In recent years, denoising problems have become intertwined with the development of deep generative models. In particular, diffusion models are trained like denoisers, and the distribution they model coincide with denoising priors in the Bayesian picture. However, denoising through diffusion-based posterior sampling requires the noise level and covariance to be known, preventing blind denoising. We overcome this limitation by introducing Gibbs Diffusion (GDiff), a general methodology addressing posterior sampling of both the signal and the noise parameters. Assuming arbitrary parametric Gaussian noise, we develop a Gibbs algorithm that alternates sampling steps from a conditional diffusion model trained to map the signal prior to the family of noise distributions, and a Monte Carlo sampler to infer the noise parameters. Our theoretical analysis highlights potential pitfalls, guides diagnostic usage, and quantifies errors in the Gibbs stationary distribution caused by the diffusion model. We showcase our method for 1) blind denoising of natural images involving colored noises with unknown amplitude and spectral index, and 2) a cosmology problem, namely the analysis of cosmic microwave background data, where Bayesian inference of "noise" parameters means constraining models of the evolution of the Universe.

Fibre-optic Distributed Acoustic Sensing (DAS) is an emerging technology for vibration measurements with numerous applications in seismic signal analysis, including microseismicity detection, ambient noise tomography, earthquake source characterisation, and active source seismology. Using laser-pulse techniques, DAS turns (commercial) fibre-optic cables into seismic arrays with a spatial sampling density of the order of metres and a time sampling rate up to one thousand Hertz. The versatility of DAS enables dense instrumentation of traditionally inaccessible domains, such as urban, glaciated, and submarine environments. This in turn opens up novel applications such as traffic density monitoring and maritime vessel tracking. However, these new environments also introduce new challenges in handling various types of recorded noise, impeding the application of traditional data analysis workflows.

We propose a general framework for denoising high-dimensional measurements which requires no prior on the signal, no estimate of the noise, and no clean training data. The only assumption is that the noise exhibits statistical independence across different dimensions of the measurement. Moreover, our framework is not restricted to a particular denoising model. We show how it can be used to calibrate any parameterised denoising algorithm, from the single hyperparameter of a median filter to the millions of weights of a deep neural network. We demonstrate this on natural image and microscopy data, where we exploit noise independence between pixels, and on single-cell gene expression data, where we exploit independence between detections of individual molecules. Finally, we prove a theoretical lower bound on the performance of an optimal denoiser. This framework generalizes recent work on training neural nets from noisy images and on cross-validation for matrix factorization.