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There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition and decision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain features of the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of such features, this paper presents an image formation method that formulates the SAR imaging problem as a sparse signal representation problem. For problems of complex-valued nature, such as SAR, a key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. Since features of the SAR reflectivity magnitude are usually of interest, the approach is designed to sparsely represent the magnitude of the complex-valued scattered field. This turns the image reconstruction problem into a joint optimisation problem over the representation of magnitude and phase of the underlying field reflectivities. The authors develop the mathematical framework for this method and propose an iterative solution for the corresponding joint optimisation problem. The experimental results demonstrate the superiority of this method over previous approaches in terms of both producing high-quality SAR images and exhibiting robustness to uncertain or limited data.