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Octonion Phase Retrieval

Roman Jacome, Kumar Vijay Mishra, Brian M. Sadler, Henry Arguello

TL;DR

This work addresses octonion phase retrieval (OPR) for eight-channel multispectral signals by introducing octonion Wirtinger flow (OWF), a gradient-based recovery method that uses a pseudo-real-matrix representation to handle octonion non-associativity. The authors establish that OPR has only a trivial right-octonion phase ambiguity, and they formulate an OWF algorithm with spectral initialization to solve the phaseless reconstruction problem in the real-representation domain. Empirical results on synthetic and real multispectral data show OWF achieves high-accuracy recovery, outperforming naive concatenation-based approaches, and remains robust under noise. This advances hypercomplex signal processing for optical imaging and multispectral phase retrieval, enabling reliable reconstruction from magnitude-only measurements.

Abstract

Signal processing over hypercomplex numbers arises in many optical imaging applications. In particular, spectral image or color stereo data are often processed using octonion algebra. Recently, the eight-band multispectral image phase recovery has gained salience, wherein it is desired to recover the eight bands from the phaseless measurements. In this paper, we tackle this hitherto unaddressed hypercomplex variant of the popular phase retrieval (PR) problem. We propose octonion Wirtinger flow (OWF) to recover an octonion signal from its intensity-only observation. However, contrary to the complex-valued Wirtinger flow, the non-associative nature of octonion algebra and the consequent lack of octonion derivatives make the extension to OWF non-trivial. We resolve this using the pseudo-real-matrix representation of octonion to perform the derivatives in each OWF update. We demonstrate that our approach recovers the octonion signal up to a right-octonion phase factor. Numerical experiments validate OWF-based PR with high accuracy under both noiseless and noisy measurements.

Octonion Phase Retrieval

TL;DR

This work addresses octonion phase retrieval (OPR) for eight-channel multispectral signals by introducing octonion Wirtinger flow (OWF), a gradient-based recovery method that uses a pseudo-real-matrix representation to handle octonion non-associativity. The authors establish that OPR has only a trivial right-octonion phase ambiguity, and they formulate an OWF algorithm with spectral initialization to solve the phaseless reconstruction problem in the real-representation domain. Empirical results on synthetic and real multispectral data show OWF achieves high-accuracy recovery, outperforming naive concatenation-based approaches, and remains robust under noise. This advances hypercomplex signal processing for optical imaging and multispectral phase retrieval, enabling reliable reconstruction from magnitude-only measurements.

Abstract

Signal processing over hypercomplex numbers arises in many optical imaging applications. In particular, spectral image or color stereo data are often processed using octonion algebra. Recently, the eight-band multispectral image phase recovery has gained salience, wherein it is desired to recover the eight bands from the phaseless measurements. In this paper, we tackle this hitherto unaddressed hypercomplex variant of the popular phase retrieval (PR) problem. We propose octonion Wirtinger flow (OWF) to recover an octonion signal from its intensity-only observation. However, contrary to the complex-valued Wirtinger flow, the non-associative nature of octonion algebra and the consequent lack of octonion derivatives make the extension to OWF non-trivial. We resolve this using the pseudo-real-matrix representation of octonion to perform the derivatives in each OWF update. We demonstrate that our approach recovers the octonion signal up to a right-octonion phase factor. Numerical experiments validate OWF-based PR with high accuracy under both noiseless and noisy measurements.
Paper Structure (6 sections, 2 theorems, 10 equations, 3 figures)

This paper contains 6 sections, 2 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Desiderata for Octonion Algebra
  3. OPR and Trivial Ambiguity
  4. Recovery Algorithm
  5. Numerical Experiments
  6. Summary

Key Result

Proposition 1

koltchinskii2015bounding Assume $\beta_\ell$ where $\ell=1,\dots,m$ to be independent copies of $\beta$. Denote a family of functions that satisfy a uniform small-ball estimation by $\mathcal{F}$. For a constant $\tau>0$, we have $Q_{\mathcal{F}}(\tau)=\operatorname{inf}_{f\in \mathcal{F}} \mathbb{P

Figures (3)

  • Figure 1: (a) Success rate of OWF and concatenated WF for different value of sampling complexity $m/n$ with $n=100$. (b) Convergence rate of OWF for $m/n=20$ for 2000 iterations.
  • Figure 2: Success rate of OWF for different values of sampling complexity $m/n$ with $n=30$ with measurements under additive Gaussian noise
  • Figure 3: Reconstruction with real data. (a) Statistical performance for 64 images of the spectral image dataset. (b) RGB representation of the 8-channel spectral image and its individual 8 components on the right panel. (c) OWF-reconstructed image and its components; recovered image's PSNR $= 39.01$ dB (d) GD-reconstructed image and its components; recovered image's PSNR $= 24.16$ dB. (e) Recovered spectral signature.

Theorems & Definitions (2)

  • Proposition 1: Lower bound on quadratic stochastic process
  • Theorem 2: Trivial ambiguity of right-octonion phase factor