Construction Of Wavelet - Tight Frame With Minimum Energy Vector

In recent years, wavelet analysis developed rapidly and became an emerging mathematical subject. It not only has a strong theoretical basis, but also has this significance of a wide application. It is a breakthrough on the basis of Fourier analysis. In 1910, Haar proposed the first orthonormal basis, namely Haar orthonormal basis. In 1987, Mallat introduced the idea of multi-scale analysis of the wavelet analysis and proposed the idea of multi-resolution analysis. Moreover, he gave the general method of constructing orthogonal wavelet basis, namely Mallat algorithm. After, Daubechies proved that Haar wavelet is the only wavelet that has three character of orthogonality, symmetry and compact support. In 1992, Zou and other researchers extended wavelet transform from "2-band" to "multi-band"; In 1993, Goodman and other researchers extended single wavelet scaling function to multi-wavelet scaling function which can generate multi-resolution space and establish multi-wavelet theory frame so that we can obtain greater freedom. Scholars pursued the situation that three character of orthogonality, symmetry and compact support can set up at the same time, which lead the upsurge of multi-wavelet theory. In the application, with the rapid development of the network, the problems including the network bandwidth fluctuation, signal decoding error, blockage caused by the substitution may occur. In order to overcome this problem, people decomposed signal into several description in different channels for transmission. Moreover, these description must balance energy with each other and have some redundancy. This is referred to description coding MDC technology and lead to the development of wavelet frame and multi-wavelet. But there is not a lot of research on multi-wavelet frame and the structure of minimum energy frame.In 2000, C. K. Chui and W. J. He constructed the 2-band minimum energy frame with compact support. In 2002, S. Z. Yang, Y. Y. Tang and Z. X. Cheng gave the method of constructing multi-band orthogonal multi-wavelets with compact support. In 2003, Han and Mo successfully constructed dual wavelet frame with vector-valued scaling function. In 2007, Averbuch and other searchers gave the perfect reconstruction condition of wavelet frames. In the same year, Y. D. Huang and Z. X. Cheng gave the equivalent conditions of minimum energy frame and the necessary and sufficient conditions of the minimum energy frame. In 2010, W. Guo and L. Z. Peng gave the method to construct low-pass filter and high-pass filter of multi-wavelet frame and the parameterizations of the low-pass filter in multi-wavelet frame. In 2011, Y. T. He constructed a multi-band minimum energy frame with compact support.In this paper, based on the previous multi-band wavelet tight frames and vector-valued wavelets, we introduce the method to construct M-band minimum energy vector-valued wavelets tight frames, which on the one hand maintain the advantages of wavelet orthogonal basis, on the other hand have a good deal of the shortcoming of wavelet orthogonal basis that the three properties, including compact support, symmetry and continuity, cannot be established. We review the development and application of wavelet analysis and minimum energy frames; then give the concept of M-band minimum energy vector-valued wavelets tight frames, multi-resolution analysis associated with M-band minimum energy vector-valued wavelets tight frames and equivalent conditions of M-band minimum energy vector-valued wavelets tight frames. On the basis of the equivalent conditions, we study the method to construct the low-pass filter and the high pass filter of M-band minimum energy vector-valued wavelets tight frames, and we discuss conditions of the establishment of M-band minimum energy vector-valued wavelets tight frames.