Research Of Construction Methods For Biorthonormal And Lifting Wavelet

Wavelet analysis is a new rapidly developing field in modern mathematical andinformation subject, and it has double character that is of deep in theory and com-prehensive in application. In order to improving traditional wavelet, lifting waveletappear, which not only is current and active but also can be realized with e?ectivemethod. Basing of above lifting wavelet become very important in wavelet field. Wepropose methods how to construct lifting wavelet in this paper.Lifting wavelet comesfrom biorthonormal wavelet, so we begin with biorthonormal wavelet. Then we obtaina method of constructing birorthonormal wavelet. We study it in theory and apply itin signal and figure denoising. The main works we do in this paper are:(1) We introduce the development history of wavelet analysis. In order to studyinglifting wavelet and applying wavelet, at the same time, how to transform it into signaland figure denoising, we detail classical theory about traditional wavelet: them aremultiresolution analysis and Mallet arithmetic.(2) When we use wavelet transformation, the choice of wavelet will affect theproperty of wavelet coeffcient then affect its application. After introducing the ba-sis of choosing wavelet, we analyze the symmetry and how vanishing moment affectthe construction of compactly supported biorthonormal wavelet. We deduce a newconclusion that when the construct wavelet has vanishing moment, the vanishing mo-ment of symmetrical wavelet with even length can be obtained only by even derivativesatisfying condition, and wavelet with odd length only need odd derivative satisfyingcondition. With this conclusion, we decrease the half numeration quantity.(3) Basing of Lawtion condition and traditional vanishing moment theory, weobtain a new method of constructing symmetrical compactly support biorthonormalwavelet wish vanishing moment. An example about construction of biorthonormalwavelet with length 8-8 and vanishing moment 3-3 is given. We can obtain a formulawith free variable which is about wavelet coeffcients through solving an equation.Then by solving a linear algebra problem, the domains of filter coeffcients which cangenerate biorthonormal wavelet are obtained. We can construct bior3.3 wavelet withour method, and its properties are better than bior3.3 wavelet. Some examples ofwavelet are applied to signal and figure denoise.Simulation results show that the methodis effective.(4) After study the relation of lifting wavelet and traditional wavelet in detail, bas-ing of construction perfect reconstruction filter and together with traditional construc-tion biorthonormal wavelet theory, we obtain the method which constructs compactlysupported biorthonormal wavelet based on lifting scheme. And an example about the construction of biorthonormal wavelet with length 4 - 4 and with vanishing moment2 - 1 is given. Some other examples of wavelet are applied to signal denoise. We canfind a result that with some length, wavelet based on lifting scheme denoises betterthan ordinary wavelet, we also using adaptive method to search free variable, which isdenoised best.(5) When construct the lifting wavelet, the methods choosing predicting operatorand updating operator is different. In this paper, we make a new criteria how to choosethe updating operator in order to decomposer signal into low frequency componentsand high frequency components more accurately, which is that the sum of the updatingoperator's components is 1/2 , and then we choose predicting operator according tothe least square theory. Some examples of lifting wavelet are applied to signal de-noising. We search the law of choosing predicting and updating operator, and get agood de-noising result. It proves that this method is valid.