Speckle noise inherent in synthetic aperture radar (SAR) images seriously affects the result of various SAR image processing tasks such as edge detection and segmentation. Thus, speckle reduction is critical and is used as a preprocessing step for smoothing homogeneous regions while preserving features such as edges and point scatterers. Although state-of-the-art methods provide better despeckling compared with conventional methods, their resource consumption is higher. In this letter, a sparsitydriven total-variation (TV) approach employing l0-norm, fractional norm, or l(1)-norm to smooth homogeneous regions with minimal degradation in edges and point scatterers is proposed. Proposed method, sparsity-driven despeckling (SDD), is capable of using different norms controlled by a single parameter and provides better or similar despeckling compared with the state-of-the-art methods with shorter execution times. Despeckling performance and execution time of the SDD are shown using synthetic and real-world SAR images.