Maximum Entropy Image Reconstruction from Phaseless Fourier Data

We investigate the reconstruction of a real and positive spatial pattern or "image" (1) from incomplete phaseless Fourier data Dkwith noise σk, (2) The first step is to define the set of "feasible" images, any of which is consistent with the data. This involves comparing the actualdata Dkwith the simulated data |Fk|2which wouldbe observed (apart from noise) from a trial image f. The simplest comparison measure is chisquared (3) Any trial image f for which χ2>M+3⋅3M (M = number of data) is rejected with 99% confidence: the surviving images are feasible and only these need be considered further. In N-dimensional image-space, the feasible set forms a 2M-dimensional toroid, projected linearly to infinity in any unmeasured Fourier planes. Much of the difficulty encountered with phaseless data stems from the connected topology of this constraint.