Noise in optical synthesis images. I. Ideal Michelson interferometer

We study the distribution of noise in optical images produced by the aperture synthesis technique, in which the principal source of noise is the intrinsic shot noise of photoelectric detection. The results of our analysis are directly applicable to any space-based optical interferometer. We show that the signal-to-noise ratio of images synthesized by such an ideal interferometric array is essentially independent of the details of the beam-combination geometry, the degree of array redundancy, and whether zero-spatial-frequency components are included in image synthesis. However, the distribution of noise does depend on the beam-combination geometry. A highly desirable distribution, one of uniform noise across the entire image, is obtained only when the beams from the n primary apertures are subdivided and combined pairwise on n(n - 1)/2 detectors

Cramer Rao-Type Bounds for Sparse Bayesian Learning

In this paper, we derive Hybrid, Bayesian and Marginalized Cram\'{e}r-Rao lower bounds (HCRB, BCRB and MCRB) for the single and multiple measurement vector Sparse Bayesian Learning (SBL) problem of estimating compressible vectors and their prior distribution parameters. We assume the unknown vector to be drawn from a compressible Student-t prior distribution. We derive CRBs that encompass the deterministic or random nature of the unknown parameters of the prior distribution and the regression noise variance. We extend the MCRB to the case where the compressible vector is distributed according to a general compressible prior distribution, of which the generalized Pareto distribution is a special case. We use the derived bounds to uncover the relationship between the compressibility and Mean Square Error (MSE) in the estimates. Further, we illustrate the tightness and utility of the bounds through simulations, by comparing them with the MSE performance of two popular SBL-based estimators. It is found that the MCRB is generally the tightest among the bounds derived and that the MSE performance of the Expectation-Maximization (EM) algorithm coincides with the MCRB for the compressible vector. Through simulations, we demonstrate the dependence of the MSE performance of SBL based estimators on the compressibility of the vector for several values of the number of observations and at different signal powers.Comment: Accepted for publication in the IEEE Transactions on Signal Processing, 11 pages, 10 figure