Biorthogonal Wavelet Space: Parametrization and Factorization

Abstract

In this paper we study the algebraic and geometric structure of the space of compactly supported biorthogonal wavelets. We prove that any biorthogonal wavelet matrix pair (which consists of the scaling filters and wavelet filters) can be factored as the product of primitive para-unitary matrices, a pseudo identity matrix pair, an invertible matrix, and the canonical Haar matrix. Compared with the factorization results of orthogonal wavelets, it now becomes apparent that the difference between orthogonal and biorthogonal wavelets lies in the pseudo identity matrix pair and the invertible matrix, which in the orthogonal setting will be the identity matrix and a unitary matrix. Thus by setting the pseudo identity matrix pair to be the identity matrix and using the Schmidt orthogonalization method on the invertible matrix, it is very straightforward to convert a biorthogonal wavelet pair into an orthogonal wavelet.