A Study On The Construction Of Armlet Multiwavelets And Matrix-valued Wavelets

Multiwavelet has attracted much attention because it can have attributes of compact support,orthogonality and symmetry at the same time.However,the further development of multiwavelet gets into trouble:multiwavelet construction is difficult is one reason;for single-valued singles,multiwavelet needs prefiltering process is another.To overcome second disadvantage,balanced multiwavelet and Armlet multiwavelet are introduced.In addition,the concept of matrix-valued wavelet,which directly processes vector-valued signals,was proposed by Xia Xianggen.The main difference between matrix-valued wavelet and multiwavelet is that discrete multiwavelet transform needs prefiltering,while discrete matrix-valued wavelet transform doesn't need prefiltering.Based on a thorough understanding of the current situation of wavelet theory research at home and abroad,and inspired by Professor Yang Shouzhi and Professor Chen Qingjiang,this paper studied the structural theory of Armlet multiwavelet and matrix-valued wavelet,and achieved some meaningful results.Firstly,we studied the problem that multiscale functions are balanced and their corresponding multiwavelets are Armlet multiwavelets.The construction of a paraunitary matrix is discussed,than the relationship between the balance of multiscale functions and the corresponding Armlet multiwavelets is studied by using the paraunitary two-scale similarity transformation.The construction theorem and construction example of the balanced Armlet multiwavelets is given.Secondly,the concepts of four-scale matrix-valued multiresolution analysis and matrix-valued orthogonal wavelets are introduced in L2(Rd,Cn×n),and the necessary and sufficient conditions for the existence of finite-dimensional matrix-valued orthogonal wavelets are given,and the construction algorithm a class of compactly supported finite imensional matrix-valued orthogonal wavelets are provided.Finally,the construction of ternary matrix-valued biorthogonal wavelet filters is studied.When the polynomial decomposition of matrix sequence of one of matrix-valued scaling function is a matrix polynomial,the corresponding matrix-valued biorthogonal wavelet filter formula is constructed by using the multiphase decomposition method of matrix and the lifting idea,and the properties of ternary matrix-valued wavelet packets are obtained.