Multivariate Quasi-tight Framelets with High Balancing Orders Derived from Any Compactly Supported Refinable Vector Functions

TL;DR

This paper introduces a method to construct multivariate quasi-tight framelets with high vanishing moments and balancing orders from any compactly supported refinable vector functions, enhancing multiscale analysis tools.

Contribution

It develops a new approach using the oblique extension principle to derive quasi-tight framelets with optimal properties from arbitrary refinable vector functions.

Findings

Constructed multivariate quasi-tight framelets with maximal vanishing moments.

Achieved high balancing order in associated fast framelet transforms.

Provided theoretical foundations for multivariate quasi-tight framelets.

Abstract

Generalizing wavelets by adding desired redundancy and flexibility,framelets are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported $\dm$-refinable vector function $\phi$ with multiplicity greater than one, we prove that we can always derive from $\phi$ a compactly supported multivariate quasi-tight framelet such that (i) all the framelet generators have the highest possible order of vanishing moments;(ii) its associated fast framelet transform is compact with the highest balancing order.For a refinable scalar function $\phi$, the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from $\phi$ satisfying item (i).This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.