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Framelets derived from refinable (vector) functions via the popular oblique extension principle (OEP) are of interest in both theory and applications. Though OEP can increase vanishing moments of framelet generators to improve sparsity, it has a serious shortcoming for scalar framelets: the associated discrete framelet transform is often not compact and deconvolution is unavoidable. On the other hand, in sharp contrast to the extensively studied scalar framelets, OEP-based multiframelets are far from well understood. In this paper, we prove that from any compactly supported refinable vector function having at least two entries, one can always construct through OEP a compactly supported quasi-tight multiframelet such that all framelet generators have the highest possible order of vanishing moments, and its underlying discrete framelet transform is compact and balanced. The key ingredient of our proof is a newly developed normal form of matrix-valued filters, which greatly facilitates the study of multiframelets. [ABSTRACT FROM AUTHOR]

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 × 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. [ABSTRACT FROM AUTHOR]