Scanning microscopy systems, such as confocal and multiphoton microscopy, are powerful imaging tools for probing deep into biological tissue. However, scanning systems have an inherent trade-off between acquisition time, field of view, phototoxicity, and image quality, often resulting in noisy measurements when fast, large field of view, and/or gentle imaging is needed. Deep learning could be used to denoise noisy microscopy measurements, but these algorithms can be prone to hallucination, which can be disastrous for medical and scientific applications. We propose a method to simultaneously denoise and predict pixel-wise uncertainty for scanning microscopy systems, improving algorithm trustworthiness and providing statistical guarantees for deep learning predictions. Furthermore, we propose to leverage this learned, pixel-wise uncertainty to drive an adaptive acquisition technique that rescans only the most uncertain regions of a sample, saving time and reducing the total light dose to the sample. We demonstrate our method on experimental confocal and multiphoton microscopy systems, showing that our uncertainty maps can pinpoint hallucinations in the deep learned predictions. Finally, with our adaptive acquisition technique, we demonstrate up to 16X reduction in acquisition time and total light dose while successfully recovering fine features in the sample and reducing hallucinations. We are the first to demonstrate distribution-free uncertainty quantification for a denoising task with real experimental data and the first to propose adaptive acquisition based on reconstruction uncertainty.

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