We develop a maximum-likelihood (ML) algorithm for estimation and correction (autofocus) of phase errors induced in synthetic-aperture-radar (SAR) imagery. Here, M pulse vectors in the range-compressed domain are used as input for simultaneously estimating M − 1 phase values across the aperture. The solution involves an eigenvector of the sample covariance matrix of the range-compressed data. The estimator is then used within the basic structure of the phase gradient autofocus (PGA) algorithm, replacing the original phase-estimation kernel. We show that, in practice, the new algorithm provides excellent restorations to defocused SAR imagery, typically in only one or two iterations. The performance of the new phase estimator is demonstrated essentially to achieve the Cramér–Rao lower bound on estimation-error variance for all but small values of target-toclutter ratio. We also show that for the case in which M is equal to 2, the ML estimator is similar to that of the original PGA method but achieves better results in practice, owing to a bias inherent in the original PGA phase estimation kernel. Finally, we discuss the relationship of these algorithms to the shear-averaging and spatial correlation methods, two other phase-correction techniques that utilize the same phase-estimation kernel but that produce substantially poorer performance because they do not employ several fundamental signal-processing steps that are critical to the algorithms of the PGA class.
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