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In this letter, a novel constant false alarm rate (CFAR) detection algorithm for high-resolution synthetic aperture radar (SAR) images is proposed with the generalized gamma distribution (GΓD) modeling the background. At first, the method of log-cumulants is introduced for estimating the parameters of the GΓD. In addition, a closed-form expression for the detection threshold of the CFAR algorithm is derived, which refers to the inverse incomplete gamma function. Finally, comparing with the algorithms using the Weibull, KA, and GA0 distributions for background, the advantages of the proposed algorithm, including maintaining the false alarm rate and the efficiency, are validated with an actual high-resolution SAR image.