Bounds Estimate Of Sub-band Operators And Saddle Point Wavelets

Wavelet analysis is a new offset of mathematics which was developing very fast in 1980s. During the last two decades, the theory of wavelet still providing a reconstruction formula, has been growing rapidly, since several new applications such as nonlinear sparse approximation (e.g., image compression), coarse quan-tization, data transmission with erasures, and wireless communication, have been developed. Compare to other bases, wavelet bases have some advantages such as localize character, mathematical micro-telescope feature and adaptive feature. The main contents of this thesis are the construction and application of multi-band wavelets. We have investigated sub-band operator to directly evaluate or examine the performances of filters. Specifically, we define the sub-band operator, give the exact bounds of sub-band operators and design some excellent wavelets, namely, saddle point wavelets. The outline of this thesis is as follows:In chapter one, the history of wavelet development, the motivation and the primary results of the thesis are introduced.In chapter two, based on MRA, multi-band wavelet frame is introduced and some methods to construct multi-band scaling functions are summarized.In chapter three, according to the UEP regulation, we consider in detail the method to construct multi-band wavelets with symmetry. Because most useful properties are only related to the scaling filter, a typically two-step construction procedure is introduced. The first step is to design the scaling filter carefully, and the second step is to choose the wavelet filters from the given scaling filter. So, odd-band wavelets are constructed based on two-step construction. More-over, this step-by-step algorithm is easily implemented by computer program. We construct 4-band biorthogonal wavelets with some structure, in which the half filters can be determined by exchanging position and changing the sign of the other half filters. We propose an algebraic approach to construct multi- band wavelets by solving constraint equations, which can partially overcome above disadvantages of the classical two-step construction. Moreover, we can construct innumerable wavelet bases, among which we can select the best ones for practical applications.In chapter four, firstly, some definitions and properties of Toeplitz matrix and circular matrix are introduced. Secondly, a concept of sub-band operator is defined and some properties are obtained such as the bounds estimate. Finally, the method to calculate the bounds of the sub-band operators is described by computing the maximum of a function. The exact bounds of the sub-band operators of 2-band and 4-band biorthogonal wavelets with specially structure are obtained.In chapter five, compared to orthogonal wavelets, biorthogonal multi-band wavelets have many advantages, which are especially striking due to their sig-nificant impact on applications, especially in signal and image processing. It is found that the size of bound of a sub-band operator is very sensitive to the performance of wavelet transforms. So a model to minimize the bounds is built and then a class of wavelets, namely, saddle point wavelets, is designed. Ex-periments show that some saddle point wavelets with the filters of even lengths perform better than the well-known 9-7-tap wavelet in terms of image compres-sion. In addition, the design method of saddle point wavelets can be extended to multi-band biorthogonal wavelets, and the exact value of bound is an important index for evaluating a non-orthogonal wavelet system.In chapter six, conclusions and future work are summarized.The main innovations of this thesis are as follows:1. General methods to construct odd-band wavelets are proposed based on the two-step construction. Moreover, this step-by-step algorithm is easily implemented by computer program.2.4-band biorthogonal wavelets with some structure are constructed, in which the half filters can be determined by exchanging position and changing the sign of the other half filters. We propose an algebraic approach to construct multi-band wavelets by solving constraint equations.3. A concept of sub-band operator is introduced, a 2-circular matrix method is developed and the exact bounds of the sub-band operators are ob-tained. The method to calculate the bounds of the sub-band operators is de-scribed by computing the maximum of a function. In addition, it is found that the size of bound of a sub-band operator is very sensitive to the performance of wavelet transforms. So a model to minimize the bounds is built and then a class of wavelets, namely, saddle point wavelets, is designed. In addition, the design method of saddle point wavelets can be extended to multi-band biorthogonal wavelets, and the exact value of bound is an important index for evaluating a non-orthogonal wavelet system.