Optimum coherent imaging of a limited field of view in the presence of angular and aperture noise

Abstract

An optimal procedure is established for the reconstruction of the angular object distribution in a given field of view (FOV). The object is coherently illuminated and located in the far zone of the receiving aperture. The procedure is “uniformly” optimal in the sense of minimizing the statistical r.m.s. difference between the object distribution, modeled as a random function of the angular coordinates and its reconstructed image, for each direction belonging to the FOV. The observable complex amplitude distribution of the field on the aperture is due in the general case not only to the incidentfield scattered by the object but also to background disturbance, or “angular noise”, randomly distributed inside and outside the FOV, and is affected by “measurement noise”, that is random errors introduced in measuring the aperture field. The reconstruction algorithm consists of summing a truncated series of special functions—prolate spheroidal for the linear case and their generalizations for two dimensional apertures—weighted by appropriate coefficients. These coefficients depend upon the observed aperture field and upon the relative power densities associated with the object field and the various types of noise. The series is truncated to a number of terms (“effective degrees of freedom” of the image) determined through an information theoretical method: each term of the series, suitably ordered, provides an information gain less than the preceding one, and the information gain goes rapidly to zero. A relationship between information transfer and mean squared error for each term in the image series is established. Numerical examples are discussed.