The Construction Of The Multi-wavelets And Tight Wavelet Frames And Their Related Research

Wavelet analysis has been a hot topic in the fields of applied mathematics and engineering.In recent years,with the information and computer technology rapid advance,multi-wavelets and wavelet frames can keep the some desirable properties of scalar wavelets,and are widely applied in signal processing and big data analysis.The study of multi-wavelets and wavelet frames has drawn a great deal of interest in various areas,and has become a hot topic in recent years.Based on the multiresolution analysis(MRA),we study the construction of the multiwavelets and tight wavelet frames as well as two related problems,such as the clustering and Gibbs phenomenon.The main organizations and innovations of the thesis are as follows:(1).Based on the related theories of the pseudo splines,a new symmetric compactly supported orthogonal multi-wavelets is constructed.The smoothed pseudo splines can be considered as an extension of pseudo splines.First,we study that the shifts of a smoothed pseudo spline are linearly independent,which is a necessary and sufficient condition for the construction of the biorthogonal wavelet system.Based on this result,we generalize the results of Riesz wavelets and derive biorthogonal wavelets from smoothed pseudo splines.Second,by studying the relationship between matrix sequences of multi-scaling function and matrix sequences of multi-wavelets,we derive the explicit formulations of the symmetric compactly supported orthogonal multi-wavelets function with dilation 3 and 4-coefficient.Last,we give the examples to illustrate our general constructive schemes.(2).Based on Unitary Extension Principle(UEP)and Oblique Extension Principle(OEP),we construct the symmetric tight frame systems with three generators.First,based on the smoothed pseudo spline,by using the unitary extension principle,we study the construction of the symmetric tight frame systems with three generators,and discuss the desired approximation order of the tight frame.Second,under the condition of the linear independence of the shifts of a pseudo spline,by using the oblique extension principle,we construct symmetric tight wavelet frames generated with three generators,which have the higher possible order of vanishing moments.Last,we give two examples to illustrate our constructive schemes.(3).To improve clustering algorithm of the big data by wavelet and tight wavelet,we propose a novel ensemble clustering algorithm.For huge and unstructured date analysis,we introduce some theory of the wavelet and tight wavelet frame on manifolds and discrete setting graphs.By using the sparsity of tight wavelet frame,we discuss the semi-supervised clustering of the data sets by tight wavelet transforms.Based on the problem of clustering algorithm improvement,we propose a novel ensemble method by utilizing a dense representation model for clustering,which reduces the size of input data and further improves the stability and accuracy of the clustering task.(4).For the Gibbs phenomenon in wavelet and wavelet frame applications,we discuss the existence of the Gibbs phenomenon for the bi-orthogonal wavelets and subdivision.First,by studying the properties of the bi-orthogonal wavelet kernel,we discuss the existence of the Gibbs phenomenon for the bi-orthogonal wavelets expansions of a function with jump discontinuity.Second,due to the close relationship between subdivision and wavelet analysis,we also prove that the Gibbs phenomenon occurs for the ternary and p-ary subdivision schemes.Last,we give two examples to illustrate our results.