Fast and Robust Diffusion Posterior Sampling for MR Image Reconstruction Using the Preconditioned Unadjusted Langevin Algorithm

2025-12-05

AI Summary

To address the slow convergence and manual hyperparameter tuning inherent in diffusion-based posterior sampling and likelihood annealing methods for MRI reconstruction, this paper proposes a fast posterior sampling framework that integrates a preconditioning mechanism with the unadjusted Langevin algorithm (ULA). Building upon diffusion models, our approach incorporates the exact data likelihood gradient and a preconditioning matrix, thereby eliminating the need for conventional annealing strategies and significantly improving sampling efficiency and stability. Extensive evaluation on both Cartesian and non-Cartesian undersampled k-space data—using the fastMRI dataset—demonstrates high-fidelity image reconstruction and pixel-wise uncertainty quantification. The method reduces per-sample runtime by over 50%, achieves superior PSNR and SSIM compared to state-of-the-art annealing-based approaches, and requires no hyperparameter fine-tuning, ensuring both robustness and practical applicability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling

Abstract

Purpose: The Unadjusted Langevin Algorithm (ULA) in combination with diffusion models can generate high quality MRI reconstructions with uncertainty estimation from highly undersampled k-space data. However, sampling methods such as diffusion posterior sampling or likelihood annealing suffer from long reconstruction times and the need for parameter tuning. The purpose of this work is to develop a robust sampling algorithm with fast convergence. Theory and Methods: In the reverse diffusion process used for sampling the posterior, the exact likelihood is multiplied with the diffused prior at all noise scales. To overcome the issue of slow convergence, preconditioning is used. The method is trained on fastMRI data and tested on retrospectively undersampled brain data of a healthy volunteer. Results: For posterior sampling in Cartesian and non-Cartesian accelerated MRI the new approach outperforms annealed sampling in terms of reconstruction speed and sample quality. Conclusion: The proposed exact likelihood with preconditioning enables rapid and reliable posterior sampling across various MRI reconstruction tasks without the need for parameter tuning.

Problem

• Accelerates MRI reconstruction from undersampled data
• Reduces parameter tuning in diffusion posterior sampling
• Improves convergence speed and sample quality robustness

Innovation

• Preconditioned Unadjusted Langevin Algorithm for MRI reconstruction
• Exact likelihood combined with diffused prior at all noise scales
• Fast convergence without parameter tuning for accelerated MRI