Partial parameterization of orthogonal wavelet matrix filters

TL;DR

This paper introduces a method for constructing a broad family of orthogonal matrix wavelet filters using a novel parameterization linked to shift-invariant operators, with explicit examples for 2x2 filters.

Contribution

It presents a new procedure for generating orthogonal matrix wavelet filters based on the representation of SO(2d), expanding the design space for such filters.

Findings

Characterization of full rank filters obtainable with the method

Explicit formulas for 2x2 filters derived from Lie algebra perturbations

A new parameterization approach for orthogonal wavelet matrices

Abstract

In this paper we propose a procedure which allows the construction of a large family of FIR d x d matrix wavelet filters by exploiting the one-to-one correspondence between QMF systems and orthogonal operators which commute with the shifts by two. A characterization of the class of filters of full rank type that can be obtained with such procedure is given. In particular, we restrict our attention to a special construction based on the representation of SO(2d) in terms of the elements of its Lie algebra. Explicit expressions for the filters in the case d = 2 are given, as a result of a local analysis of the parameterization obtained from perturbing the Haar system.