Cramér-Rao bounds for variance of Fourier magnitude measurements

Abstract

Many imaging modalities measure magnitudes of Fourier components of an object. Given such data, reconstruction of an image from data that is also noisy and sparse is especially challenging, as may occur in some forms of intensity interferometry, Fourier telescopy, and speckle imaging. In such measurements, the Fourier magnitudes must be positive, and moreover must be less than 1 given the usual normalization, scaling the magnitudes so that the magnitude is one at zero spatial frequency in the u-v plane data. The Cramér-Rao formalism is applied to single Fourier magnitude measurements to ascertain whether a reduction in variance is possible given these constraints. An extension of the Cramér-Rao formalism is used to address the value of relatively general prior information. The impact of this knowledge is also shown for simulated image formation for a simple disk, with varying measurement SNR and sampling in the (u,v) plane.