The Theories Of N-D Generalized Quasi-real Wavelets & Wavelet Moments And Their Applications

Wavelets analysis (Multiresolution analysis) is regarded as the most important development of Fourier analysis in the 20th century. In the history of Mathematics, it is also one of the important discoveries. As a new research field, the theory and the application of the wavelets analysis attract more and more attention of many subjects. In this paper, we do some research in both aspects.In theory, we break the restriction of the ladder spaces and the restriction of 2-band in the the classical multiresolution analysis (MRA). We expand MRA to the n-d quasi-real generalized multiresolution analysis (Q-GMRA), which increases more freedom of wavelets. Under this theory frame, we study the definition, computation and construction algorithm. And in the 3 dimension space, we unite the wavelets analysis together with the moments theory for the first time, and give the definition and properties of 3-d wavelet moments. In application, we use the M-band GMRA (the special case of 2-d Q-GMRA GMRA) to reduce the noise of the huge image datasets of the digital human slices. We apply the improved algorithm of the 2-d wavelet moments to the gait description and recognition, and apply the 3-d wavelet moments to the representation and survey of 3-d objects.The major contributions of this paper are listed in the following:1. Extend 1-d 2-band GMRA to n-d Q-GMRA. We present the definitions and relative theorems of the n-d orthonormal and bi-orthogonal Q-GMRA. Moreover weprove that the generalized Mallat algorithm still holds on.For the length limit, here as an example we only list the definition and relative theorems of the orthonormal Q-GMR.A.Definition 1. An n-d orthonormal Q-GMRA consists of a sequence of successive approximation spaces {Vj}jzz in L2(Rn). They satisfy the following conditions:1. Vj Vj+i, ;eZ;2. for each Vj, there exists a function { j(x),x Rn} 6 Vj such that {a" Vj(ajx-kbj)}fcezn is an orthonormal basis for Vj, where aj,bj > 0 R and ajbj-14/bjaj-1 = mj N, and j(x) is called level scaling function of Vj.Theorem 2. Suppose {a' (pj(ajX - kbj)}k zn 's an orthonormal basis for Vj and the bi-scaling equation between the adjacent spaces isx - kbj) (1)where aj,bj > 0 R and ajbj-1/bjaj_i = mj N. Then kbj-1)}k Zn is an orthonormal basis for Vj-1 if and only if|Hj( + 2t /mj)|2 = mjn, a.e. (2)where Theorem 3. Given an n-d orthonormal Q-GMRA, j(x) is the level scalingfunction of Vj and the bi-scaling equation (1) holds. Suppose sj-1(aj-1x) Vj,1 s mjn - 1. It indicates is an orthonormal basis for Wj₁, if and only ifj) = m"d's/, 0 < s, / < m" - 1 (3)where G0j(u;) = Hj(u), Gi(w) = ffisexp(-iA;u;), 1 s mjn - L.2Since the Q-GMRA still remain the bi-scaling relation between the adjacent spaces, the generalized Mallat algorithm still holds on.Given a practial signal f L2(Rn), 3Vj0 s.t. / 6 Vj0. We only need to consider the relation between the adjacent spaces from Vj0. Later we note ckj = (f,ajn/2 (ajx - kbj)), dk,sj = (f,a (ojZ - kbj)}. Next we give the Mallat, algorithm in the orthonormal case. Decomposition algorithm isComposition algorithm is2. In the sense of the L2-norm, we discuss the computation algorithm and the construction method of the Q-GMRA. And we give three forms of definitions and relative equivalent theorems of vanishing moments. As a special case of the orthonormal Q-GMRA, the optimal 2-d m-band GMRA gets better processing results in the huge datasets of the digital human slices.For the computation of the orthonormal and bi-orthogonal Q-GMRA, we discuss it in detail for the low-pass filters finite or infinite. When the filters are infinite, the computing complexity is increasing as the linear to the numbers of the original signals. And in every level, it needs to solve a quadratic equation in the orthonormal Q-GMRA(or quartic equation in the bi-orthogonal case). When the filters are finite, it, needs to solve a maximum value question for a quartic fun...