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.

翻译：超复数信号处理在众多光学成像应用中具有重要意义。特别地，谱图像或彩色立体数据常采用八元数代数进行处理。近年来，八波段多光谱图像相位恢复问题日益凸显，其目标是从无相位测量中恢复八个波段的信息。本文针对这一尚未被解决的、经典相位恢复问题的超复数变体展开研究。我们提出八元数维廷格流算法，用于从仅含强度信息的观测中恢复八元数信号。然而，与复值维廷格流不同，八元数代数的非结合性及其导致的导数定义缺失，使得向OWF的扩展变得非平凡。我们通过采用八元数的伪实矩阵表示，在每次OWF更新中执行导数运算来解决这一难题。理论分析表明，该方法能以右八元数相位因子的不确定性恢复原始信号。数值实验验证了基于OWF的相位恢复方法在无噪及含噪测量条件下均能实现高精度重建。