Abstract

We investigate the reconstruction of a real and positive spatial pattern or "image" (1) from incomplete phaseless Fourier data D_kwith noise σ_k, (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 D_kwith the simulated data |F_k|²which wouldbe observed (apart from noise) from a trial image f. The simplest comparison measure is chisquared (3) Any trial image f for which ${\chi }^2>\text{M}+3\cdot 3\sqrt{\text{M}}$ (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.

© 1983 Optical Society of America