Hypercomplex signal processing (HSP) offers powerful tools for analyzing and processing multidimensional signals by explicitly exploiting inter-dimensional correlations through Clifford algebra. In recent years, hypercomplex formulations of the phase retrieval (PR) problem, wheren a complex-valued signal is recovered from intensity-only measurements, have attracted growing interest. Hypercomplex phase retrieval (HPR) naturally arises in a range of optical imaging and computational sensing applications, where signals are often modeled using quaternion- or octonion-valued representations. Similar to classical PR, HPR problems may involve measurements obtained via complex, hypercomplex, Fourier, or other structured sensing operators. These formulations open new avenues for the development of advanced HSP-based algorithms and theoretical frameworks. This chapter surveys emerging methodologies and applications of HPR, with particular emphasis on optical imaging systems.

翻译：超复数信号处理通过克利福德代数显式利用维度间相关性，为分析和处理多维信号提供了强大工具。近年来，相位恢复问题的超复数表述——即从仅含强度的测量中恢复复值信号——引起了日益增长的研究兴趣。超复数相位恢复自然出现在一系列光学成像与计算感知应用中，这些应用中的信号常采用四元数或八元数值表示进行建模。与经典相位恢复类似，超复数相位恢复问题可能涉及通过复数、超复数、傅里叶或其他结构化感知算子获得的测量数据。这些表述为开发基于超复数信号处理的先进算法与理论框架开辟了新途径。本章系统综述了超复数相位恢复的新兴方法与应用，特别聚焦于光学成像系统。