Synthetic Aperture Radar (SAR) images are widely used in remote sensing due to their all-weather, all-day imaging capabilities. However, SAR images are highly susceptible to noise, particularly speckle noise, caused by the coherent imaging process, which severely degrades image quality. This has driven increasing research interest in SAR despeckling. Sparse representation-based denoising has been extensively applied in natural image processing, yet SAR despeckling requires addressing non-Gaussian noise and ensuring sparsity in the transform domain. In this work, we propose an innovative SAR despeckling approach grounded in compressive sensing theory. By applying the Log-Yeo-Johnson transformation, we convert gamma-distributed noise into an approximate Gaussian distribution, facilitating sparse representation. The method incorporates noise and sparsity priors, leveraging a non-local sparse representation through auxiliary matrices: one capturing varying noise characteristics across regions and the other encoding adaptive sparsity information.