Linear array synthetic aperture radar (LASAR) is a promising radar 3-D imaging technique. In this paper, we address the problem of sparse recovery of LASAR image from under-sampled and phase errors interrupted echo data. It is shown that the unknown LASAR image and the nuisance phase errors can be constructed as a bilinear measurement model, and then the under-sampled LASAR imaging with phase errors can be mathematically transferred into sparse signal recovery by solving an ill-conditioned constant modulus linear program (ICCMLP) problem. Exploiting the prior sparse spatial feature of the observed targets, a new super-resolution sparse autofocus recovery algorithm is proposed for under-sampled LASAR 3-D imaging. The algorithm is an iterative minimize estimation procedure, wherein it converts the ICCMLP into two independent convex optimal problems, and joints l1-norm reweights least square regularization and semi-definite relax to find the optimal solutions. Simulated and experimental results confirm that the proposed method outperforms the classical autofocus techniques in under-sampled LASAR imaging.
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