Cramér-Rao lower bound and maximum-likelihood estimation in ptychography with Poisson noise

We investigate the performance of ptychography with noisy data by analyzing the Cramér-Rao lower bound. The lower bound of ptychography is derived and numerically computed for both top-hat plane wave and structured illumination. The influence of Poisson noise on the ptychography reconstruction is discussed. The computation result shows that, if the estimator is unbiased, the minimum variance for Poisson noise is mostly determined by the illumination power and the transmission function of the object. Monte Carlo analysis is conducted to validate our calculation results for different photon flux numbers. Furthermore, the performance of the maximum-likelihood method and the approach of amplitude-based cost-function minimization is studied in the Monte Carlo analysis.