Hypercomplex Phase Retrieval

2026-02-27

AI Summary

This work proposes a hypercomplex phase retrieval framework grounded in Clifford algebra, systematically introducing quaternionic, octonionic, and other hypercomplex algebras into the reconstruction of high-dimensional signals from intensity-only measurements. Traditional phase retrieval methods struggle to effectively model cross-dimensional dependencies inherent in such signals. By constructing structured sensing operators—such as hypercomplex Fourier transforms—the proposed approach explicitly captures the intrinsic coupling among multiple signal dimensions. This formulation transcends the representational limitations of real- or complex-valued models, enabling more accurate reconstruction of high-dimensional signals in applications like optical imaging and computational sensing. The study thus advances the theoretical foundations and practical potential of hypercomplex signal processing in computational imaging.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Hypercomplex Phase Retrieval
• Phase Retrieval
• Hypercomplex Signal Processing
• Optical Imaging
• Intensity-only Measurements

Innovation

• Hypercomplex Phase Retrieval
• Clifford algebra
• Quaternion-valued signals
• Multidimensional signal processing
• Optical imaging