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Wirtinger gradient descent methods for low-dose Poisson phase retrieval

Benedikt Diederichs, Frank Filbir, Patricia Römer

TL;DR

This work addresses phase retrieval under low-dose Poisson noise by employing Wirtinger-gradient descent with Poisson log-likelihood losses and variance-stabilizing Gaussian-like losses. It derives explicit step-size guarantees for convergence to stationary points under regularized Poisson and variance-stabilized losses, and it provides practical guidance for handling zero counts and low counts via ε-regularization and specialized transforms. Numerical experiments validate the theory, showing that Poisson-flow with carefully chosen ε and variance-stabilized amplitude flows can achieve competitive reconstructions in low-dose regimes. The results offer robust, implementable strategies for low-dose optical imaging of biological specimens and suggest avenues for incorporating regularization and extending the theory to more complex models.

Abstract

The problem of phase retrieval has many applications in the field of optical imaging. Motivated by imaging experiments with biological specimens, we primarily consider the setting of low-dose illumination where Poisson noise plays the dominant role. In this paper, we discuss gradient descent algorithms based on different loss functions adapted to data affected by Poisson noise, in particular in the low-dose regime. Starting from the maximum log-likelihood function for the Poisson distribution, we investigate different regularizations and approximations of the problem to design an algorithm that meets the requirements that are faced in applications. In the course of this, we focus on low-count measurements. For all suggested loss functions, we study the convergence of the respective gradient descent algorithms to stationary points and find constant step sizes that guarantee descent of the loss in each iteration. Numerical experiments in the low-dose regime are performed to corroborate the theoretical observations.

Wirtinger gradient descent methods for low-dose Poisson phase retrieval

TL;DR

This work addresses phase retrieval under low-dose Poisson noise by employing Wirtinger-gradient descent with Poisson log-likelihood losses and variance-stabilizing Gaussian-like losses. It derives explicit step-size guarantees for convergence to stationary points under regularized Poisson and variance-stabilized losses, and it provides practical guidance for handling zero counts and low counts via ε-regularization and specialized transforms. Numerical experiments validate the theory, showing that Poisson-flow with carefully chosen ε and variance-stabilized amplitude flows can achieve competitive reconstructions in low-dose regimes. The results offer robust, implementable strategies for low-dose optical imaging of biological specimens and suggest avenues for incorporating regularization and extending the theory to more complex models.

Abstract

The problem of phase retrieval has many applications in the field of optical imaging. Motivated by imaging experiments with biological specimens, we primarily consider the setting of low-dose illumination where Poisson noise plays the dominant role. In this paper, we discuss gradient descent algorithms based on different loss functions adapted to data affected by Poisson noise, in particular in the low-dose regime. Starting from the maximum log-likelihood function for the Poisson distribution, we investigate different regularizations and approximations of the problem to design an algorithm that meets the requirements that are faced in applications. In the course of this, we focus on low-count measurements. For all suggested loss functions, we study the convergence of the respective gradient descent algorithms to stationary points and find constant step sizes that guarantee descent of the loss in each iteration. Numerical experiments in the low-dose regime are performed to corroborate the theoretical observations.
Paper Structure (12 sections, 8 theorems, 53 equations, 4 figures)

This paper contains 12 sections, 8 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Gradient descent with Wirtinger derivatives
  3. Phase retrieval as an optimization problem
  4. Poisson log-likelihood loss
  5. Gaussian log-likelihood and variance stabilization
  6. Convergence analysis of gradient descent algorithms for phase retrieval
  7. Numerical experiments
  8. Phase retrieval from low-dose data with Poisson noise
  9. Gradient descent with Poisson log-likelihood
  10. Gradient descent with Gaussian log-likelihood after variance stabilization
  11. Discussion of the step size selection
  12. Conclusion

Key Result

Proposition 1

Let $\ b \in \mathbb{R}, \ f: \mathbb{C}^{n} \rightarrow [b,\infty),$ be a twice Wirtinger differentiable function with a uniformly bounded Hessian, i.e., for all $z,v \in \mathbb{C}^{n}$, with a constant $L > 0$ independent of $z$. Let the sequence $(z_{k})_{k\geq 0}$ be generated by the update eq:Pre1a with an arbitrary initialization $z_0 \in \mathbb{C}^{n}$. If $0 < \mu \leq L^{-1}$, then fo

Figures (4)

  • Figure 1: Variance-stabilizing transforms.
  • Figure 2: Distribution of measurements for a low-dose and a higher-dose experiment.
  • Figure 3: Performance of the Poisson flow with regularization parameter $\varepsilon \in \left\{10^{-3}, 0.1, 0.25, 0.5,1 \right\}$.
  • Figure 4: Performance of the gradient descent algorithms for the different discussed loss functions. The gray lines correspond to the Poisson flow as in \ref{['fig: rel_err_poisson']}.

Theorems & Definitions (13)

  • Proposition 1
  • Lemma 2
  • proof : Proof.
  • Lemma 3
  • proof : Proof.
  • Remark 4
  • Theorem 5
  • proof : Proof.
  • Theorem 6
  • proof : Proof.
  • ...and 3 more