Frame Wavelet

As an ideal time-frequency analysis tool, wavelet analysis is based on Fourier analysis and development together. It effectively makes up for the short comings of Fourier analysis and is seen as the work of multiharmonic analysis in half a century. Besides,because of its good adaptability and mathematical microscope feature, wavelet analysis is widely used in signal or image processing, intelligent computing, quantum theory, network information security and other lots of fields. At present, the study about wavelet analysis theory is getting more and more deep and its application range is getting more and more wide.Wavelet transform is a time-frequency localization tools, including continu-ous wavelet transform and discrete wavelet transform,and wavelet frame theory is composed as the main part of to constitute the main part of the discrete wavelet transform. Duffin and Schaeffer was proposed the concept of the frame of Hilbert space for the first time in 1952 when then studied non-harmonic Fourier analysis. Then wavelet transform was introduced into the frame theory and the concept of wavelet frame was proposed by Daubechies, Meyer and others. Wavelet frame is a generalization and expansion of wavelet bases, reducing the requirements in terms of orthogonality of wavelet bases and introducing the redundancy, which makes the wavelet frame design having a large degree of freedom. At the same time,its re-dundancy is better than the property of the stability and robustness about noise. Because the wavelet frame has good characteristics of both wavelets and frames, it has become a focus of attention of scholars both at home and abroad and shown great potential in image processing, edge detection, noise reduction and so on. Al-though the discussion about wavelet frame has been involved in lots of aspects,the theory study still needs further improvement and the wavelet frame that has good property needs us to design.This article primarily studies the relevant properties of the wavelet frame, the structure of dual wavelet frames and Parseval wavelet frames.In addition, the wavelet frame with high vanishing moments is characterized and several conclusions are given,which is helpful for the further development of wavelet frame theory.The paper consists of four parts.The first chapter is the introduction.The development process and the latest research results about wavelet analysis and wavelet frame is described in this part.The second chapter is about wavelet frame and associated properties. Firstly, the definition of frame is given and the relationship between frames and Riesz basis is discussed.Then,the dual frame is defined through frame operator and the signal reconstruction formula is given.Finally,some properties about the multi-resolution analysis of frame are discussed.The third chapter mainly studies the construction of dual wavelet frames.The construction of dual wavelet frames and Parseval wavelet frames are discussed by the multi-resolution analysis.Moreover, a specific matrix expansion method is given.The four chapter is about the characterization about the wavelet frame with higher vanishing moments. Firstly,the high vanishing moment properties of wavelets frame is studied.Then the dual frame wavelet with high vanishing moments is con-structed by the OEP.Besides,the construction algorithm of dual frame wavelet is given in this paper.