A Geometric Algebra Framework for a Multidimensional Analytic Signal

TL;DR

This paper introduces a unified geometric algebra framework for defining multidimensional analytic signals, integrating previous approaches into a single mathematical structure using idempotents.

Contribution

It presents a novel geometric algebra-based framework that generalizes and unifies existing multidimensional analytic signal definitions.

Findings

Unified framework for multidimensional analytic signals

Incorporation of hypercomplex and monogenic signals

Use of geometric algebra and idempotents

Abstract

This work examines the problem of extending the one-dimensional analytic signal, which is ubiquitous throughout signal processing, to higher dimensional signals. Bulow et al. and Felsberg et al. have previously used techniques from Clifford algebra and analysis to extend the one-dimensional analytic signal to higher dimensions. However, each author sets forth a different definition of a multidimensional analytic signal. Herein we follow an observation of Brackx et al. and adopt a general definition of an analytic signal that encompasses both the hypercomplex signal of Bulow et al. and the monogenic signal of Felsberg et al. within the same mathematical framework. The crux of our approach is captured by the following statement: A multidimensional analytic signal is generated by an idempotent. We develop this notion more specifically using examples from geometric algebra.