Interpolatory Dual Framelets with a General Dilation Matrix

TL;DR

This paper introduces a new method for constructing interpolatory dual framelets with high-order vanishing moments using a general dilation matrix, enhancing wavelet analysis with explicit formulas and symmetry considerations.

Contribution

The paper proposes a novel, easily implementable approach to construct interpolatory dual framelets with high-order vanishing moments for any dilation matrix, including methods for symmetry analysis.

Findings

Constructed dual framelets with high-order vanishing moments.

Provided explicit formulas and linear system solutions for high-pass filters.

Demonstrated theoretical results with several examples.

Abstract

Interpolatory filters are of great interest in subdivision schemes and wavelet analysis. Due to the high-order linear-phase moment property, interpolatory refinement filters are often used to construct wavelets and framelets with high-order vanishing moments. In this paper, given a general dilation matrix $\mathsf{M}$, we propose a method that allows us to construct a dual $\mathsf{M}$-framelet from an arbitrary pair of $\mathsf{M}$-interpolatory filters such that all framelet generators/high-pass filters (1) have the interpolatory properties; (2) have high-order vanishing moments. Our method is easy to implement, as the high-pass filters are either given in explicit formulas or can be obtained by solving specific linear systems. Motivated by constructing interpolatory dual framelets, we can further deduce a method to construct an interpolatory quasi-tight framelet from an arbitrary interpolatory filter. If, in addition, the refinement filters have symmetry, we will perform a detailed analysis of the symmetry properties that the high-pass filters can achieve. We will present several examples to demonstrate our theoretical results.