A Class Of Shearlet On Z~2

In the latter half of 20th century,it had become an important method for differ-ent dimensions of the signal and image analyzed by using wavelet transform.However,the options of the orientation of wavelet transform are few,and it' s not the optimal representation for the straight line in image or the singularity in the curve.Moreover,it cannot help us achieve a good sparse representation.Until 2007,Guo etc.construct-ed a nearly optimal representation multidimensional function called shearlet by using a synthetic expansion affine system.Shearlet inherited the advantages of translational invariance and multidirection in curvelet and contourlet.It will generate a series of differ-ent shearlet bases by dilation,shearing and translation on the basis functions.In case of two-dimensional signals,shearlet transform not only detects all the singularity,but also adaptively tracks the orientation of the singular curve.Meanwhile,shearlet transform can precisely represent the singular characteristic of functions.As a result,it can acquire a better sparse representation property for images.Discrete wavelet transform can be used as signal processing tools in the time domain of visualization unstable waveform.In his monograph,M.W.Frazier conducted a thorough research for the properties and structures of one-dimensional discrete orthogonal wavelet base.On the basis of his monograph,this thesis extends the discrete wavelet on Z to Z~2,then we construct a class of discrete s-hearlet in Z~2.At the same time we explore the characteristics of the discrete shearlet on Z~2 by using time frequency analysis,matrix analysis,operator theory and other tools.As a result we propose the algorithm to construct the Parseval frame on Z~2 through the multi-scale method.In the two-dimensional data,the discrete orthogonal wavelet base and discrete frame may conduct a well representation for the general sequence of points.However,they can' t represent the two-dimensional data containing singularities when dealing with image denoising,edge processing,and sparse matrix representation as well as the image compression effectively and precisely.By taking advantage of the properties of shearlet,we construct an orientational Parseval frame which we add the shear factor on general Parseval frame.These consequences make it possible for image denoising and image compression.Moreover,we can extend this Parseval frame to higher dimension by using tensor product,which allows for higher dimensional da.ta processing.Chapter 1 briefly introduces the background of frames and shearlet as well as the structure and the work of this thesis.Chapter 2 lists the concept and some basic facts of frame,shearlet and the properties of two related Hilbert spaces.Chapter 3 lists some operators in lP(Z~2)and then discuss thier properties.Chapter 4 is the main work of this thesis.In this chapter we construct a class of sheaxlet on Z~2,then construct the Paxseval frame in Z~2(Z~2),then we discuss the algorithm.Chapter 5 reveals an application in l~2(Z~2)which uses the tensor product to construct the Parseval frame.