| The wavelet theory is still active as a subject. The scalar wavelets ( 2-channel wavelet) have been succeeded in both theory and application, of which the mathematical characters and the application performance have drawn enough attention. However, the scalar wavelets is only a part of the wavelet theory, and the multi-channel wavelet is more profound relatively, of which many mathematical characters and application performance have not been known until nowadays. So in this dissertation, the multi-channel wavelet became the subject of our studies, and some multi-channel wavelets with good mathematical characters and application performance were constructed. Concretely constructing multi-channel wavelet with good mathematical characters and application performance is a difficult work, though commonly theories about multi-channel wavelets are abundant. It is a difficult work to verify whether the multi-channel wavelets behave excellent application performance or not, even though we can construct it with good mathmatical characters. Moreover, it's not easy to compare 2-channel wavelets with multi-channel wavelets objectively, because the applicative research and the theoretical value are youthful relatively.This dissertation theoretically and practically studies the 4-channel orthonormal wavelet with symmetrical filter banks, including multiwavelets and muti-bank wavelets. We begin with a general multiwavelet system with symmetrical filter banks, whose scale and wavelet functions have no linear phase. After that, we focus on discussing the 4-bank wavelets. Previously published related works mainly come from "cross"4-bank wavelets proposed by professor Guoqiu Wang and"shift"4-bank wavelets proposed by professor Lizhong Peng, the main idea of which is that three filter banks can gain from one filter bank by changing symbols and (or) positions. We propose another new form of "shift"4-bank and compare the properties and performances of these three kinds of 4-bank wavelets in image compression coding application. Researches indicate that between 2-bank and 4-bank wavelets there is a difference the length of 4-bank wavelets with good performances usually is above 30. Experimental results also show that the 4-bank wavelet constructed in this paper performs better than those proposed by professorGuoqiu Wang and Lizhong Peng. Furthermore, research results show that the multi-channel wavelets do well in close-grained texture image compression coding. Some results preponderate over those of the scalar 9-7 tap wavelets. In this dissertation, we use SPIHT algorithm as the compression-coding algorithm. However, SPIHT does not favor the properties of the 4-bank wavelet transform. How to construct more efficient compression coding algorithms is left as our future work.This dissertation is organized as follows. In chapter 1, we introduce the history of the wavelets and briefly discuss the application of wavelets in image compression. In chapter 2, basic notions of multi-channel orthonormal wavelets are introduced. Meanwhile, the Mallat scheme is extended to M-channel space, which is the preliminary of the wavelet theory and application. In chapter 3, we propose a general technique to construct a new class of balanced orthonormal multiwavelets without linear phase by adopting M=2 and r=2. In chapter 4, we adopt M=4 and r=1, hence the usual 4-bank wavelet. We put forward a new form of "shift" 4-bank structure, and construct 4-bank wavelets family different from those proposed by professor Guoqiu Wang and professor Lizhong Peng. Furthermore, we calculate all these filter banks with different length according to image compression coding technique(regularity and vanishing). In chapter 5, we finally introduce the basic technologies about image compression. Besides, we compare the performances of all these three kinds of 4-bank wavelets and the scalar 9-7 tap wavelets in image compression.
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