具有一般膨胀矩阵的插值对偶框架
Interpolatory Dual Framelets with a General Dilation Matrix
AI总结：
本文提出一种从任意插值滤波器对构造具有插值性和高阶消失矩的对偶框架的方法，并进一步推导出构造插值拟紧框架的方法，同时分析了对称性性质。
摘要：
插值滤波器在细分格式和小波分析中具有重要价值。由于具有高阶线性相位矩性质，插值细化滤波器常被用于构造具有高阶消失矩的小波和框架。本文针对一般膨胀矩阵 $\mathsf{M}$，提出了一种方法，使得我们可以从任意一对 $\mathsf{M}$-插值滤波器构造一个对偶 $\mathsf{M}$-框架，且所有框架生成元/高通滤波器（1）具有插值性质；（2）具有高阶消失矩。我们的方法易于实现，因为高通滤波器要么以显式公式给出，要么可以通过求解特定的线性方程组获得。受构造插值对偶框架的启发，我们进一步推导出一种从任意插值滤波器构造插值拟紧框架的方法。此外，如果细化滤波器具有对称性，我们将对高通滤波器所能达到的对称性质进行详细分析。我们将给出几个例子来证明我们的理论结果。
英文摘要：
Interpolatory filters are of great interest in subdivision schemes and wavelet analysis. Due to the high-order linear-phase moment property, interpolatory refinement filters are often used to construct wavelets and framelets with high-order vanishing moments. In this paper, given a general dilation matrix $\mathsf{M}$, we propose a method that allows us to construct a dual $\mathsf{M}$-framelet from an arbitrary pair of $\mathsf{M}$-interpolatory filters such that all framelet generators/high-pass filters (1) have the interpolatory properties; (2) have high-order vanishing moments. Our method is easy to implement, as the high-pass filters are either given in explicit formulas or can be obtained by solving specific linear systems. Motivated by constructing interpolatory dual framelets, we can further deduce a method to construct an interpolatory quasi-tight framelet from an arbitrary interpolatory filter. If, in addition, the refinement filters have symmetry, we will perform a detailed analysis of the symmetry properties that the high-pass filters can achieve. We will present several examples to demonstrate our theoretical results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
