Theory And Application Of Wavelet Frames

Wavelet transform provides a tool for time-frequency localization. It overcomes the flaw of Fourier analysis, which can clearly show the frequency characteristic of signal but can't reflect its local information in time domain. However it is important to describe the local property of signal both in theory research and in application fields. So wavelet is denoted as "Mathematical Microscope."Wavelet transform consists of the continuous wavelet transform and the discrete wavelet transform. In the discrete wavelet transform, wavelet frames are the main part. Wavelet frames have been successfully applied in many fields;there still are many theory basis and application potential powers, which need to be perfected and exploited.For further comprehending and application of wavelet frames. This article first introduces the basic theory of wavelet frames. Then, it summarizes and extends the construction of wavelet frames. At last, the frames' applications in signal denoising are given.The dissertation is divided into six chapters:Chapter 1 is preface, which expain the origin of thought, the characteristics and the prospects for development on wavelet frames. In 1952, Duffin and Schaffer proposed conception about frame in the Hilbert Space. Later Daubechies, Grossman and Meyer unified the theory of the wavelet transform and frame to define the affine frame (or call wavelet frame). Its redundancy has the unique superiorly and waits for the further exploration in application domains, such as the signal analysis, imagery processing and soon.In the chapter 2, we introduce the Multi-resolution analysis and wavelet system. Constructions and applications of wavelet are based on Multi-resolution analysis. Along with the emergence of Multi-resolution analysis, the difficulty in constructing wavelet obtains a perfect solution. The Multi-resolution analysis formula is gradually developed in the actual applications. These practical applications promote the wavelet theory development. Wavelet frames based on Multi-resolution analysis guarantee the existence of fast and numerically stable algorithms that implement the decomposition and the reconstruction."Good" wavelet systems are characterized by desirable properties. Such assymmetry, high number of vanishing moments and interpolating that may compete with each other. The different properties of wavelet systems function have the specific function in the different actual applications. Therefore, the symmetry, the high number of vanishing moments and interpolating and so on hold the important status in the wavelet research. This chapter also elaborates these properties and their values for applications.In the third chapter, we summarizes analyze some basic properties of wavelet frames and discuss the filter bank theory which relates with tight wavelet frames. The so-called wavelet frame refers to a sequence obtained by dilating and translating a function in L2(R), if the sequence forms a frame in L2(R). If the two frame bounds are equal, then we will call the frame a tight frame. Tight wavelet frames are easy to realize the perfect reconstruction. Their constructions and applications often relate with the filter bank. The wavelet frames are under-development in frame theory, but they are the most valuable in application fields. There are some experts, scholars on harmonic analysis and wavelet analysis who have conducted the widespread research and obtained many results.The fourth chapter discusses construction about the wavelet frames. Furthermore, we extend some results. This chapter mainly constructs the wavelet frames by three different methods: in time domain;in frequency range;by frame perturbation. Among these methods, in frequency fields tight wavelet frames via UEP can guarantee a perfect reconstruction and a fast and stable algorithm. So this method occupies a prominent position. It is the extension of method that is used to construct orthogonal wavelet via conjugate-mirror filter bank.Chapter 5 studies the wavelet frames' applications in digital signal processing. Frames can be very redundant. This redundancy may lead to robustness, in the sense that it can afford to store the wavelet coefficients < f,i//mn > with low precision, and still reconstruct / with comparatively much higher precision. This redundancy has the original superiority in de-noising, image fusion, digital watermark, encryption, code and so on. This chapter mainly elaborates signal de-noising by wavelet frames and gives its algorithms. The more redundant frame is, the less the wavelet coefficients error is, so de-noising result is to be better. But it simultaneously enlarges the computation quantity. Better algorithms in each application domain need to be expoited.Sixth chapter is conclusion and prospect.