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 (DPS) 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 and DPS 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.

翻译：目的：非调整朗之万算法（Unadjusted Langevin Algorithm, ULA）结合扩散模型，可从高度欠采样的k空间数据生成具有不确定性估计的高质量MRI重建结果。然而，扩散后验采样（DPS）或似然退火等采样方法存在重建时间长及参数调优需求的问题。本研究旨在开发一种具有快速收敛特性的鲁棒采样算法。理论与方法：在后验采样的逆扩散过程中，精确似然函数在所有噪声尺度下与扩散先验相乘。为克服收敛缓慢的问题，引入了预条件技术。该方法在fastMRI数据上进行训练，并在健康志愿者的回顾性欠采样脑数据上进行测试。结果：在笛卡尔与非笛卡尔加速MRI的后验采样中，新方法在重建速度与样本质量上均优于退火采样与DPS。结论：所提出的带预条件的精确似然函数可在无需参数调优的情况下，实现多种MRI重建任务的快速可靠后验采样。