  As a generalization of orthonormal wavelets in $L_2(R)$, tight framelets
(also called tight wavelet frames) are of importance in wavelet analysis and
applied sciences due to their many desirable properties in applications such as
image processing and numerical algorithms. Tight framelets are often derived
from particular refinable functions satisfying certain stringent conditions.
Consequently, a large family of refinable functions cannot be used to construct
tight framelets. This motivates us to introduce the notion of a quasi-tight
framelet, which is a dual framelet but behaves almost like a tight framelet. It
turns out that the study of quasi-tight framelets is intrinsically linked to
the problem of the generalized matrix spectral factorization for matrices of
Laurent polynomials. In this paper, we provide a systematic investigation on
the generalized matrix spectral factorization problem and compactly supported
quasi-tight framelets. As an application of our results on generalized matrix
spectral factorization for matrices of Laurent polynomials, we prove in this
paper that from any arbitrary compactly supported refinable function in
$L_2(R)$, we can always construct a compactly supported one-dimensional
quasi-tight framelet having the minimum number of generators and the highest
possible order of vanishing moments. Our proofs are constructive and
supplemented by step-by-step algorithms. Several examples of quasi-tight
framelets will be provided to illustrate the theoretical results and algorithms
developed in this paper.
