Construction Of Multi Wavelet And Vector-valued Wavelet

Wavelet analysis has been a new branch of mathematics since 1980s. As a time-frequency analysis method, it has been widely applied in image compression and denoising. Comparing with uni-wavelet, multi-wavelet can simultaneously has properties like compactly supported, orthogonality and symmetry etc, which decide that multi-wavelet can get broad application and research in signal processing.This paper makes some systematic researches on the theory and application of single and multi-wavelet. The main work is summarized as follows:We describe the theory of uni-wavelet and multi-wavelet in detail, and give the decomposition and the reconstruction algorithm of wavelet transform.Based on the theory of multi-wavelet, we give a method to construct m dimensional orthogonal multi-wavelet.Based on the theory of multi-wavelet, we research the theory of vector-valued wavelet, and give a algorithm to construct supported orthogonal vector-valued wavelet.Finally, motivated by the notion of orthogonal frames, we describe sufficient conditions for the construction of orthogonal MRA wavelet frames from a suitable scaling function. These constructions naturally lead to filter banks, through these filter banks, the orthogonal wavelet frames give rise to a vector-valued discrete wavelet transform. The novelty of these constructions lies in their potential for use with vector-valued data.