Phase Retrieval from Intensity Data degraded by Shot-Noise

In the paper by one of us (H.A.F.) [1], the phase problem was reviewed for circumstances under which the stochastic character of the image could be ignored. A statistical description becomes imperative for images obtained under low-intensity illumination as occurs e.g. in low-dose electron microscopy. Under these circumstances the images are heavily degraded by shot-noise. Often(e.g. [2,3])images are described as a signal plus signal-independent noise. Such a treatment does not apply to shot-noise, the quanta arriving in non-overlapping image cells are statistically independent Poisson-distributed random variables [4]. The variances of the data-counts also contain information about the signal. In this contribution we discuss the retrieval of the object wave function (w.f.) from shot-noise degraded images. Due to the stochastic imaging process, the reconstructed object w.f. is also stochastic. The purpose of this contribution is to establish the statistical characterization of the object w.f. We assume axial coherent quasi-monochromatic illumination. The analysis is presented for one lateral dimension only, the extension to two dimensions is obvious.