Generalized Matrix Spectral Factorization with Symmetry and Applications to Symmetric Quasi-Tight Framelets

TL;DR

This paper develops a generalized spectral factorization method for symmetric 2x2 matrices of Laurent polynomials and applies it to construct symmetric quasi-tight framelets with two generators, useful in signal processing.

Contribution

It introduces a new generalized spectral factorization approach with symmetry constraints and provides constructive proofs for designing symmetric quasi-tight framelets.

Findings

Established necessary and sufficient conditions for symmetric quasi-tight framelets.

Provided constructive algorithms for spectral factorization and framelet construction.

Demonstrated the theoretical results with illustrative examples.

Abstract

Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for $2\times 2$ matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry.