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Monte Carlo Maximum Likelihood Reconstruction for Digital Holography with Speckle

Xi Chen, Arian Maleki, Shirin Jalali

TL;DR

This work tackles the challenge of speckle in coherent imaging by formulating a principled maximum likelihood reconstruction that accounts for realistic aperture models. It introduces PGD‑MC, a matrix‑free optimization framework that uses conjugate gradient for likelihood terms and Monte Carlo trace estimation to avoid explicit inversions of the covariance $oldsymbol{ abla}(oldsymbol{x})$, enabling scalable, high‑resolution reconstructions in digital holography. The method accommodates flexible priors via denoisers (e.g., BM3D, DnCNN, Deep Decoder) and demonstrates superior accuracy and robustness over prior Plug‑and‑Play approaches across aperture types, noise levels, and numbers of looks. The results highlight practical viability for fast, physically accurate holographic imaging in the presence of speckle, with code released for reproducibility and further development.

Abstract

In coherent imaging, speckle is statistically modeled as multiplicative noise, posing a fundamental challenge for image reconstruction. While maximum likelihood estimation (MLE) provides a principled framework for speckle mitigation, its application to coherent imaging system such as digital holography with finite apertures is hindered by the prohibitive cost of high-dimensional matrix inversion, especially at high resolutions. This computational burden has prevented the use of MLE-based reconstruction with physically accurate aperture modeling. In this work, we propose a randomized linear algebra approach that enables scalable MLE optimization without explicit matrix inversions in gradient computation. By exploiting the structural properties of sensing matrix and using conjugate gradient for likelihood gradient evaluation, the proposed algorithm supports accurate aperture modeling without the simplifying assumptions commonly imposed for tractability. We term the resulting method projected gradient descent with Monte Carlo estimation (PGD-MC). The proposed PGD-MC framework (i) demonstrates robustness to diverse and physically accurate aperture models, (ii) achieves substantial improvements in reconstruction quality and computational efficiency, and (iii) scales effectively to high-resolution digital holography. Extensive experiments incorporating three representative denoisers as regularization show that PGD-MC provides a flexible and effective MLE-based reconstruction framework for digital holography with finite apertures, consistently outperforming prior Plug-and-Play model-based iterative reconstruction methods in both accuracy and speed. Our code is available at: https://github.com/Computational-Imaging-RU/MC_Maximum_Likelihood_Digital_Holography_Speckle.

Monte Carlo Maximum Likelihood Reconstruction for Digital Holography with Speckle

TL;DR

This work tackles the challenge of speckle in coherent imaging by formulating a principled maximum likelihood reconstruction that accounts for realistic aperture models. It introduces PGD‑MC, a matrix‑free optimization framework that uses conjugate gradient for likelihood terms and Monte Carlo trace estimation to avoid explicit inversions of the covariance , enabling scalable, high‑resolution reconstructions in digital holography. The method accommodates flexible priors via denoisers (e.g., BM3D, DnCNN, Deep Decoder) and demonstrates superior accuracy and robustness over prior Plug‑and‑Play approaches across aperture types, noise levels, and numbers of looks. The results highlight practical viability for fast, physically accurate holographic imaging in the presence of speckle, with code released for reproducibility and further development.

Abstract

In coherent imaging, speckle is statistically modeled as multiplicative noise, posing a fundamental challenge for image reconstruction. While maximum likelihood estimation (MLE) provides a principled framework for speckle mitigation, its application to coherent imaging system such as digital holography with finite apertures is hindered by the prohibitive cost of high-dimensional matrix inversion, especially at high resolutions. This computational burden has prevented the use of MLE-based reconstruction with physically accurate aperture modeling. In this work, we propose a randomized linear algebra approach that enables scalable MLE optimization without explicit matrix inversions in gradient computation. By exploiting the structural properties of sensing matrix and using conjugate gradient for likelihood gradient evaluation, the proposed algorithm supports accurate aperture modeling without the simplifying assumptions commonly imposed for tractability. We term the resulting method projected gradient descent with Monte Carlo estimation (PGD-MC). The proposed PGD-MC framework (i) demonstrates robustness to diverse and physically accurate aperture models, (ii) achieves substantial improvements in reconstruction quality and computational efficiency, and (iii) scales effectively to high-resolution digital holography. Extensive experiments incorporating three representative denoisers as regularization show that PGD-MC provides a flexible and effective MLE-based reconstruction framework for digital holography with finite apertures, consistently outperforming prior Plug-and-Play model-based iterative reconstruction methods in both accuracy and speed. Our code is available at: https://github.com/Computational-Imaging-RU/MC_Maximum_Likelihood_Digital_Holography_Speckle.
Paper Structure (30 sections, 42 equations, 4 figures, 4 tables, 3 algorithms)

This paper contains 30 sections, 42 equations, 4 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Forward model in coherent imaging with speckle
  4. Maximum likelihood estimation for image reconstruction
  5. Projected gradient descent to solve the optimization problem
  6. Challenges in MLE-based reconstruction
  7. Randomized matrix-Free gradient evaluation for MLE
  8. Conjugate gradient
  9. Monte Carlo-based approximation of matrix diagonals
  10. Numerical Results
  11. Experimental setup
  12. Sub-area of hologram spectrum for reconstruction
  13. Reconstruction performance
  14. Impact of prior model
  15. Impact of aperture models
  16. ...and 15 more sections

Figures (4)

  • Figure 1: Visualization of apertures in the imaging forward model \ref{['eq:forward_1']}.
  • Figure 2: (a) Sub-area of hologram spectrum used for the corresponding low-resolution despeckling; (b) Average PSNR convergence curves of PGD-MC and CPnP-EM, with different noise levels $\sigma_z=15,25,50$, and numbers of looks $L=1,2,4$, where prior model is Deep Decoder.
  • Figure 3: Reconstructions of PGD-MC (top) and CPnP-EM (bottom) under different experimental settings. In (b)-(d) Deep Decoder is used to model signal prior. (a) Different prior models, circular aperture radius $\frac{2r}{H} = 1.0$, single look, and additive noise $\sigma_z=25$. (b) Different aperture models (circular and annular), single look, additive noise $\sigma_z=25$. (c) Different noise levels $\sigma_z=15,25,50$, circular aperture radius $\frac{2r}{H}=1.0$, single look. (d) Different numbers of looks $L=1,10,20$, circular aperture radius $\frac{2r}{H}=1.0$, additive noise $\sigma_z=25$.
  • Figure 4: Effect of MC samples $K$ on reconstruction performance (left). Number of CG iterations required to converge at each PGD iteration (middle), test image is peppers, aperture radius $\frac{2r}{H}=1.0$, single look $L=4$, additive noise $\sigma_z = 25$. Time cost per iteration for gradient approximation in PGD-MC and CPnP-EM (right), image sizes are $128{\times}128$, $256{\times}256$, and $512{\times}512$, with $K=5$ MC samples.