基于小波分析的偏微分方程图像处理方法/Image Processing Methods Using PDE Based on Wavelet
图像科学是一门集多学科于一体的交叉学科，与相关学科的基础理论在该学科的成功应用密不可分。在图像处理中，无论是图像模型的建立，图像特征的描述，图像处理算子的设计，还是图像优化处理中的泛函极小化，最终都可归结为一个数学理论问题。特别是近年来，以小波分析和偏微分方程(Partial derivative equation,PDE)为代表的数学工具活跃在图像处理的各个研究领域，“图像科学”正在形成，并逐步为人们所接受。该文旨在以小波分析和偏微分方程为主要工具，对底层图像处理中图像恢复、边缘提取等问题进行实验研究，得出它们在图像处理中联合使用效果较好。
Image science is an inter-discipline subject and has a great relationship with the successful applications of multi-subject, especially mathematics. In image processing, no matter image modeling, representation of image contents, description of image feature,design of image processing operator, or energy functional minimization of image optimization procedure, all can be upgraded to a mathematic problem. Especially in recent years, as a representation of mathematic tools, wavelet analysis and partial differential equation (PDE) are active in many image-processing fields, making the name of "Image Science" gradually being accepted by people. The aim of this thesis is to study the applications of wavelet and partial differential equation in image processing, and try to get a good effect when they are both used in image processing.
As a new mathematics tool, wavelet analysis is a perfect integration of functional analysis, Fourier analysis, spline analysis, harmonic analysis and numerical analysis. It has been applied in computer vision, image processing, object recognition and other fields and a great number of advances have been made both in theories and techniques. It also plays an important role in image restoration, feature extraction, as well as object recognition. However, in other fields of image processing, there is still a big gap between people's expectation about wavelet analysis and its applications. As an important part of this thesis. it focuses on the definition and the construction of optimal biorthogonal wavelets base, proves the definition for optimal biorthognal wavelets base is reasonable by analyzing the data on image processing, gives a base on image processing for using optimal biorthognal wavelets base constructed formerly and partial differential equation.
The combination of PDE and wavelet analysis will provide a solid base establishment of image science. Currently their theoretical relationship is very weak. Whether it can play the same role as Fourier analysis did is unknown. We analyze the data on image peocessing for one dimensional and multidimensional case on wavelet analysis and partial differential equation based image processing methods in this thesis, improve the original Weickert model: redesign diffusion tensor D using muti-resolution analysis character for wavelet transform. Theory analysis based we analysis and compare the filter result on improved model and original model. The result indicates the improved model has a better filter result than the original model and P-M model: the edge information and detail for image is better than theirs. Finally, the summary and further research directions were given.