Spectral Factorization of Matrices of Laurent Polynomials and Construction of Quasi-tight Framelets
Chenzhe Diao · ERA: Education and Research Archive (University of Alberta) · 2018
As a generalization of orthonormal wavelets, tight framelets (also called tight wavelet frames) are of importance in both wavelet analysis and applied sciences due to their many desirable properties in applications. However, tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Hence, we generalize the notion of tight framelets to quasi-tight framelets, which is essentially a dual framelet system, but behaves quite similar to tight framelets. This thesis makes a comprehensive study of the construction of Oblique Extension Principle (OEP) based compactly supported quasi-tight framelets. For univariate cases, we show that the construction of quasi-tight framelets is much more flexible than that of tight framelets. As a matter of fact, we can always derive a quasi-tight framelet system with high order of vanishing moments from refinable functions associated with any arbitrary compactly supported refinement masks. Also, it is much easier to design moment correcting filters for the quasi-tight framelet filter banks. We provide detailed algorithms to construct quasi-tight framelets in Chapter 2 and Chapter 3, where the highest order of vanishing moments and the smallest number of framelet generators can easily be achieved. Symmetry is also a desirable property in the construction of framelet systems. So we construct univariate (anti-)symmetric quasi-tight framelets in Chapter 4. We completely characterize the OEP-based (anti-)symmetric compactly supported quasi-tight framelet systems with two generators. For the multivariate framelets, it is known in the literature that the problems of constructing tight framelets / dual framelets with vanishing moments from general (nonseparable) refinable functions are quite hard. We propose solutions to the problems using quasi-tight framelets. We constructively prove that it is very easy to derive multivariate quasi-tight framelets with directionality/high order of vanishing moments, from any arbitrary M-refinable function, with any dilation matrix M. The constructions of quasi-tight framelets are directly linked to the mathematical problem of (indefinite) spectral factorizations of matrices of Laurent polynomials. We study/solve the spectral factorization problem in different settings in each Chapter 2 to 5.