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Reducing speckle fluctuations in multichannel SAR images is essential in many applications of synthetic aperture radar (SAR) imaging such as polarimetric classification or interferometric height estimation. While single-channel despeckling has widely benefited from the application of deep learning techniques, extensions to multichannel SAR images are much more challenging. This article introduces MuChaPro, a generic framework that exploits existing single-channel despeckling methods. The key idea is to generate numerous single-channel projections, restore these projections, and recombine them into the final multichannel estimate. This simple approach is shown to be effective in polarimetric and/or interferometric modalities. A special appeal of MuChaPro is the possibility to apply a self-supervised training strategy to learn sensor-specific networks for single-channel despeckling.