System theory and orthogonal multi-wavelets

TL;DR

This paper unifies the characterization of scalar and multi-wavelets using system theory, showing that their masks correspond to transfer functions of conservative linear systems, thus revealing a systematic parametrization of wavelet classes.

Contribution

It provides a complete, unifying framework for scalar and multi-wavelets based on system theory, connecting wavelet masks to transfer functions of linear systems.

Findings

Wavelet masks are identified with transfer functions of conservative linear systems.

All classes of wavelet and multi-wavelet masks can be systematically parametrized.

No intrinsic differences between Daubechies wavelets and other multi-wavelet constructions.

Abstract

In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function $$ F(z)=A+B z (I-Dz)^{-1} \, C, \quad z \in \mathbb{D}=\{z \in \mathbb{C} \ : \ |z| < 1\}, $$ of a conservative linear system. The complex matrices $A,\ B, \ C, \ D$ define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by $A,\ B, \ C, \ D$ allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.