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Interpolatory Dual Framelets with a General Dilation Matrix

Ran Lu

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.

Interpolatory Dual Framelets with a General Dilation Matrix

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 , we propose a method that allows us to construct a dual -framelet from an arbitrary pair of -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.

Paper Structure

This paper contains 11 sections, 7 theorems, 143 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Backgrounds on Framelets
  3. Interpolatory Filters
  4. Our Contributions and Paper Structure
  5. Construction of Interpolatory Dual Framelets
  6. Interpolatory Dual Framelets with High-order Vanishing Moments
  7. Interpolatory Quasi-tight Framelets with High-order Vanishing Moments
  8. Analysis on the Symmetry Properties of Interpolatory Dual Framelets Derived from Theorem \ref{['thm:df:int']}
  9. Interpolatory Filters and Their Symmetry Properties
  10. Analysis on the Symmetry Properties of the High-pass Filters
  11. Illustrative Examples for $d=2$

Key Result

Theorem 1.1

Let $a,\tilde{a},b_1,\dots,b_s,\tilde{b}_1,\dots,\tilde{b}_s\in l_{0}(\mathbb{Z}^d)$ be such that $\widehat{a}(0)=\widehat{\tilde{a}}(0)=1$. Define $\phi,\tilde{\phi},\psi_1,\dots,$$\psi_s,\tilde{\psi}_1,\dots,\tilde{\psi}_s$ by for all $\xi\in\mathbb{R}^d$. Then $(\{\phi; \psi_1,\dots,\psi_s\},\{\tilde{\phi}; \tilde{\psi}_1,\dots,\tilde{\psi}_s\})$ is a dual $\mathsf{M}$-framelet in $L_{2}(\math

Figures (3)

  • Figure 1: The graphs of the interpolatory standard $M_{\sqrt{2}}$-refinable functions $\phi$ and $\tilde{\phi}$ of the filters $a$ and $\tilde{a}$ in Example \ref{['ex1']}.
  • Figure 2: The graphs of the interpolatory standard $2I_2$-refinable functions $\phi$ and $\tilde{\phi}$ of the filters $a$ and $\tilde{a}$ in Example \ref{['ex2']}.
  • Figure 3: The graph of the interpolatory standard $M_{\sqrt{3}}$-refinable function $\phi$ of the filter $a$ in Example \ref{['ex3']}.

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 3.1
  • proof
  • ...and 6 more