Adaptive Lifting Scheme Via BP Neural Networks

Wavelet is the most important mathematical branch after Flourier analysis.Wavelet's development is deeply rooted in field such as Pure Math.,Physics and Engineering, meanwhile wavelet is also a interdisciplinary between Math.(harmonic analysis), Scientific Computing and Signal Processing,it provides a group of conceptions,methods and algorithms for processing nonstationary signals.First generation wavelet is constructed by shifting and dilation of scaling function and wavelet function, wavelets constructed by this method can describe signals in different places with different resolutions. Its feature of describing signals at different resolutions makes it very suitable for nonstationary signals ayalysis. But the difficulties exist in first generation wavelet's construction and selection of special wavelet for certain signal obstacled its more wild application.In the middle of 1990s,Sweldens proposed the Lifting Scheme for biorthogonal wavelets construction, the proposition of Lifting Scheme solved the difficulties of traditional wavelet's factoring factorization construction. Lifting Scheme have brought some changes into wavelet theory as follows. First,Lifting Scheme allows us to construct a similar biorthogonal wavelet based on a original biorthogonal wavelet,this makes it possible to construct more suitable biorthogonal wavelet according to application needs. Second, Lifting Scheme makes adaptive construction of biorthogonal wavelet possible.In real applications, we can construct new biorthogonal wavelet according to signals'features by using some kinds of methods.This method is essentially adjusting wavelet to be suitable for a certain signal.Third, Lifting Scheme implies nonlinear wavelet is possible. Traditional wavelet construction is based on shifting and dilation of scaling function and wavelet function. Those operations essentially belong to a linear space, which means first generation wavelet is linear. However, Lifting Scheme allows nonlinear lifting steps to be used in it, this kind of nonlinearity gives Lifting wavelet excellent nonlinear properties and makes Lifting wavelet more powerful in describing some nonlinear structure. Those so called nonlinear wavelet can not be expressed by linear combination of shifting and dilation of scaling function and wavelet function, i.e., Lifting Scheme is out of first generation wavelets'boundary, it has completely different features and construction methods. Thus, we give it a new name—Second Generation Wavelet. As far as the relationship of first and second generation wavelet is concerned, Daubechies pointed out that any first generation wavelet can be decomposed into basic lifting steps of lifting scheme, and those lifting steps have the form of polynomial. From anther point of view, Polynomial's linearity is the reason why first generation is linear. On the contrary does not hold, i.e. , many second generation wavelet can not be expressed in first generation wavelet's theory frame. As second generation wavelet's construction methods are concerned, we have mainly two methods.First one is to start from a biorthogonal wavelet and adjust it via lifting steps. Second one is to split signal into several disjoint union sets and apply lifting steps between those sets. Then a completely new biorthogonal wavelet is got by combining those lifting steps. A typical example for this method is so called Lazy Wavelet, Lazy Wavelet firstly splits the original signal into odd subsequence and even subsequence, then apply lifting steps between those two sets. Generally speaking, first generation wavelet's lifting decomposition is done using Lzay Wavelet theory.Linear approximation ability is a basic criterion for judging a wavelets transform is good or not. Good linear approximation ability usually means low-frequency coefficients containing high-energy and high-frequency coefficients containing low-energy, and the high-frequency coefficients are sparse. Those features are important no matter in data compression or signal detection and target recognition.Adaptive lifting scheme is a flexible scheme by introducing adaptive methods into lifting scheme. Adaptive methods have achieved wide useage in many fields such as sound code,JPEG 2000.Existing adaptive lifting schemes are mainly Piella adaptive lifting scheme and Chan's ENO adaptive lifting scheme.This article gives a completely novel adaptive lifting scheme--Adaptive lifting scheme via BP neural networks. Based on BP neural networks'excellent self-learning ability, nonlinearity and optimization approximation ability, the BP neural networks adaptive lifting scheme has excellent linear approximation ability. We compared the linear approximation ability of the lifting scheme which applies BP neural networks lifting to Daubechies 9/7 wavelets and Daubechies 9/7 wavelets. It turns out that BP neural networks lifting scheme has more powerful linear approximation ability.