An Invitation to Hypercomplex Phase Retrieval: Theory and Applications

TL;DR

This work introduces hypercomplex phase retrieval (HPR), extending classical phase retrieval to quaternion and octonion signals to exploit cross-dimensional correlations via Clifford algebra. It develops quaternion and octonion PR formulations, including real-valued and quaternion-valued sensing matrices, spectral initialization, and Wirtinger-flow–type gradient updates, with detailed descriptions of gradient calculations in quaternion calculus and real-representation techniques for octonions. The paper then surveys emerging HPR applications, notably hypercomplex Fourier PR, hypercomplex STFT PR, and hypercomplex wavelet PR, illustrating how these transforms and coding strategies improve phaseless reconstruction and enable 3-D and multispectral imaging scenarios. Finally, it outlines future directions, including co-design of diffractive elements, bispectrum/HPR, FrFT-based PR, and data-driven hypercomplex reconstruction, underscoring the practical impact on optical imaging and high-dimensional sensing.

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.

Figures (4)

• Figure 1: (a) Recovery performance of QWF for different sample complexity and WF algorithm by concatenating the four dimensions of the quaternion signal. For the quaternion setting, the sensing matrix was drawn from a quaternion normal distribution. For the concatenation, the same was drawn from a complex normal distribution. The experiments are performed for $n=500$. (b) Illustration of QPR recovery using a red-green-blue (RGB) image. Here, we set each color channel as the imaginary parts of a quaternion signal and reconstruct it with QWF. For comparison, by concatenating each color channel and applying WF, the recovery is poorer than QPR. In both cases, reconstruction was performed in patches of $N=32$ such that $n=1024$ and $m/n = 15$.
• Figure 2: Reconstruction of real data with OWF. (a) RGB representation of the eight-channel spectral image and its individual eight components on the right panel. (b) The OWF-reconstructed image and components. The PSNR of the recovered image is 39.007 [dB]. (c) The GD-based reconstructed image and components. The PSNR of the recovered image is 24.1615 [dB]. Recovered octonion-valued numbers from two coordinates (c) (15,15) and (d) (10,10), labeled 'P1' and 'P2', respectively jacome2023octonion.
• Figure 3: Success rate of OWF for different values of sampling complexity $m/n$ with $n=30$ with measurements under additive Gaussian noise with a signal-to-noise ratio varying from 0 to 30 jacome2023octonion.
• Figure 4: (a) Illustration of the physical measurement set-up of Fourier QPR in optical imaging where a coherent illumination diffracts from an object. A DOE codifies the scene. Considering far-zone propagation, the sensor measures the magnitude of the coded scenes via QDFT. (b) Quaternion modeling of Fourier phase retrieval, where the sensing matrix contains both the per-channel codification of the DOE and the quaternion Fourier transform. (c) Fourier QPR recovery of an RGB image with a different number of coding elements from $d=4$ to $d=8$ using QWF. Here, $m/n=L=10$ and $n=128^2$. (d) Phase transition of QWF for Fourier QPR by varying the sample complexity $m/n$ and different values of $d$. Here, we set $n=1000$.