On The Lifting Factorization Of Biorthogonal Wavelets

Because of the success applications of wavelets, several methods ofbuilding wavelets speed up in the field of mathematics and the field of signal processing.The lifting scheme discovered by Wim Sweldens is one of them. The lifting schemebuilds wavelets in the time-field, its basic idea is to construct wavelet filters by finitepredict operators and update operators. It is a smart tool to build wavelet factorizations.The second generation wavelet can be constructed by this scheme, that is mapping thefirst generation wavelet to the second generation wavelet by uaing lifting to factor thewavelet transform into finite lifting steps. The lifting scheme is a way of realizingwavelet transforms, but the properties of the transofrmed signal are decided by theproperties of the original signal. Finally, on the problems of edges, nonequispacedsampling, diffcult curves and so on, which the traditional wavelets can not divide, thesecond generation wavelets have lots of advantages.The polyphase matrices of traditional wavelet filters can be factored into products ofseveral finite matrices. These matrices can be regarded as predict operators and updateoperators. One important advantage of using the lifting scheme to realize waveletfiltering is that it factor the filter into some basic steps, each of which is invertible andits inverse transform is very simple. In this paper, the lifting scheme and the liftingfactorization of biorthogonal wavelet filters are discussed. The conclusions voted in thispaper are classical conclusions in this field. They stated the research level and theproceeding direction. At the base of this, this paper extend some of the conclusions andgive some new results at the same time. This paper is consisted by three chapters.The chapter 1 is an introduction. The first part of this chapter summaries theemergence and development of the wavelet analysis. The second part introduces theimportant conception in the wavelet analysis, the conception and the properties ofMultiresolution Analysis. The third part introduces the algorithms of wavelet analysisand synthesis so. as to be compared with the algorithms of lifting analysis andreconstruction. Finally, the forth part states the consistence of this paper.The chapter 2 states the necessity and advantages of the lifting scheme firstly. Thenthe theory about the lifting scheme of wavelets is introduced. The lifting scheme ofwavelets is one of the methods to construct wavelets. It can construct the secondgeneration wavelet, that is mapping the first generation wavelet to the second generationwavelet by uaing lifting to factor the wavelet transform into finite lifting steps. And allthe wavelet transforms realized by Mallat algorithm can be realized by the liftingscheme, furthermore this algorithm is faster and smarter. Its basic idea is to constructwavelet filters by finite predict operators and update operators. Afterward, introducesthe adaptive lifting scheme of wavelet design. In the end, four advantages of the wavelettransform method based on the lifting scheme.The chapter 3, in the first part, the conception and the properties of biorthogonalMultiresolution Analysis is introduced. The second part notes the polyphase matrices ofbiothogonal wavelet filters P(z) and introduces the Euclidean algorithm of Laurentpolynomial. Then using the algorithm to divide the polyphase matrices of biothogonal wavelet filters. The third part gives the lifting algorithm also called the lifting scheme ofbiothogonal wavelet filters by using the division of the polyphase matrices ofbiothogonal wavelet filters P(z). Afterward examples are given to show how to factorbiorthogonal and compactly-supported wavelets analysis and synthesis transforms intolifting steps. For any finite length filter, the division of its polyphase matrix provides asimple method of giving the lifting steps of the wavelet transform. But this division isnot exclusive. Because of the nonsymmetry of the coefficients of orthogonal waveletfilters, this means that orthogonal wavelet filters do not have linear phase, which islimited in the image processing. But the coefficients of biorthogonal wavelet filters canbe symmetrical, so biorthogonal wavelet filters have linear phase.