高维紧支撑正交对称的小波

基于仿酉矩阵的对称扩充方法,该文提出了一种尺度因子为3的紧支撑高维正交对称小波构造算法.即设φ(x)∈L~2(R~d)是尺度因子为3的紧支撑d维正交对称尺度函数,P(ξ)是它的两尺度符号,p_(0,v)(ξ)为P(ξ)的相位符号.首先提出一种向量的对称正交变换,应用对称正交变换对3~d维向量(p_(0,v)(ξ))_v,v∈E_d的分量进行对称化.通过仿酉矩阵的对称扩充,给出了3~d-1个紧支撑高维正交对称小波构造.这种方法构造的小波支撑不超过尺度函数的支撑.最后给出一个构造算例.

Multidimensional Compactly Supported Orthogonal Symmetric Wavelets

An algorithm for constructing the compactly supported multidimensional orthogonal symmetric wavelets with dilation factor 3 is provided based on paraunitary matrix symmetric extension. Namely, let Φ(x)∈L²(R^d) be a d dimensional compactly supported orthogonal scaling function with dilation factor 3;  P(ξ) and (p_0, ν(ξ))_ν, ν ∈E_d, respectively, be the mask symbol and the polyphase symbol of Φ(x). Firstly, the authors propose a symmetric orthogonal transform of vectors, and take the symmetric orthogonal transform to (p_0, ν(ξ))_ν, ν ∈E_d， then get a symmetric unit vector. Secondly, based on paraunitary matrix symmetric extension, 3^d-1 compactly supported orthogonal symmetric wavelets associated with Φ(x) are obtained. The support of ψ_ν, ν ∈E_d, is not larger than that of Φ(x). In addition, the algorithm can also be used to construct orthogonal wavelets with the dilation factor M(M ≥ 3). Finally,  an example is given.