Research On The Construction Of Univariate Non-stationary Daubechies And Multivariate Non-stationary Wavelets

Wavelet analysis, proposed nearly30years, is a signal analysis and Processingtechnology, with great value in theory research and application, and a far-reaching influence.Now, the theories for univariate wavelets have developed rapidly, but among them,multivariate wavelets received special attention due to many applications involved inmulti-dimension, in which multivariate wavelets is divided into non-separable and separable.There are two methods for non-separable wavelets construction. One is to construct waveletswith special formal through definition or properties directly, and give a fixed sampling matrix.The other is to construct the polyphase matrixs and build the filters (coefficient mask) basedon elements of the polyphase matrixs. Compared to separable wavelets, non-separablewavelets have more design freedom and avoid man-made priority axis resulting in signaldistortion. But in nature, most of the signal is the non-stationary signal which base ontime-varying, the above stationary wavelets based on algebraic polynomial does not considerthe smooth features of the signal. In dealing with these non-stationary signals, the signal willbecome increasingly weak, and the signal features will be more and more fuzzy, or evendestroy the integrity of the signal features. In recent years, the non-stationary waveletconstructed by exponential spline can adapt to the frequency changes of the non-stationarysignal, which can attune changes with the signal feature, protect the signal feature very welland be more suitable for human visual system. Therefore, the construction and research of thenon-stationary wavelets has been the focus of many experts and scholars in the field oftheoretical and applied of Wavelet analysis.Vonesch uses exponent polynomial vanishing moment instead of the algebra vanishingmoment of Daubechies wavelet, and constructs non-stationary generalized Daubechieswavelet. In Chapter2, according to the structure of the generalized Daubechies wavelet, weadd an odd polynomial, construct a class of non-stationary Daubechies wavelet. In Chapter3we use the construction methods of Belogay and Wang, select the sample matrix, providingthe coefficient mask be only two rows of non-zero and neighboring, the filter adoptingexponent polynomial, through the vanishing moment conditions and orthogonality, weconstructed a non-stationary orthogonal scaling function which has two rows and any smooth.In Chapter4, according to the iterative construction method of Karoui, we select a class ofmatrices satisfying certain conditions through low-dimensional filter iteration ofmulti-dimensional filter on each non-stationary space, and then use the sample matrix, weconstructed multi-dimensional non-stationary orthogonal wavelet which is non-separable. There are different methods of constructing between Chapter3and Chapter4, in Chapter3we use the first construction method and in Chapter4we use the second, in two-dimensional,the results of Chapter4does not contain the results of Chapter3.