摘要
经典的小波理论是基于二带的情形研究的,然而在大量的实际应用中,需要对信号作更为精细的分析和处理以得到满意的效果,仅考虑二带小波是不够的,还需要进一步使用多带小波；另一方面,多带小波还具有一些二带小波所不具有的性质,如既正交同时又具有线性相位等。所以多带小波的理论与构造已经成为信息领域一个新的研究热点。本文在经典的二带小波理论基础上,研究了三带正交小波的若干性质并对其进行了严格证明,特别是证明了三带情况下尺度函数离散矩与连续矩的关系,并给出了代数方法构造三带正交小波的几个算例,该方法将尺度函数的构造问题转化为求解滤波器系数的约束方程组,克服了已有算法过于复杂或不能保证对称性质等缺陷。
小波框架及小波紧框架是近年来新的理论研究热点。Daubechies和Meyer分别在其著作《小波十讲》和《小波算子》中提出了生成框架的母小波的充分必要条件,Wang和Weiss对单变量小波紧框架进行了刻画,而后Bin Han给出了多元情况下小波紧框架的显式刻画。此外,Ron和Shen还给出了所有小波框架的一般刻画,并且该理论也可以特殊化到小波紧框架的情况。近来,Daubechies和Bin Han等构造出基于多分辨分析的小波紧框架,并对对偶小波框架的存在性和构造方法进行了讨论。考虑到多分辨分析在小波框架的构造中有着非常重要的作用,本文在小波紧框架和框架多分辨分析的基础上,提出了一个构造多带对偶小波紧框架尺度函数的充分条件,并给出了详细证明。
The typical wavelet theory is researched around two-band wavelets, but in order to acquire satisfactory outcoming with analyzing and processing the signal precisely, it is not enough to use only two-band wavelets, and M-band wavelets is needed. Besides, M-band wavelets have some good properties that two-band wavelets don't have, such as being orthogonal and liner phase simultaneously. So it has been a hotspot in information fields to research M-band wavelets'theory and construction. In this paper, we studied some properties of three-band orthogonal wavelets based on the classical two-band wavelets theory. Especially, we proved the relationship between discrete and continuous moments of the scaling functions in case three. Then an algebraic method and a few examples to construct the three-band orthogonal wavelets were given. This method converts the structure of scaling function to solving the constraint equations of the filter coefficients. It overcame that the existing algorithms were too complex or could not ensure the symmetry etc.
Wavelet frames and tight wavelet frames become new research hotspots in recent years. Daubechies and Meyer advanced necessary and sufficient conditions for mother wavelet to generate frames in "Ten Lectures on Wavelets" and "Ondelettes et Operateurs". Characterizations of univariate tight wavelet frames are implicit in the works of Wang and Weiss. Then an explicit characterization of tight wavelet frames in the multivariate case was obtained by Bin Han. Independently of these, Ron and Shen gave a general characterization of all wavelet frames, and specialized this to the case of tight wavelet frames. Recently, Daubechies and Bin Han constructed MRA-based tight wavelet frames, and discussed the existence of dual wavelet frames and its construction method. Considering MRA plays a very important role during the construction of the wavelet frames, we presented and proved a sufficient condition for constructing the scaling functions of M-band dual wavelet frames based on tight wavelet frames and FMRA.
引文
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