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Listening to the Noise: Blind Denoising with Gibbs Diffusion

David Heurtel-Depeiges, Charles C. Margossian, Ruben Ohana, Bruno Régaldo-Saint Blancard

TL;DR

GDiff is introduced, a general methodology addressing posterior sampling of both the signal and the noise parameters and showcasing the method for blind denoising of natural images involving colored noises with unknown amplitude and spectral index and a cosmology problem, where Bayesian inference of"noise"parameters means constraining models of the evolution of the Universe.

Abstract

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.

Listening to the Noise: Blind Denoising with Gibbs Diffusion

TL;DR

GDiff is introduced, a general methodology addressing posterior sampling of both the signal and the noise parameters and showcasing the method for blind denoising of natural images involving colored noises with unknown amplitude and spectral index and a cosmology problem, where Bayesian inference of"noise"parameters means constraining models of the evolution of the Universe.

Abstract

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.
Paper Structure (42 sections, 5 theorems, 34 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 42 sections, 5 theorems, 34 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related work.
  4. Method
  5. Diffusion Models as Denoising Posterior Samplers
  6. Blind Denoising with Gibbs Diffusion
  7. Gibbs Sampling.
  8. Algorithm and practical details.
  9. Properties of the Gibbs Sampler
  10. Existence of the stationary distribution.
  11. Error in the Gibbs sampler.
  12. Applications
  13. Common technical details.
  14. Denoising for Natural Images
  15. Data and setting.
  16. ...and 27 more sections

Key Result

Proposition 2.1

For a forward SDE $({\boldsymbol{z}})_{t\in [0, 1]}$ defined by Eq. eq:forward_sde, for all $t \in [0, 1]$, ${\boldsymbol{z}}_t$ reads with $a : [0, 1] \rightarrow \mathbb{R}_+$ a decreasing function with $a(0) = 1$, and $b : [0, 1] \rightarrow \mathbb{R}_+$ an increasing function with $b(0) = 0$.See App. app:proof_lemma for expressions of $a$ and $b$ as functions of $f$ and $g$. Moreoever, for $

Figures (12)

  • Figure 1: Graphical model: we observe ${\boldsymbol{y}} = {\boldsymbol{x}} + {\boldsymbol{\varepsilon}}$ and aim to infer both ${\boldsymbol{x}}$ and ${\boldsymbol{\phi}}$ in a Bayesian framework.
  • Figure 2: We train diffusion models to define a stochastic linear interpolation between the signal prior distribution $p({\boldsymbol{x}})$ and the noise distribution $p({\boldsymbol{{\boldsymbol{\varepsilon'}}}}) \sim \mathcal{N}({\boldsymbol{0}}, {\boldsymbol{\Sigma}}_{{\boldsymbol{\varphi}}})$.
  • Figure 3: Example of blind denoising with GDiff on an ImageNet sample for $\sigma = 0.2$ and $\varphi = 0.4$. Left: Noisy example ${\boldsymbol{y}}$ next to the noise-free image ${\boldsymbol{x}}$, a denoised sample $\hat{{\boldsymbol{x}}}$ and an estimate of the posterior mean $\mathbb{E}_{}\left[ {\boldsymbol{x}}\,|\,{\boldsymbol{y}} \right]$ (with PSNR on top when relevant). Right: Inferred posterior distribution over the noise parameters.
  • Figure 4: Posterior accuracy diagnostic of GDiff with simulation-based calibration on natural images blind denoising.
  • Figure 5: Left: Observed maps and maps reconstructed with GDiff. The true dust ${\boldsymbol{x}}$ and CMB ${\boldsymbol{\varepsilon}}$ maps compose the observed mixture ${\boldsymbol{y}}$. We reconstruct the dust $\hat{{\boldsymbol{x}}}$ and CMB $\hat{{\boldsymbol{\varepsilon}}}$ with our diffusion model. The global unit is arbitrary. Right: Inferred cosmological parameters.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 1.1
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof