The CRLB and Maximum likelihood in ptychography with Poisson noise

We investigate the performance of ptychography with noisy data by analyzing the Cram\'{e}r Rao Lower Bound. The lower bound of ptychography is derived and numerically computed for both 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 probe function. Monte Carlo analysis is conducted to validate our calculation results for different signal-to-noise ratios. The performance of the maximum likelihood method and the approach of amplitude-based cost function minimization is studied in the Monte Carlo analysis also.