UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matching

Reconstructing high-quality images from substantially undersampled k-space data for accelerated MRI presents a challenging ill-posed inverse problem. While supervised deep learning has revolutionized this field, it relies heavily on large datasets of fully sampled ground-truth images, which are often impractical or impossible to acquire in clinical settings due to long scan times. Despite advances in self-supervised/unsupervised MRI reconstruction, their performance remains inadequate at high acceleration rates. To bridge this gap, we introduce UPMRI, an unsupervised reconstruction framework based on Projected Conditional Flow Matching (PCFM) and its unsupervised transformation. Unlike standard generative models, PCFM learns the prior distribution of fully sampled parallel MRI data by utilizing only undersampled k-space measurements. To reconstruct the image, we establish a novel theoretical link between the marginal vector field in the measurement space, governed by the continuity equation, and the optimal solution to the PCFM objective. This connection results in a cyclic dual-space sampling algorithm for high-quality reconstruction. Extensive evaluations on the fastMRI brain and CMRxRecon cardiac datasets demonstrate that UPMRI significantly outperforms state-of-the-art self-supervised and unsupervised baselines. Notably, it also achieves reconstruction fidelity comparable to or better than leading supervised methods at high acceleration factors, while requiring no fully sampled training data.