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：超复数信号处理在许多光学成像应用中具有重要价值。特别是，光谱图像或彩色立体数据常通过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。本文首次解决了相位恢复（PR）这一经典问题中尚未涉及的超复数变体。我们提出八元数Wirtinger流（OWF）算法，用于从仅有强度观测中恢复八元数信号。然而，与复数值Wirtinger流不同，八元数代数的非结合特性及其导数运算的缺失使得OWF扩展具有非平凡性。我们利用八元数的伪实矩阵表示法在每次OWF迭代中执行导数运算。理论证明该方法可恢复至多一个右八元数相位因子的八元数信号。数值实验验证了基于OWF的相位恢复方法在无噪声与有噪声测量条件下均具有高精度。