Because of the effect of speckles in synthetic aperture radar (SAR) images, its reduction has been considered by many researchers to obtain reliable information. This paper proposes a method based on the curvelet transform to reduce speckles in SAR images. This study is based on the thresholding neural network (TNN) technique, which has been previously used in wavelet transformation. In addition, an adaptive learning TNN with remarkable time saving was introduced. Comparing the obtained results from the method with conventional speckle filters such as Lee, Kuan, Frost, and Gamma filters, curvelet-based, nonadaptive despeckling, wavelet-based TNN despeckling, and curvelet-based particle swarm optimization show better achievement of the proposed algorithm. For instance, noise mean value, noise standard deviation, mean square difference, equivalent number of looks, and β (an edge-preserving criterion) improved 2%, 9%, 21%, 35%, and 9%, respectively.