Discretization of quaternionic continuous wavelet transforms

TL;DR

This paper introduces a method to discretize quaternionic continuous wavelet transforms by leveraging discretization techniques from 2D wavelet transforms, resulting in a discrete frame for quaternion-valued functions.

Contribution

It presents a novel scheme to construct bases and frames for quaternionic Hilbert spaces from complex-valued counterparts, extending wavelet discretization to quaternionic analysis.

Findings

A discretization scheme for quaternionic wavelet transforms is developed.

A discrete frame for quaternion-valued square integrable functions is constructed.

The approach generalizes 2D wavelet discretization techniques to quaternionic settings.

Abstract

A scheme to form a basis and a frame for a Hilbert space of quaternion valued square integrable function from a basis and a frame, respectively, of a Hilbert space of complex valued square integrable functions is introduced. Using the discretization techniques for 2D-continuous wavelet transform of the $SIM(2)$ group, the quaternionic continuous wavelet transform, living in a complex valued Hilbert space of square integrable functions, of the quaternion wavelet group is discretized, and thereby, a discrete frame for quaternion valued Hilbert space of square integrable functions is obtained.