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.

翻译：近年来，去噪问题与深度生成模型的发展日益交织。特别是，扩散模型的训练方式类似于去噪器，其所建模的分布在贝叶斯框架中与去噪先验相吻合。然而，基于扩散的后验采样进行去噪需要已知噪声水平和协方差，这阻碍了盲去噪的实现。我们通过引入吉布斯扩散（GDiff）克服了这一限制，这是一种解决信号和噪声参数联合后验采样的通用方法。在假设任意参数化高斯噪声的前提下，我们开发了一种吉布斯算法，该算法交替执行两个采样步骤：从一个条件扩散模型中采样，该模型训练用于将信号先验映射到噪声分布族；以及一个蒙特卡洛采样器，用于推断噪声参数。我们的理论分析揭示了潜在的缺陷，指导了诊断性使用，并量化了由扩散模型引起的吉布斯平稳分布误差。我们展示了该方法在以下两个方面的应用：1）涉及未知振幅和谱指数的彩色噪声的自然图像盲去噪；2）一个宇宙学问题，即宇宙微波背景数据分析，其中“噪声”参数的贝叶斯推断意味着对宇宙演化模型的约束。