Hypercomplex Phase Retrieval

TL;DR

This chapter surveys emerging methodologies and applications of HPR, with particular emphasis on optical imaging systems.

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.

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 jacome2024octonion.
• 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 jacome2024octonion.
• 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 $n=1000$.