  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.
