Vese and also Mumford-Shah model by adding an internal energy, and new numerical scheme greatly speed up their convergence.4. As to knowledge based image segmentation, a new snakes-searching based active shape model is proposed. Among the active searching process in active shape model, traditional way is to isolatedly adjust positions of separated landmarks, while in our new model, we proposed to regard the energy functional of snakes as the optimal measure of active searching, thus intrinsic relationship between landmarks is considered.