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

TL;DR

This paper reformulates the construction of quaternion-valued wavelets on the plane as a feasibility problem and uses the Douglas-Rachford algorithm to generate new wavelet examples with desirable properties.

Contribution

It introduces a novel feasibility-based approach and applies the Douglas-Rachford algorithm to construct nonseparable, multiresolution quaternion wavelets with symmetry features.

Findings

New nonseparable quaternion wavelets constructed

Application of Douglas-Rachford algorithm to wavelet design

Symmetric quaternion-valued scaling functions developed

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.