Compactly supported multivariate dual multiframelets with high vanishing moments and high balancing orders

TL;DR

This paper develops a method to construct multivariate dual multiframelets with high vanishing moments and balancing orders, addressing the complexity of multivariate polynomial matrix factorization and enhancing sparse, compact discrete transforms.

Contribution

It introduces a systematic way to generate OEP-based dual multiframelets with optimal vanishing moments from any pair of compactly supported refinable vector functions.

Findings

Constructed dual multiframelets with maximal vanishing moments.

Achieved compact and sparse discrete framelet transforms.

Addressed the polynomial matrix factorization challenge in multivariate framelets.

Abstract

Comparing with univariate framelets, the main challenge involved in studying multivariate framelets is that we have to deal with the highly non-trivial problem of factorizing multivariate polynomial matrices. As a consequence, multivariate framelets are much less studied than univariate framelets in the literature. Among existing works on multivariate framelets, multivariate multiframelets are much less considered comparing with the exitensively studied scalar framelets. Hence multiframelets are far from being well understood. In this paper, we focus on multivariate dual multiframelets (or dual vector framelets) obtained through the popular oblique extension principle (OEP), which are called OEP-based dual multiframelets. We will show that from any given pair of compactly supported refinable vector functions, one can always construct an OEP-based dual mltiframelet, such that its generators have the highest possible order of vanishing moments. Moreover, the associated discrete framelet transform is compact and sparse.