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A Stochastic ADMM Algorithm for Large-Scale Ptychography with Weighted Difference of Anisotropic and Isotropic Total Variation

Kevin Bui, Zichao Di

TL;DR

A novel approach is presented by introducing a class of variational models that incorporate the weighted difference of anisotropic–isotropic total variation that enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge.

Abstract

Ptychography, a prevalent imaging technique in fields such as biology and optics, poses substantial challenges in its reconstruction process, characterized by nonconvexity and large-scale requirements. This paper presents a novel approach by introducing a class of variational models that incorporate the weighted difference of anisotropic--isotropic total variation. This formulation enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge. To tackle the large-scale nature of the problem, we propose an efficient stochastic alternating direction method of multipliers, which guarantees convergence under mild conditions. Numerical experiments validate the superiority of our approach by demonstrating its capability to successfully reconstruct complex-valued images, especially in recovering the phase components even in the presence of highly corrupted measurements.

A Stochastic ADMM Algorithm for Large-Scale Ptychography with Weighted Difference of Anisotropic and Isotropic Total Variation

TL;DR

A novel approach is presented by introducing a class of variational models that incorporate the weighted difference of anisotropic–isotropic total variation that enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge.

Abstract

Ptychography, a prevalent imaging technique in fields such as biology and optics, poses substantial challenges in its reconstruction process, characterized by nonconvexity and large-scale requirements. This paper presents a novel approach by introducing a class of variational models that incorporate the weighted difference of anisotropic--isotropic total variation. This formulation enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge. To tackle the large-scale nature of the problem, we propose an efficient stochastic alternating direction method of multipliers, which guarantees convergence under mild conditions. Numerical experiments validate the superiority of our approach by demonstrating its capability to successfully reconstruct complex-valued images, especially in recovering the phase components even in the presence of highly corrupted measurements.
Paper Structure (18 sections, 8 theorems, 119 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 8 theorems, 119 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. AITV model
  4. $u$-subproblem
  5. $\omega$-subproblem
  6. $v$-subproblem
  7. $z$-subproblem
  8. Convergence Analysis
  9. Numerical Results
  10. Gaussian noise
  11. Poisson noise
  12. Conclusion
  13. Proof of Lemma \ref{['lemma:prox_l1l2']}
  14. Proofs of Section \ref{['sec:convergence']}
  15. Proof of Lemma \ref{['lemma:bounded_sequence']}
  16. ...and 3 more sections

Key Result

Lemma 2.1

Given $x' \in \mathbb{C}^n$, $\lambda >0$, and $\alpha \geq 0$, we have the following cases: Then $x^*$ is an optimal solution to

Figures (4)

  • Figure 1: Schematic of a ptychography experiment.
  • Figure 2: Total computational time (seconds) vs. magnitude/phase SSIMs for the blind algorithms. Each algorithm is ran for 300 epochs for the complex image given by Figures \ref{['fig:cameraman_mag']}-\ref{['fig:cameraman_phase']}, where the measurements are corrupted with Poisson noise with $\zeta = 0.01$ (SNR $\approx 40$)
  • Figure 3: Changes in magnitude and phase SSIMs with respect to the AITV regularization parameter $\lambda \in \{0.01, 0.05, 0.15, 0.25, 0.50, 1.00\}$. Each AITV algorithm is ran for 300 epochs in the blind case for the complex image given by Figures \ref{['fig:cameraman_mag']}-\ref{['fig:cameraman_phase']}, where the measurements are corrupted with Poisson noise with $\zeta = 0.01$ (SNR $\approx 40$).
  • Figure 4: Magnitude and phase SSIMs over different Poisson noise level for the complex image given by Figure \ref{['fig:cameraman_mag']}-\ref{['fig:cameraman_phase']} for the blind case.

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Remark 2
  • proof
  • ...and 8 more