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This letter presents the despeckling of single-look complex (SLC) synthetic aperture radar (SAR) images using nonquadratic regularization. The objective function consists of an image model, a gradient, and a prior model. The Huber-Markov random field (HMRF) models the prior. A numerical solution is achieved through extensions of half-quadratic regularization methods using complex-valued SAR data. The proposed method using the HMRF prior together with nonquadratic regularization shows the superior results on SLC synthetic and actual SAR images.