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.

翻译：相位恢复问题在光学成像领域具有诸多应用。受生物标本成像实验的启发，我们主要考虑低剂量照明场景，其中泊松噪声起主导作用。本文讨论基于不同损失函数的梯度下降算法，这些函数针对受泊松噪声影响的数据（特别是低剂量场景）进行适配。从泊松分布的最大对数似然函数出发，我们研究了问题的不同正则化与逼近方法，以设计满足实际应用需求的算法。在此过程中，我们重点关注低计数测量。针对所有提出的损失函数，我们考察了相应梯度下降算法收敛至驻点的性质，并确定了能保证每次迭代损失下降的恒定步长。通过低剂量场景下的数值实验验证了理论观察结果。