Learning Clean Generative Processes from Noisy Measurements

We study how to learn a generative model of clean images using only noisy measurements with a known noise law. Diffusion processes and their generalizations beyond Gaussian noise require posterior predictions at every noise level. We first show that it is possible to learn these predictions from independent noisy views using Noise2Noise regression, but only at or above the noise level of the training data. To learn the predictions below that level, needed to generate clean images, we impose self-consistency: a prediction should equal its expected value after continuing the model’s sampling process. We derive conditions under which self-consistency uniquely determines the continuation to the clean image distribution and verify them for Poisson, binomial, Gamma, and negative-binomial measurements. Experiments on CIFAR-10, microscopy data, and synthetic aperture radar show that the required posterior predictions can be learned from noisy views alone and that self-consistency repairs the continuation beyond the noise level of the data, which otherwise collapses.