Hypercomplex signal processing (HSP) provides state-of-the-art tools to handle multidimensional signals by harnessing intrinsic correlation of the signal dimensions through Clifford algebra. Recently, the hypercomplex representation of the phase retrieval (PR) problem, wherein a complex-valued signal is estimated through its intensity-only projections, has attracted significant interest. The hypercomplex PR (HPR) arises in many optical imaging and computational sensing applications that usually comprise quaternion and octonion-valued signals. Analogous to the traditional PR, measurements in HPR may involve complex, hypercomplex, Fourier, and other sensing matrices. This set of problems opens opportunities for developing novel HSP tools and algorithms. This article provides a synopsis of the emerging areas and applications of HPR with a focus on optical imaging.

翻译：超复数信号处理（HSP）通过克利福德代数利用信号维度的内在相关性，为处理多维信号提供了先进工具。近年来，相位恢复（PR）问题的超复数表示引起了广泛关注，该问题通过仅利用强度的投影来估计复值信号。超复数相位恢复（HPR）出现在许多光学成像和计算传感应用中，这些应用通常涉及四元数和八元数值信号。与传统相位恢复类似，HPR中的测量可能涉及复数、超复数、傅里叶以及其他传感矩阵。这类问题为开发新型HSP工具和算法提供了机遇。本文综述了HPR的新兴领域及其应用，重点关注光学成像。