  In this paper, SAR image reconstruction with joint phase error estimation
(autofocusing) is formulated as an inverse problem. An optimization model
utilising a sparsity-enforcing Cauchy regularizer is proposed, and an
alternating minimization framework is used to solve it, in which the desired
image and the phase errors are optimized alternatively. For the image
reconstruction sub-problem (f-sub-problem), two methods are presented capable
of handling the problem's complex nature, and we thus present two variants of
our SAR image autofocusing algorithm. Firstly, we design a complex version of
the forward-backward splitting algorithm (CFBA) to solve the f-sub-problem
iteratively. For the second variant, the Wirtinger alternating minimization
autofocusing (WAMA) method is presented, in which techniques of Wirtinger
calculus are utilized to minimize the complex-valued cost function in the
f-sub-problem in a direct fashion. For both methods, the phase error estimation
sub-problem is solved by simply expanding and observing its cost function.
Moreover, the convergence of both algorithms is discussed in detail. By
conducting experiments on both simulated scenes and real SAR images, the
proposed method is demonstrated to give impressive autofocusing results
compared to other state of the art methods.
