Phase-retrieval error: a lower bound

Estimation theory is applied to problems in which it is desired to estimate an object given measurements of the intensity of its Fourier transform and object constraint information. The Cramer–Rao lower bound on the mean-squared error of the object estimate is found. The lower bound is independent of the phase-retrieval algorithm and is therefore indicative of the best possible estimation performance. Objects with complex Gaussian statistics were simulated and corresponding noisy Fourier intensity measurements computed. An iterative phase-retrieval algorithm was then used to estimate the object. Comparison of the error with the lower bound indicates that estimation theory is useful in predicting phase-retrieval error.