System theory and orthogonal multi-wavelets

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+Bz(I−Dz)−1C,z∈D={z∈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.