Speckle noise which is inherent to Synthetic Aperture Radar (SAR) imaging obstructs various image exploitation tasks such as edge detection, segmentation, change detection, and target recognition. Therefore, speckle reduction is generally used as a first step which has to smooth out homogeneous regions while preserving edges and point scatterers. Traditional speckle reduction methods are fast and their memory consumption is insignificant. However, they are either good at smoothing homogeneous regions or preserving edges and point scatterers. State of the art despeckling methods are proposed to overcome this trade-off. However, they introduce another trade-off between denoising quality and resource consumption, thereby higher denoising quality requires higher computational load and/or memory consumption. In this paper, a local pixel-based total variation (TV) approach is proposed, which combines l2-norm and l1-norm in order to improve despeckling quality while keeping execution times reasonably short. Pixel-based approach allows efficient computation model with relatively low memory consumption. Their parallel implementations are also more efficient comparing to global TV approaches which generally require numerical solution of sparse linear systems. However, pixel-based approaches are trapped to local minima frequently hence despeckling quality is worse comparing to global TV approaches. Proposed method, namely mixed norm despeckling (MND), combines l2-norm and l1-norm in order to improve despeckling performance by alleviating local minima problem. All steps of the MND are parallelized using OpenMP on CPU and CUDA on GPU. Speckle reduction performance, execution time and memory consumption of the proposed method are shown using synthetic images and TerraSAR-X spot mode SAR images.