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

TL;DR

This paper introduces a preconditioned Unadjusted Langevin Algorithm for MRI reconstruction that achieves faster, more reliable posterior sampling without parameter tuning, outperforming existing methods in speed and quality.

Contribution

It develops a robust, preconditioned sampling algorithm that significantly accelerates MRI posterior sampling while maintaining high reconstruction quality and eliminating parameter tuning.

Findings

Outperforms annealed sampling and DPS in speed and sample quality

Enables rapid and reliable MRI posterior sampling across different data types

Does not require parameter tuning for different MRI reconstruction tasks

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 (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.