Wavelet Designing Based On Slant Transform And Application

Wavelet analysis is a new subject developed at the 80's of the last century,it is the production of mathematic theory and practical application. Wavelet transform is a tool of signal analysis in recent years, and it becomes a very efficient method and has important applications in the image code because of its well time-frequency trait and multi-resolution analysis trait.The Walsh analysis is the development of the Fourier analysis. It is very suitable for data signal process and applications in many field such as signal processing and image processing in that only two numbers, +1 and -1, and the same property with sine function and cosine function are included in the Walsh function. The Walsh transform is faster than the fast Fourier transform, it is more appropriate for 2-D data processing and real time data processing. The Walsh matrix and slant matrix are the orthogonal matrixes which have a close relationship with each other. Slant matrix is a transmutation of the Walsh matrix, and they are both the wavelet matrixes.The aim of this thesis is to contact the Walsh transform with wavelet transform. The wavelet designing based on slant transform are presented in this thesis. A new design idea of M-bond multiwavelet is proposed to change the complexity of the procedure in deriving filter of the traditional wavelet. The constructing process of filter group, based on the fast algorithm of slant transform, is simple, convenient and more efficient in realization. However, for traditional wavelet, the constructing process is not only inconvenient, but also complicated. In addition, the form of filter group is out of order, and the implement of filter group is more difficult than the method presented in the thesis. M-bond multiwavelet discussed in this thesis is applied more abroad in practice. Two numerical experiments about wavelet application based on the slant transform are presented in this thesis. The one is about image denoising, and the other is image compressing. According to the numerical experiments, the results of the image denoising and image compressing show that this scheme is more excellent and efficient than the traditional wavelet.