Octonion Phase Retrieval

TL;DR

This paper introduces a novel octonion phase retrieval method using Wirtinger flow, addressing the challenges of non-associativity in octonion algebra, and demonstrates high-accuracy signal recovery in optical imaging applications.

Contribution

It develops the first octonion Wirtinger flow algorithm for phase retrieval, overcoming algebraic challenges with a pseudo-real-matrix approach, and validates its effectiveness through numerical experiments.

Findings

Accurately recovers octonion signals from intensity measurements.

Effective under both noiseless and noisy conditions.

Addresses non-associativity in octonion calculus.

Abstract

Signal processing over hypercomplex numbers arises in many optical imaging applications. In particular, spectral image or color stereo data are often processed using octonion algebra. Recently, the eight-band multispectral image phase recovery has gained salience, wherein it is desired to recover the eight bands from the phaseless measurements. In this paper, we tackle this hitherto unaddressed hypercomplex variant of the popular phase retrieval (PR) problem. We propose octonion Wirtinger flow (OWF) to recover an octonion signal from its intensity-only observation. However, contrary to the complex-valued Wirtinger flow, the non-associative nature of octonion algebra and the consequent lack of octonion derivatives make the extension to OWF non-trivial. We resolve this using the pseudo-real-matrix representation of octonion to perform the derivatives in each OWF update. We demonstrate that our approach recovers the octonion signal up to a right-octonion phase factor. Numerical experiments validate OWF-based PR with high accuracy under both noiseless and noisy measurements.