Research On The Structure Theory Of Armlet Multiwavelets And Three Dimensional Eight Direction Wavelets

Wavelet analysis is a new method of analysis that has been considered as one of the hot topics in the preface scientific research since its introduction.Wavelet structure problem is the key issue of wavelet analysis.It is well known that a single wavelet can not have good properties at the same time,such as tight support,symmetry and orthogonality;however,as a generalization of a single wavelet,a multiwavelet can combine many excellent features simultaneously.At present,the construction theory of multiwavelets is still one of the hot topics in the forefront of wavelet theory research.Especially in the study of Armlet multiwavelet,Professor Yang Shouzhi explored the construction method of Armlet multiwavelets.Moreover,Professor Yang Shouzhi further explored and constructed the concept of bi-directional wavelet with the help of the methods and techniques of exploring multiple wavelet to further develop bi-directional wavelet theory.Inspired by Professor Yang Shouzhi's ideas,this paper further studies and generalizes the theory of Armlet multiwavelets and bi-directional wavelet structures.Some meaningful achievements have been obtained,including the following four innovations:In chapter 2,based on the basic theory of Armlet multiwavelet and drawing lessons from Professor Yang Shouzhi's techniques of constructing multiwavelets and Armlet multiwavelets,the multi-scale functions of the two-bracketed orthogonality support balanced interpolation and the Armlet multiwavelet functions with interpolation are given Construction method,and the corresponding multiwavelet sampling theorem.Two examples are given to verify the theoretical results.In chapter 3,based on the study of bi-directional wavelet theory by Prof.Yang Shouzhi.Due to its complex construction method and reasoning process,the research on the approximation order of multi-scale bi-directional functions is still scarce.There is no general method of construction.In this chapter,the definition of the approximation order of bidirectional multi-scale functions is given first,and then some conditions for its establishment are given.Finally,the related calculation methods are given.In chapter 4,based on the bi-directional wavelet theory proposed by Professor Yang Shouzhi and the construction theory of biorthogonal bi-directional wavelet,I proposed a three-dimensional octave wavelet;three-dimensional octahedron is Professor Yang Shou-bi bi-directional wavelet extension.In this chapter,the a-scale refinement function in three-dimensional space is obtained through the construction of tensor product,and the matrix whose corresponding eigenvalue is 1 is explored.We discuss the case of obtaining a compact support solution for a-dimensional three-dimensional octagonal refinement equation.In the meantime,we find that the three-dimensional octagonal a-scale plus finer wavelet function has a tight support interval.In chapter 5,On the basis of chapter 4,combined with professor Yang shouzhi's proposed biorthogonal bidirectional bidirectional wavelet structure,to explore and give three dimensions of orthogonal multiresolution analysis of orthogonal biorthogonal.And the definition and structure of the three-dimensional eight-dimensional function and wavelet function with biorthogonality.