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Diffusion models (DMs) are a powerful frame- work for image generation and restoration. How- ever, existing DMs are primarily trained in a su- pervised manner by using a large corpus of clean images. This poses fundamental challenges in many real-world scenarios, where acquiring noise- free data is hard or infeasible. While some meth- ods are capable of training DMs using noisy data, they are effective only when the amount of noise is very mild or when additional noise-free data is available. In addition, existing methods for training DMs from incomplete measurements re- quire access to multiple complementary acquisi- tion processes, a significant practical limitation. Here we introduce the first approach for learning DMs for image restoration using only noisy mea- surement data from a single operator. First, we show that DMs, and more broadly minimum mean squared error denoisers, exhibit a weak form of scale equivariance linking rescaling in signal am- plitude to changes in noise intensity. We then leverage this theoretical insight to develop a de- noising score-matching strategy that generalizes robustly to noise levels below the training data, thereby enabling the learning of DMs from noisy measurements. For problems involving measure- ments both noisy and incomplete, we integrate our method with equivariant imaging, a complemen- tary self-supervised learning framework that ex- ploits the inherent invariants of imaging problems. This allows training DMs for image restoration from single-operator noisy measurements. We validate the effectiveness of our approach through extensive experiments on image denoising, demo- saicing, inpainting, and MRI reconstruction along with comparisons with the state of the art. 

Edge-resolved transient imaging (ERTI) is a method for non-line-of-sight imaging that combines the use of direct time of flight for measuring distances with the azimuthal angular resolution afforded by a vertical edge occluder. Recently conceived and demonstrated for the first time, no performance analyses or optimizations of ERTI have appeared in published papers. This paper explains how the difficulty of detection of hidden scene objects with ERTI depends on a variety of parameters, including illumination power, acquisition time, ambient light, visible-side reflectivity, hidden-side reflectivity, target range, and target azimuthal angular position. Based on this analysis, optimization of the acquisition process is introduced whereby the illumination dwell times are varied to counteract decreasing signal-to-noise ratio at deeper angles into the hidden volume. Inaccuracy caused by a coaxial approximation is also analyzed and simulated.