Hypercomplex Phase Retrieval

TL;DR

Hypercomplex phase retrieval (HPR) leverages hypercomplex signal processing and Clifford algebra to improve signal recovery from intensity measurements, with applications in optical imaging and computational sensing.

Contribution

This paper surveys emerging methodologies and applications of hypercomplex phase retrieval, highlighting its potential in optical imaging and advanced signal processing.

Findings

HPR effectively models signals in quaternion and octonion forms.

HPR algorithms utilize structured sensing operators like Fourier transforms.

HPR opens new avenues for optical imaging applications.

Abstract

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.