The Research into a Two-directional matrix-valued Quarternary Wave Wraps with multi-scale and Applications

Abstract. In this paper, we introduce a class of vector-valued four-dimensional wavelet packets according to a dilation matrix, which are generalizations of univariate wavelet packets. The defini-tion of biorthogonal vector four-dimensional wavelet packets is provided and their biorthogonality quality is researched by means of time-frequency analysis method, vector subdivision scheme and functional analysis method. Three biorthogonality formulas regarding the wavelet packets are es-tablished. Finally, it is shown how to draw new Riesz bases of space 2 4 ()nL R from these wavelet packets. The sufficient condition for the existence of four-dimensional wavelet packets is estab-lished based on the multiresolution analysis method