拉普拉斯金字塔的扩展
Scaling Laplacian Pyramids
AI总结：
本文研究了基于拉普拉斯金字塔的Laurent多项式矩阵的可扩展性，提出新的构造方法用于紧波let滤波器组和框架。
摘要：
基于拉普拉斯金字塔的Laurent多项式（LP$^2$）矩阵由Laurent多项式列向量生成，长期以来与信号处理中的拉普拉斯金字塔算法相关。本文研究了此类矩阵何时可扩展，即右乘Laurent多项式对角矩阵是否得到paraunitary矩阵。可扩展性概念最近在有限框架理论中被引入，可视为框架的预处理方法。本文将当前关于可扩展框架的研究扩展到多相表示的滤波器组设置中。此外，作为主要结果的应用，我们提出了新的构造方法用于紧波let滤波器组和紧波let框架。
英文摘要：
Laplacian pyramid based Laurent polynomial (LP$^2$) matrices are generated by Laurent polynomial column vectors and have long been studied in connection with Laplacian pyramidal algorithms in Signal Processing. In this paper, we investigate when such matrices are scalable, that is when right multiplication by Laurent polynomial diagonal matrices results in paraunitary matrices. The notion of scalability has recently been introduced in the context of finite frame theory and can be considered as a preconditioning method for frames. This paper significantly extends the current research on scalable frames to the setting of polyphase representations of filter banks. Furthermore, as applications of our main results we propose new construction methods for tight wavelet filter banks and tight wavelet frames.
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