Listening to the Noise: Blind Denoising with Gibbs Diffusion (2402.19455v2)

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.

• The paper presents Gibbs Diffusion, integrating diffusion models with Gibbs sampling to address blind denoising challenges.
• It efficiently estimates both signal and noise parameters through Monte Carlo sampling and diagonalizes covariance in Fourier space.
• Empirical results on natural images and cosmic microwave background data reveal superior performance compared to established methods.

Overview of "Listening to the Noise: Blind Denoising with Gibbs Diffusion"

The paper "Listening to the Noise: Blind Denoising with Gibbs Diffusion" presents an innovative approach named Gibbs Diffusion (GDiff) to tackle the challenge of blind denoising, which involves inferring both the original signal and the characteristics of noise from observed data. This research intersects fields such as machine learning, physics, and image processing, offering a new perspective on applying deep generative models for denoising tasks.

Key Contributions

The primary contribution of the work is the introduction of GDiff, a combination of diffusion models and a Gibbs sampler. This method enhances the blind denoising process by concurrently sampling both signal and noise parameters. The methodology is constructed on the foundation of diffusion models which have been shown to effectively approximate complex data distributions and are already prominent in state-of-the-art generative models. Notably, GDiff manages the intricacies of noise covariance by incorporating a Monte Carlo sampling approach, allowing inference on noise parameters under a Bayesian framework. The flexibility of GDiff is demonstrated in two significant applications: denoising of natural images with colored noise and a cosmological analysis of cosmic microwave background (CMB) data.

Methodology

The methodology employs a diffusion model trained to reverse a stochastic process that progressively adds noise to data, effectively serving as a mechanism to sample from the posterior distribution over the denoised signal. This requires modeling noise variance and covariance, and the paper details a conditional model that accounts for unknown noise parameters. The practical implementation involves a Gibbs sampler alternately drawing from conditional distributions over signals and noise parameters. To handle high-dimensional noise covariance within a manageable parameter space, the researchers emphasize the diagonalization of the covariance matrix in Fourier space, which reduces computational complexity considerably.

Theoretical Insights

The authors provide a comprehensive theoretical analysis of their approach, addressing potential pitfalls and ensuring the robustness of the Gibbs Diffusion model. This includes conditions for the stationary distribution of the Gibbs sampler, as well as diagnostics for tuning the inference and quantifying the propagation of errors. Importantly, they highlight the existence of a reverse stochastic process which aids in the denoising and posterior sampling procedure, supported by the theoretical backing from prior work on stochastic differential equations.

Numerical Results and Applications

The authors demonstrate the effectiveness of their method with empirical results. For natural images contaminated with noises of unknown amplitude, GDiff, despite operating blindly, achieves superior denoising performance compared to established methods like BM3D and DnCNN in certain metrics such as peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM). Additionally, the cosmological application showcases the method's utility in decomposing CMB data into foreground and background components, a complex task due to intricate interferences in cosmological observations. These results underscore the potential of GDiff to enhance scientific inferences in cosmology by accurately modeling the signal and noise interplay.

Implications and Speculations

Practically, GDiff expands the toolkit available for addressing noisy signal reconstruction problems, particularly in domains where noise characteristics are inherently unknown or variable. Theoretically, it situates itself as a bridge integrating diffusion processes and Bayesian inference, which may inspire further explorations into more versatile generative models capable of posterior sampling beyond Gaussian assumptions. The versatility demonstrated by addressing both image processing and cosmological data highlights its applicability across a breadth of scientific disciplines.

From a speculative viewpoint, future works may build upon this foundation by integrating more sophisticated adaptation schemes for Gibbs sampling or exploring extensions that relax Gaussian noise assumptions. Additionally, refining the computational efficiency of diffusion models, particularly for real-time applications, remains an open challenge.

In summary, "Listening to the Noise: Blind Denoising with Gibbs Diffusion" introduces a robust, flexible framework balancing state-of-the-art techniques in generative modeling with the demands of nuanced scientific data analysis, paving the way for more dynamic and adaptive noise processing strategies.