The Research On Some Problems For Wavelet Analysis With Its Applications

Wavelet analysis and its applications have become more and more important in the field of numerical analysis and information science . In this paper, we focus on several problems in this fields,mainly some theoretical problems and algorithms, and verify the significance when these approaches are used in signal and image processing.This paper is composed of the algorithm design of dual-tree binary coefficients complex wavelets,MGM algorithm for linear systems that can be diagonalized by generalized discrete Fourier transform (GFT) and type-II discrete sine transform matrices, wavelet and MGM algorithms for Toeplitz systems and their applications in signal and image processing. The main theory results includes: (1) Using the properties of Hilbert transform, perfectly reconstruction and new type of lifting scheme, a new type of dual-tree binary coefficients complex wavelet with linear phase is achieved. (2) For linear systems that can be diagonalized by GFT and DST-II matrices, an efficient MGM method is proposed, convergence is proved. (3) We discuss the algebraic structure when Toeplitz matrix is transformed by multi-band wavelet,show that Toeplitz matrix is composed of generating function is transformed to a band and sparse matrix when wavelet applied to this matrix, based on the above results, an efficient solution of Toeplitz equations is obtained, and the computational complex is O(N),where N is the order of matrix. (4) Using wavelet, we design new interpolation and prolongation operators to develop fast algorithm for solving Toeplitz equations via multi-band wavelet and MGM, efficiency of the method is verified by numerical experimentation. The mainly application results includes: (1) Efficient regularization restoration methods for signal and image via the combination of wavelet, PCG and MGM are proposed, simulation show that the presented algorithms are very highly efficiently and precision. (2) We first design an multi-variant shrinkage denosing method, jointly using complex wavelet,diffusion equation and image enhancement, a new method via time field and frequency domain for image denoised is also proposed, simulation experimentations show that significant denoising results are achieved. Moreover, for a parabolic variational inequality, we also propose a domain decomposition method for the Euler-Galerkin approximation of a parabolic variational inequality. It is proved that the convergence rate of the algorithm is independent of the space meshsize h and the time stepsize.