Quaternionic B-Splines

TL;DR

This paper introduces quaternionic B-splines, extending classical B-splines to quaternionic order for multi-channel signal and image analysis, establishing their properties, basis, and convergence behaviors.

Contribution

It defines quaternionic B-splines via Fourier transforms and differential equations, explores their properties, and demonstrates their use in multiresolution analysis and convergence to quaternionic Gaussians.

Findings

Quaternionic B-splines form a Riesz basis for their span.

They generate a multiresolution analysis.

They converge pointwise and in L^p to quaternionic Gaussian functions.

Abstract

We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the algebra of quaternions) for the purposes of multi-channel signal and image analysis. The functions $B_q$ are defined first by their Fourier transforms, then as the solutions of distributional differential equation of quaternionic order. The equivalence of these definitions requires properties of quaternionic Gamma functions and binomial expansions, both of which we investigate. The relationship between $B_q$ and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of $B_q$ is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and $L^p$ convergence of the quaternionic B-splines to quarternionic Gaussian functions.