DeepMURE: Deep Self-Supervised Despeckling via Monte Carlo Multiplicative Unbiased Risk Estimation

Synthetic Aperture Radars (SAR) are key to all-weather, day-and-night Earth observation. Yet, the direct use of SAR images is hindered by multiplicative speckle. While deep learning has significantly advanced SAR despeckling, the reliance on pristine, speckle-free ground truth for supervised training remains a critical bottleneck in real-world applications. To overcome this fundamental limitation, we propose DeepMURE, a novel and general-purpose self-supervised training framework that effectively bypasses the need for clean reference data. DeepMURE mathematically reconstructs the supervised Mean Squared Error (MSE) directly from noisy observations by leveraging multi-perturbation, higher-order Taylor approximations, explicitly incorporating Jacobian and Hessian corrections to guarantee stable and accurate loss convergence. To fully exploit the potential of this framework in SAR applications, we also introduce SpeckleFormer, a specialized joint spatial-frequency-domain transformer architecture. SpeckleFormer features a Frequency domain Token Block (FTB) to effectively handle Gamma-distributed speckle in homogeneous and heterogeneous regions, alongside a Target Preservation Module (TPM) that explicitly preserves the radiometric response of strong scatterers. Comprehensive evaluations on hybrid Sentinel-1 and high-resolution real ICEYE datasets demonstrate that DeepMURE enables networks to closely approach their fully supervised counterparts. Furthermore, the combined DeepMURE and SpeckleFormer pipeline achieves state-of-the-art performance, delivering superior despeckling with detail preservation as validated by both full-reference and blind quality metrics. Finally, validation on downstream remote sensing task as well ablations confirm that the proposed restoration pipeline effectively enhances the accuracy and reliability of subsequent semantic applications, highlighting its utility for operational Earth observation workflows.