Scaling Laplacian Pyramids

TL;DR

This paper explores the conditions under which Laplacian pyramid matrices are scalable to paraunitary matrices, extending the concept to filter banks and proposing new methods for tight wavelet constructions.

Contribution

It extends the theory of scalable frames to polyphase filter bank representations and introduces new construction techniques for tight wavelet filter banks.

Findings

Characterization of scalability conditions for LP$^2$ matrices.

New methods for constructing tight wavelet filter banks.

Application of scalability concepts to wavelet frame design.

Abstract

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.