Precomputable (Cramer-Rao) lower bounds for the integrated mean-square error are presented for the phase retrieval problem. The results are based upon the assumption that the phase is a sample function of a Gaussian random process and that the intensity measurements include additive white Gaussian noise. The bound, which is obtained from an information kernel which is the solution to a Fredholm integral equation, clearly demonstrates the effect of prior statistics, measurement noise and the nature of the linear transformation introduced prior to detection. Several examples are given; in particular, the role of diffraction is shown to be quite important for successful phase retrieval. Finally, the results are extended to include independent measurements of intensity in several planes.
Stanley R. Robinson,
"Fundamental Performance Limitations For The Phase Retrieval Problem", Proc. SPIE 0351, Wavefront Sensing, (1 August 1983); doi: 10.1117/12.933913; https://doi.org/10.1117/12.933913