Relative Entropy and the Dempster-Laird-Rubin EM Algorithm: Application to the Autofocus Problem in ISAR Imaging

Abstract

The popular Dempster-Laird-Rubin expectation-maximization (EM) algorithm obtains maximum likelihood estimates by postulating a refinement of the data (the complete data) which contains information about the unknown parameters in excess of that contained in the actual (incomplete) data. Each iteration of the algorithm involves (1) an expectation step in which the complete data log—likelihood function is averaged over the complete data distribution conditioned on the incomplete data and a prior estimate of the unknowns, and (2) a maximization step in which the new estimate of the unknown parameter vector is taken as the value which maximizes the expectation in (1). It is shown that an iteration may also be described as the minimization of the Kullback-Leibler “distance” between the complete data distribution conditioned on the incomplete data and the prior parameter vector estimate and the complete data distribution conditioned on the current unknown parameter vector estimate, ie., the entropy of the former distribution relative to the latter one. The insights which this alternative interpretation provides are applied in adapting the EM algorithm to the problem of estimating the focusing parameters in automatic focusing of inverse synthetic aperture radar (ISAR) images.