Solving Inverse Problems with Score-Based Generative Priors learned from Noisy Data

Abstract

We present SURE-Score: an approach for learning score-based generative models using training samples corrupted by additive Gaussian noise. When a large training set of clean samples is available, solving inverse problems via score-based (diffusion) generative models trained on the underlying fully-sampled data distribution has recently been shown to outperform end-to-end supervised deep learning. In practice, such a large collection of training data may be prohibitively expensive to acquire in the first place. In this work, we present an approach for approximately learning a score-based generative model of the clean distribution, from noisy training data. We formulate and justify a novel loss function that leverages Stein's unbiased risk estimate to jointly denoise the data and learn the score function via denoising score matching, while using only the noisy samples. We demonstrate the generality of SURE-Score by learning priors and applying posterior sampling to ill-posed inverse problems in two practical applications from different domains: compressive wireless multiple-input multiple-output channel estimation and accelerated 2D multi-coil magnetic resonance imaging reconstruction, where we demonstrate competitive reconstruction performance when learning at signal-to-noise ratio values of 0 and 10 dB, respectively.

References in corpus (13)

• Denoising Diffusion Probabilistic Models
• Generative Modeling by Estimating Gradients of the Data Distribution
• fastMRI: An Open Dataset and Benchmarks for Accelerated MRI
• Generalized SURE for Exponential Families: Applications to Regularization
• Compressed Sensing Based Multi-User Millimeter Wave Systems: How Many Measurements Are Needed?
• Denoising Diffusion Restoration Models
• Solving Inverse Problems in Medical Imaging with Score-Based Generative Models
• Bayesian MRI Reconstruction with Joint Uncertainty Estimation using Diffusion Models
• Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images
• Self-Score: Self-Supervised Learning on Score-Based Models for MRI Reconstruction
• DDM${}^2$: Self-Supervised Diffusion MRI Denoising with Generative Diffusion Models
• Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistent
• Statistical Efficiency of Score Matching: The View from Isoperimetry

Cited by in corpus (2)

• Robust multi-coil MRI reconstruction via self-supervised denoising
• Self-supervised feature learning for cardiac Cine MR image reconstruction