Construction Of J^th-stage Discrete Wave Packet Frames

This thesis considers the theory and application of J^th-stage periodic wave packet frames for the space l²?Z_N?of all N-periodic complex-valued sequences.Since frames are of many nice properties,it have been widely used in the characterization of func-tion spaces,signal or image processing,sampling theory,digital communication and many other fields.From then on,the studies on wavelet frames have also been exten-sively investigated.The“Gaussian wave packets”,in which its corresponding families of functions are obtained by applying certain collections of dilations,modulations and translations to the Gaussian function,have been successfully applied in physics.More-over,the data in both the physics and engineering is often discrete and periodic,which makes many authors return to focus on the discrete sequence spaces l²?Z?or l²?Z_N?.The related problems of this spaces have been investigated by many authors.There-fore,it is more natural and fundamental to study the discrete versions for wave packet systems.Our purpose in this thesis is to construct a class of J^th-stage discrete periodic wave packet frame for l²?Z_N?using the technique of the p^th-stage orthonormal wavelet basis,whose filter bank is a more general sequence comparing with the classic extension principles proposed by Ron and Shen.Then,we present a new model,termed GaborTV,to restore images via combining Gabor frames and total variation.At last,numerical experiments demonstrate the efficiency of our proposed method.Firstly,we introduce the research background and status of frames theory and main work of this thesis.Then,we review the theory of wavelet frames and the algorithm of imaging denoising used by tight wavelet frames.Secondly,we propose the definition of discrete wave packet systems for the periodic sequence space l²?Z_N?using the theory of frames and operators and the technique of the p^th-stage orthonormal wavelet basis.Further,we give the definition of discrete periodic wave packet transforms and construct a first-stage discrete periodic wave packet frame for l²?Z_N?.Moreover,we provide a sufficient condition for the system to be a first-stage discrete periodic wave packet frame for the sequence space l²?Z_N?and establish a necessary condition and a sufficient condition for the system to be a frame for the sequence space l²?Z_N?in this thesis.Thirdly,we introduce the J^th-stage Parseval frame filter sequences on the basis of a first-stage discrete periodic wave packet frame for the periodic sequence space l²?Z_N?and the first-stage discrete periodic wave packet transforms.Then,we give the construction of J^th-stage discrete periodic wave packet frame for the periodic sequence space l²?Z_N?by iterating the filter sequence and establish the associated decomposition and reconstruction algorithms for these wave packet systems using the discrete periodic wave packet transforms.In addition,we give an illustrative example to demonstrate the validity of the proposed scheme.Fourthly,we present the definition of discrete Gabor systems for the discrete space C^Nand propose a new model by keeping the low pass filter and modifying the high pass filters and combining Gabor frame and total variation?the GaborTV model?.Then,numerical experiments demonstrate the efficiency of our proposed method.Finally,we give some conclusions and possible problems for future study at the end of this thesis.