The goal of the present research is to despeckle SAR images, which is critical for segmentation and target recognition in satellite SAR images. When a despeckling algorithm is applied to a SAR image, important information such as the edges, corners, textures, and object parts will degrade. Curvelet transform is a recently proposed form of multi-scale analysis that achieves better performance of wavelet and Gabor transforms in edge and curve detection. This is a geometric transform that is useful for SAR image processing. For unsupervised texture images, segmentation is different and distinct from the textures, so the textures at the boundary noises will disappear. Curvelet transform has produced good results in the detection of curved edges with higher accuracy in finding the orientation than wavelet transforms. The present study uses fast discrete curvelet transform (FDCT) based on wresting and uses unsupervised adaptive threshold learning to develop a new despeckling algorithm for SAR images. In the proposed algorithm, each segment of the SAR image can be learned for selection of its adaptive threshold. Simulation results demonstrate that the proposed algorithm performs better than similar methods.