Wavelet And Partial Differential Equation Based Image Processing Methods And Their Applications

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 set up their relationship both in application and theory。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. For instance, the Embedded Zertree wavelet, ETW, proposed by Shaprio etc. has been widely used in image compression and the video coding algorithm based on wavelet transform has been used in several international standards about image compression. 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 applications of wavelet transforms, including wavelet analysis based medical MRI image restoration, image smoothing in the wavelet domain, multi-wavelets and image features extraction. In the medical image processing application, based on the tool of wavelets, we analyzed the causes producing DC artifacts, and the suppression methods were given; after introducing several traditional denoising methods, a new bilateral filter in wavelet domain was proposed; based on the constraint of phase congruency, a multi-wavelet based image denoising method was put forward; for the composite edge detection, we first constructed a new multi-wavelet function, and then, with this function, a new composite edge detector was proposed. Theoretically, this detector can reach any detection accuracy for composite edges. Experiments demonstrated that our method had a higher accuracy for the detection of composite edges than Canny's operator, Prewitt's operator and the single wavelet edge operator proposed by Mallet-Zhong. It has been more than 300 years since people first did the research on partial differential equation (PDE). The earliest partial differential equation problem was introduced in mechanics, geometry and physics. In recent years, many partial differential equation problems are also appeared in life science and economics. How to apply partial differential equation in image processing is another important part of the thesis. One advantage using partial differential equation in image processing is there are many numerical computational methods available. Experiments demonstrated that partial differential equation plays an important role in the development of image science. For example, the total variation model can represent a large number of image types; the thermal diffusion equation can simulate image degeneration phenomena, and the reverse equation can be successfully used in image restoration; level set can track the evolving process of wave front and used in image segmentation and target recognition. This thesis addressed image smoothing and segmentation, proposed a new morphological erosion operator, and with this operator, a new edge detector was proposed. Based on the eroding results of gradient data, a new anisotropic diffusion scheme was designed. To solve the parameter selection problem in C-V image segmentation, a new level-set based segmentation method was proposed, which can segment image automatically while its parameter selection is quite easy. The combination of PDE and wavelet analysis will provide a solid base for the establishment of image science. Currently their theoretical relationship is very weak. Whether it can play the same role as Fourier analysis did is unknown. This thesis discussed the correlation between wavelet analysis and PDE in image processing based on the researches on them. As a brief combination of them, a new multi-resolution analysis based level set image segmentation scheme was proposed. Furthermore, wavelet based numerical homogenization and the heatlet decomposing of linear PDE were studied. Finally, the summary and further research directions were given.