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Quaternionic Reweighted Amplitude Flow for Phase Retrieval in Image Reconstruction

Ren Hu, Pan Lian

TL;DR

This work extends phase retrieval to the quaternion domain by formulating an amplitude-based objective $F(oldsymbol{z})= rac{1}{n} \sum_{j=1}^{n} \left(|\langle \boldsymbol{\alpha}_j, \boldsymbol{z}\rangle|-\psi_j\right)^2$ and developing the Quaternionic Reweighted Amplitude Flow (QRAF) framework, along with variants that accelerate and stabilize convergence. It also introduces the Quaternionic Perturbed Amplitude Flow (QPAF) with linear convergence guarantees. Through extensive synthetic and real color-image experiments, the authors show that QRAF and its variants consistently outperform existing quaternionic PR methods (QWF, QTWF, QTAF) in recovery accuracy and efficiency, while QPAF provides a theoretically principled alternative with robust performance. The results demonstrate practical impact for color image reconstruction from phaseless measurements, including effective RGB processing via phase-factor estimates and robust performance under limited measurements.

Abstract

Quaternionic signal processing provides powerful tools for efficiently managing color signals by preserving the intrinsic correlations among signal dimensions through quaternion algebra. In this paper, we address the quaternionic phase retrieval problem by systematically developing novel algorithms based on an amplitude-based model. Specifically, we propose the Quaternionic Reweighted Amplitude Flow (QRAF) algorithm, which is further enhanced by three of its variants: incremental, accelerated, and adapted QRAF algorithms. In addition, we introduce the Quaternionic Perturbed Amplitude Flow (QPAF) algorithm, which has linear convergence. Extensive numerical experiments on both synthetic data and real images, demonstrate that our proposed methods significantly improve recovery performance and computational efficiency compared to state-of-the-art approaches.

Quaternionic Reweighted Amplitude Flow for Phase Retrieval in Image Reconstruction

TL;DR

This work extends phase retrieval to the quaternion domain by formulating an amplitude-based objective and developing the Quaternionic Reweighted Amplitude Flow (QRAF) framework, along with variants that accelerate and stabilize convergence. It also introduces the Quaternionic Perturbed Amplitude Flow (QPAF) with linear convergence guarantees. Through extensive synthetic and real color-image experiments, the authors show that QRAF and its variants consistently outperform existing quaternionic PR methods (QWF, QTWF, QTAF) in recovery accuracy and efficiency, while QPAF provides a theoretically principled alternative with robust performance. The results demonstrate practical impact for color image reconstruction from phaseless measurements, including effective RGB processing via phase-factor estimates and robust performance under limited measurements.

Abstract

Quaternionic signal processing provides powerful tools for efficiently managing color signals by preserving the intrinsic correlations among signal dimensions through quaternion algebra. In this paper, we address the quaternionic phase retrieval problem by systematically developing novel algorithms based on an amplitude-based model. Specifically, we propose the Quaternionic Reweighted Amplitude Flow (QRAF) algorithm, which is further enhanced by three of its variants: incremental, accelerated, and adapted QRAF algorithms. In addition, we introduce the Quaternionic Perturbed Amplitude Flow (QPAF) algorithm, which has linear convergence. Extensive numerical experiments on both synthetic data and real images, demonstrate that our proposed methods significantly improve recovery performance and computational efficiency compared to state-of-the-art approaches.
Paper Structure (27 sections, 3 theorems, 20 equations, 18 figures, 4 tables, 8 algorithms)

This paper contains 27 sections, 3 theorems, 20 equations, 18 figures, 4 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Preliminaries
  4. Quaternionic Matrices
  5. Dirac Operator and Generalized $\mathbb{HR}$ Calculus
  6. Quaternionic Reweighted Amplitude Flow
  7. Algorithm
  8. Quaternionic Weighted Maximal Correlation Initialization
  9. Quaternionic Adaptively Reweighted Gradient Flow
  10. Linear Convergence Discussion
  11. Further Developments of QRAF
  12. Incremental Algorithm: QIRAF--the critical one
  13. Accelerated Algorithm: QARAF
  14. Adapted Algorithm: QAdRAF
  15. QRWF Algorithm
  16. ...and 12 more sections

Key Result

Proposition 3.1

For an arbitrary $x\in \mathbb{H}^{d}$, consider the noiseless measurements $\psi_{i}=|\psi^{*}x|$, $1\le i\le n$. If $n\ge c_{0}|S|\ge c_{1}d$, then with probability exceeding $1-c_{3}e^{-c_{2}n}$, the initial guess $z_{0}$ obtained by the weighted maximal correlation method satisfies ${\rm dist} (

Figures (18)

  • Figure 1: Simulations of different algorithms on quaternion-valued signals
  • Figure 2: Simulations of different algorithms on signals of pure quaternions
  • Figure 3: Success rate for $d = 64$
  • Figure 4: Success rate for $d = 100$
  • Figure 5: Success rate of QRAF and its variants in various views
  • ...and 13 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1: Weighted Initialization
  • Definition 3.2: Local Regularity Condition
  • Lemma 3.3: Local error contraction
  • proof
  • Lemma 3.4
  • proof