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Synthetic aperture radar (SAR) images have two main problems: Degradation by speckle that causes low-contrast images and misinterpretation, and the large amount of data that makes the computation time an important issue. In addition to denoising, it is necessary to recover the missing data due to errors in the onboard SAR equipment, downlink equipment, and ground data registration. In this paper, we propose a method using high-order total variation (High-TV) denoising in compressive sensing (CS) framework, based on maximum a posteriori (MAP) estimation to simultaneously denoise and recover the large-size, complex-valued SAR images. Experiments on RADARSAT-1 raw data showed that the proposed method, called MAP High-TV, can suppress the speckle effectively without exhibiting a staircase effect on the images, and perfectly recovers the missed data. The performance comparison among the proposed method, MAP-MIDAL (the pioneer CS-based method), and three state of the art filters (NSM, POTDF and FANS), explores the capability of MAP High-TV for both denoising and recovering the complex-valued SAR images. [ABSTRACT FROM PUBLISHER]