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Framelets and Wavelets with Mixed Dilation Factors

Ran Lu

TL;DR

This work develops a systematic theory for framelets and wavelets with mixed dilation factors in arbitrary dimensions and multiplicities. By building a complete discrete framework—comprising mixed-dilation subdivision/transition operators, multi-level perfect reconstruction, and stability analysis—the authors extend prior scalar results to general vector-framelets and dual-framelet constructions. They then introduce discrete affine systems to study the stability and reconstruction properties in $\ell_2(\mathbb{Z}^d)$ and connect these discrete theories to continuous framelets in $L_2(\mathbb{R}^d)$, showing that stable mixed-dilation filter banks yield dual framelets in both discrete and continuous settings. The results unify and extend existing mixed-dilation theory (notably for scalar tight framelets) to arbitrary multiplicity, and establish robust connections between filter-bank formulations and $L_2$ framelets, including reconstruction, stability, and cascade properties. These contributions provide a flexible framework for constructing directionally rich, lower-redundancy framelet systems in higher dimensions with practical implications for image processing and harmonic analysis.

Abstract

As a main research area in applied and computational harmonic analysis, the theory and applications of framelets have been extensively investigated. Most existing literature is devoted to framelet systems that only use one dilation matrix as the sampling factor. To keep some key properties such as directionality, a framelet system often has a high redundancy rate. To reduce redundancy, a one-dimensional tight framelet with mixed dilation factors has been introduced for image processing. Though such tight framelets offer good performance in practice, their theoretical properties are far from being well understood. In this paper, we will systematically investigate framelets with mixed dilation factors, with arbitrary multiplicity in arbitrary dimensions. We will first study the discrete framelet transform employing a filter bank with mixed dilation factors and discuss its various properties. Next, we will introduce the notion of a discrete affine system in $l_2(\mathbb{Z}^d)$ and study discrete framelet transforms with mixed dilation factors. Finally, we will discuss framelets and wavelets with mixed dilation factors in the space $L_2(\mathbb{R}^d)$.

Framelets and Wavelets with Mixed Dilation Factors

TL;DR

This work develops a systematic theory for framelets and wavelets with mixed dilation factors in arbitrary dimensions and multiplicities. By building a complete discrete framework—comprising mixed-dilation subdivision/transition operators, multi-level perfect reconstruction, and stability analysis—the authors extend prior scalar results to general vector-framelets and dual-framelet constructions. They then introduce discrete affine systems to study the stability and reconstruction properties in and connect these discrete theories to continuous framelets in , showing that stable mixed-dilation filter banks yield dual framelets in both discrete and continuous settings. The results unify and extend existing mixed-dilation theory (notably for scalar tight framelets) to arbitrary multiplicity, and establish robust connections between filter-bank formulations and framelets, including reconstruction, stability, and cascade properties. These contributions provide a flexible framework for constructing directionally rich, lower-redundancy framelet systems in higher dimensions with practical implications for image processing and harmonic analysis.

Abstract

As a main research area in applied and computational harmonic analysis, the theory and applications of framelets have been extensively investigated. Most existing literature is devoted to framelet systems that only use one dilation matrix as the sampling factor. To keep some key properties such as directionality, a framelet system often has a high redundancy rate. To reduce redundancy, a one-dimensional tight framelet with mixed dilation factors has been introduced for image processing. Though such tight framelets offer good performance in practice, their theoretical properties are far from being well understood. In this paper, we will systematically investigate framelets with mixed dilation factors, with arbitrary multiplicity in arbitrary dimensions. We will first study the discrete framelet transform employing a filter bank with mixed dilation factors and discuss its various properties. Next, we will introduce the notion of a discrete affine system in and study discrete framelet transforms with mixed dilation factors. Finally, we will discuss framelets and wavelets with mixed dilation factors in the space .

Paper Structure

This paper contains 9 sections, 19 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Discrete Framelet Transforms with Mixed Dilation Factors
  3. Multi-level Discrete Framelet Transform and the Perfect Reconstruction Property
  4. The Stability of Framelet Filter Banks
  5. Wavelet Filter Banks with Mixed Dilation Factors
  6. Discrete Affine Systems in $l_{2}(\mathbb{Z}^d)$
  7. Framelets and Wavelets in $L_{2}(\mathbb{R}^d)$ with Mixed dilation Factors
  8. A Brief Review of Refinable Vector Functions
  9. Framlets and Wavelets in $L_{2}(\mathbb{R}^d)$ and Connections to Filter Banks

Key Result

Lemma 2.1

Let $a\in(l_{0}(\mathbb{Z}^d))^{r\times q}$ be a finitely supported matrix-valued filter and $\mathsf{M}$ be a $d\times d$ dilation matrix. Then, the following operators are well-defined: Moreover, we have $\mathcal{S}_{a,\mathsf{M}}=\mathcal{T}_{a,\mathsf{M}}^{\star}$. That is

Theorems & Definitions (42)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Proposition 3.1
  • proof
  • ...and 32 more