Generalized Gaussian Distribution And Its Application In Sparse Representation

Generalized Gaussian distribution (GGD) is a kind of important non-Gaussian distribution, In 1972 Box first discussed the characters of this distribution as well as the Bayes inference. GGD is a broader statistical distribution including Gaussian distribution. At present, GGD has been widely adopted to data modeling in many different fields such as image processing, speech signal processing, digital watermarking, blind signal separation(BSS), synthetic aperture radar (SAR), ultrasonic cardio-gram images, face recognition, power system load demand, image subband signals, and independent component analysis (ICA), etc. In these applications, the shape parameter of GGD plays a key role on the effect of data modeling, this dissertation researches some frequently-used estimation methods of the shape parameter which are based on the basic properties of GGD.The theory, methods and applications about sparse representation have been paid the utmost attention of scholars from abroad and at home in recent years, signal sparse representation can extract the intrinsic characters of signals. It has been widely used in the signal processing fields, including signal compression, feature extraction, denoising, super-resolution reconstruction and so on. Whether signals (or data) can be sparsely represent or not and the effect of sparse represent play an important role in data compressing, transfer and extraction process. However, there are still some problems to solve in compressed sensing (CS) and sparse representation theory. At present the researches of sparse representation theory and method focuses on the selection of kernel functions and the determination of its parameters, sparse decomposition algorithm and optimization algorithm, etc.This dissertation focuses on the research about GGD, including the basic properties of GGD, the methods of estimating the shape parameter in GGD, the strong convergence property of moment method, and the applications in signal sparse representation where GGD is took advantage of kernel functions, support vector machine, as well as the kernel density function of sparse kernel density estimation.The major work of this dissertation can be summarized as the following four aspects.One, the basic properties of GGD are discussed in detail, and the basic methods of estimating the shape parameter in GGD and the statistics property of moment method are studied;Two, for the character of poor signal sparse decomposition effect by taking use of Gaussian kernel functions, a new idea is put forward to the signal sparse decomposition by taking advantage of the generalized Gaussian kernel function, while the repeated weigthed boosting research (RWBS) algorithm is applied to solving the optimization problems of seeking the signal sparse decomposition, the simulation results show that, new generalized Gaussian kernel can decompose the signal to achieve higher sparsity under the same conditions of decomposition accuracy, and the RWBS algorithm is able to reduce the time complexity;Three, for deficiencies of higher complexity and poor generalization performance that radial basis function solves certain signal regression problem, a new support vector machine (SVM) based on generalized Gaussian kernel function is introduced, and the new SVM is proved to meet the Mercer condition. The simulation results show that the signal regression problems of using the generalized Gaussian SVM can acquire lower computational complexity and better generalized performance.Four, in order to further increase the accuracy and adaptability of density function estimation the generalized Gaussian kernel density function is put forward to improve the sparse kernel density estimation method of using Gaussian kernel density function. The simulation results show that the use of the generalized Gaussian density function in sparse kernel density estimation can achieve higher accuracy, and has better adaptability.The dissertation consists of the following seven chapters.Chapter one first describes the background and significance of the study about sparse decomposition taking advantage of GGD, while the course of development and research status of this problem is reviewed, and the existing main problems in the present study are pointed out, the research content of this dissertation and the significance of topics are put forward clearly according to the above statements.Chapter two studies the definition of GGD, the basic definition and properties of higher-order cumulants of generalized Gaussian random variable, and the properties of the sum of independent generalized Gaussian random variables.Chapter three studies some basic methods of estimating the shape parameter in Generalized Gaussian distribution, as well as the strong convergence property of moment estimation.Chapter Four studies the applications in signal sparse decomposition for generalized Gaussian kernel function, the emphasis the signal sparse decomposition algorithm based on fixed atoms dictionary and RWBS, and analyzes the differences of two signal decomposition algorithms based on the generalized Gaussian kernel functions and the Gaussian kernel functions by the simulation experiments.Chapter five studies the applications in the signal sparse decomposition for generalized Gaussian SVM. The generalized Gaussian SVM is firstly introduced and proved to satisfy the Mercer condition, at the same time, the simulation experiments are studies to show that the generalized Gaussian SVM has the better effect of decomposition such as the lower computational complexity and the better generalization performance than the general Gaussian SVM.Chapter six studies the applications in sparse kernel density function estimation for the generalized Gaussian kernel density function. Some basic methods of sparse kernel density estimation and its properties are summarized in this chapter, and a new method based on the generalized Gaussian kernel density function for sparse kernel density estimation is put forward, as well as, the simulation experiments are carried out to show that the new method based on generalized Gaussian density function can obtain higher estimation precision and the better ability to adapt to the various signals than the method based on Gaussian density function.Chapter seven gives a brief summary of this dissertation, and also put forward some suggestions for further researches related to the work on this dissertation.