An Invitation to Hypercomplex Phase Retrieval: Theory and Applications

TL;DR

This paper explores hypercomplex phase retrieval, leveraging Clifford algebra to improve multidimensional signal processing, with applications mainly in optical imaging, and introduces new theoretical insights and potential algorithms.

Contribution

It introduces the hypercomplex phase retrieval framework, connecting it with hypercomplex signal processing and Clifford algebra, and discusses its applications in optical imaging.

Findings

Hypercomplex PR extends traditional phase retrieval to quaternion and octonion signals.

The framework enables new algorithms for optical imaging and computational sensing.

It opens avenues for developing advanced HSP tools for multidimensional signals.

Abstract

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.