Splitting a Matrix of Laurent Polynomials with Symmetry and itsApplication to Symmetric Framelet Filter Banks
Splitting a Matrix of Laurent Polynomials with Symmetry and itsApplication to Symmetric Framelet Filter Banks
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DOI：
10.1137/s0895479802418859
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发表时间：
2004
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影响因子：
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通讯作者：
B. Han;Qun Mo
中科院分区：
文献类型：
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作者：
B. Han;Qun Mo
设M是一个2 × 2的对称真实的系数Laurent多项式矩阵.本文得到了四个罗朗多项式存在的一个充要条件（或有限脉冲响应滤波器）u1，u2，v1，v2，具有真实的系数和对称性，使得$$ \left[ \开始{矩阵} u_1（z）v z\in \CC \bs \{0 \} $$和[Su 1]（z）[Sv 2]（z）=[Su 2]（z）[Sv 1]（z），其中[Sp]（z）=p（z）/p（1/z）对于非零Laurent多项式p.我们的准则可以很容易地检查和一步一步的算法将给出构造对称滤波器u1，u2，v1，v2.作为这一结果的应用对称framelet滤波器组，我们提出了一个必要和充分条件的对称多分辨率分析紧小波框架的建设与两个compensarily支持的发电机来自一个给定的对称可加细功能。一旦这样的一个必要和充分条件得到满足，一个算法将被用来构建一个对称的框架滤波器组与两个高通滤波器，这是感兴趣的应用，如信号去噪和图像处理。作为本文结果和算法的一个例子，我们给出了几个对称框架滤波器组的例子，其中两个高通滤波器具有良好的消失矩，并来自各种对称低通滤波器，包括一些B样条滤波器。
Let M be a $2\times 2$ matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or finite-impulse-response filters) u1, u2, v1, v2 with real coefficients and symmetry such that $$ \left[ \begin{matrix} u_1(z) v z\in \CC \bs \{0 \} $$ and [Su1](z)[Sv2](z)=[Su2](z)[Sv1](z), where [Sp](z)=p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u1, u2, v1, v2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric multiresolution analysis tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two high-pass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters.