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This paper proposes an adaptive method based on the mirror-extended curvelet transform and the improved particle swarm optimization (PSO) algorithm, which reduce speckle noise and enhance edge features and contrast of synthetic aperture radar (SAR) images. First, an improved gain function, which integrates the speckle reduction with the feature enhancement, is introduced to nonlinearly shrink and stretch the curvelet coefficients. Then, a novel objective criterion for the quality of the despeckled and enhanced images is proposed in order to adaptively obtain the optimal parameters in the gain function. Finally, the PSO algorithm is employed as a global search strategy for the best despeckled and enhanced image. In order to increase the convergence speed and avoid the premature convergence, two further improvements for the classic PSO algorithm are presented. That is, a new learning scheme and a mutation operator are introduced. Experimental results demonstrate that the proposed method can efficiently reduce the speckle and enhance the edge features and the contrast of SAR images and outperforms the wavelet- and curvelet-based nonadaptive despeckling and enhancement methods.