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Quaternion-Valued Wavelets on the Plane: A Construction via the Douglas-Rachford Approach

Neil D. Dizon, Jeffrey A. Hogan

TL;DR

The work tackles the problem of constructing quaternion-valued wavelets on the plane that are nonseparable, multiresolution, compactly supported, smooth, and orthonormal. It reframes the quaternionic MRA design as a feasibility problem and solves it via the Douglas–Rachford algorithm in a product-space setting, encoding the design criteria in a quaternionic wavelet matrix $U(\xi)$ with QQMF conditions. The authors derive precise constraint sets for unitarity, vanishing moments, and symmetry, discretize the problem, and demonstrate explicit quaternionic scaling functions and wavelets obtained from DR solutions, including symmetry-enabled variants. This approach enables holistic processing of color images through quaternion-valued transforms and opens avenues for additional symmetries and design criteria in higher-dimensional wavelet theory.

Abstract

This paper presents a reformulation of the construction of nonseparable multiresolution quaternion-valued wavelets on the plane as a feasibility problem. The constraint sets in the feasibility problem are derived from the standard conditions of smoothness, compact support, and orthonormality. To solve the resulting feasibility problems, we employ a product space formulation of the Douglas-Rachford algorithm. This approach yields novel examples of nonseparable, multiresolution, compactly supported, smooth, and orthonormal quaternion-valued wavelets on the plane. Additionally, by introducing a symmetry-promoting constraint, we construct symmetric quaternion-valued scaling functions on the plane.

Quaternion-Valued Wavelets on the Plane: A Construction via the Douglas-Rachford Approach

TL;DR

The work tackles the problem of constructing quaternion-valued wavelets on the plane that are nonseparable, multiresolution, compactly supported, smooth, and orthonormal. It reframes the quaternionic MRA design as a feasibility problem and solves it via the Douglas–Rachford algorithm in a product-space setting, encoding the design criteria in a quaternionic wavelet matrix with QQMF conditions. The authors derive precise constraint sets for unitarity, vanishing moments, and symmetry, discretize the problem, and demonstrate explicit quaternionic scaling functions and wavelets obtained from DR solutions, including symmetry-enabled variants. This approach enables holistic processing of color images through quaternion-valued transforms and opens avenues for additional symmetries and design criteria in higher-dimensional wavelet theory.

Abstract

This paper presents a reformulation of the construction of nonseparable multiresolution quaternion-valued wavelets on the plane as a feasibility problem. The constraint sets in the feasibility problem are derived from the standard conditions of smoothness, compact support, and orthonormality. To solve the resulting feasibility problems, we employ a product space formulation of the Douglas-Rachford algorithm. This approach yields novel examples of nonseparable, multiresolution, compactly supported, smooth, and orthonormal quaternion-valued wavelets on the plane. Additionally, by introducing a symmetry-promoting constraint, we construct symmetric quaternion-valued scaling functions on the plane.
Paper Structure (23 sections, 26 theorems, 109 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 26 theorems, 109 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. Quaternion algebra
  5. Quaternionic Fourier transform
  6. Feasibility problems and the Douglas--Rachford algorithm
  7. Quaternionic wavelets on the plane
  8. MRA, scaling function and wavelets
  9. Quaternionic wavelet design conditions
  10. Quaternionic wavelet construction
  11. Consistency
  12. Design conditions for the wavelet matrix
  13. Discretisation by uniform sampling
  14. The quaternionic wavelet feasibility problem
  15. Relevant Hilbert modules
  16. ...and 8 more sections

Key Result

Proposition 2.2

Let $f,g \in L^2(\mathbb{R}^2, \mathbb{R}_2)$. The quaternionic Fourier transform satisfies the following properties: where $a\in\mathbb{R}^2$ and $b>0$. Furthermore, if $f \ast g$ is the convolution of $f$ and $g$, then which is also referred to as the convolution theorem for the quaternionic Fourier transform.

Figures (2)

  • Figure 1: Wavelet ensemble derived from Problem \ref{['prob:Qwaveletfeasibilityproblem']}\ref{['prob:QwaveletfeasibilityproblemA']} where $\eta=4$ and $\mu=1$, with $\tilde{\zeta}(\varphi)\approx 2.3243$.
  • Figure 5: Orthonormality checks for wavelet ensemble solutions.

Theorems & Definitions (56)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 46 more