An Invitation to Hypercomplex Phase Retrieval: Theory and applications

Hypercomplex signal processing (HSP) provides state-of-the-art tools to handle multidimensional signals by harnessing the 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.

Introduction

In many engineering applications, it is useful to represent the signals of interest as hypercomplex numbers, that is, they are elements of some algebras over the field of real numbers. While vector spaces allow only addition and scalar multiplication, algebras permit both addition and multiplication among their elements. In particular, hypercomplex representation enables multidimensional signal and image processing applications by harnessing Clifford algebra to exploit intrinsic correlation within the different signal dimensions. HSP finds applications in several important problems including optical imaging, array processing, wireless communications, filtering, and neural networks; see, e.g., [1] and [2] and references therein. Recently, HSP has been investigated for tackling the long-standing problem of PR in high-dimensional settings that usually arise in optical imaging. In this article, we provide an introduction to the fundamental concepts of HPR and corresponding sample problems in optical imaging.