On Constructions And Applications Of Multivariate Wavelet

In this paper, the constructions and applications of multivariate wavelets is mainly discussed.This paper is consisted of six chapters. In chapter one, the developments of wavelets and neural networks is briefly reviewed. Our main research results are in chapter 2, chapter 3,chapter 4, and chapter 5 (i.e. the parts of 1-5, in this abstract). The last chapter is conclusion.1. Construction of non-tensor product, compactly supported, symmetrical prewaveletTaking the bivariate B-spline basis of spline space S3k+12k(1mn) as scale function, we give a construct method of non-tensor product, compactly supported, prewavelet with symmetrical property. For the shifts of the scale function are not orthogonal, our prewavelet can not realize finite decomposion and finite reconstruct. In fact, the shifts of the scale function must be orthogonal if the wavelet with finite decomposition and finite reconstruct is not vector.Let (x) be the B-spline basis of spline space S3k+12k(1mn) and P(z) be the filter symbol associated with . The orthonormal decomposition of the scale space V1 is V1 = V0 W0, where W0 denote the orthonormal complement of V0 in V1 (i.e. wavelet space). Denotewhere E = {(0,0), (0,1), (1,0), (1,1)}.lemma 1 LetQ(z) be the trigonometric polynomial satisfying f(w) - Q(z)(f)(w/2), then the function f(x) belongs to W0 if only and ifwhere H(z) = Q(z)P(Z-l}B(z) = H(z2)z, B(z) = k2 < (x),zk.V\ is actually consisted of the integer shifts of { (2x - ), E}. (2x - ) V1, E, assume that there exist (x) W0 such thatwhere{al}l 2, {6f}(ez2 )> then the key of constructing wavelets is to solve the functions p , E, satisfying formula (2) on the premise of lemma 1.Therefore, the symbols of : Q (z)(= ), E, are deduced , whichsatisfyQ"(z) = z B(z*) - P(z)(z P(z-1)B(z)]( , n E (3)In fact, (X),/Z 6 E, are linearly dependent. The following lemmas point out that the compactly supported, non-tensor product function , F = E\{(0,0)}, form the prewavelet of L2(R2).lemma 2 Let T = zero on T, then, for j e Z2, f/ie functions (0,0)(x - j) can be linearly expressed bylemma 3 If P(0,0)(z2) nave no zero on T, then, for j Z2, { (x -l), e F, l Z2} is linearly independent.2. Construction of bivariate non-tensor product compactly supported biorthogonal waveletIn image processing, the feature with finite decomposition and finite reconstruct is very important. Therefore when the bivariate B-spline basis of spline space S mn) is also taken as scale function, a method of constructing non-separable compactly supported biorthogonal wavelet is proposed. At first in terms of the theoretics of Groebner basis, the symmetrical dual scale function is obtained and thenthe restricted transition operator is applied to directly validate the properties of L2 and Riesz basis of the dual scale function. Finally under the condition that the scale function and the dual scale function are known, a concrete method of matrix extension is constructed to get the corresponding biorthogonal wavelets. Our method of matrix extension is easily achieved.Let (p(x) be the B-spline basis of Sf+1 (An), M0'0) (z) be the refinement filter symbol of (z), and M(')(Z) be the refinement filter symbol of of the dual scale function .In this paper, the fundamental idea of construct biorthogonal wavelet is firstly to construct the function <> such that and =0,...