Product-of-Gaussian-Mixture Diffusion Models for Joint Nonlinear MRI Reconstruction

Magnetic resonance imaging (MRI) reconstruction under realistic acquisition conditions can be fundamentally viewed as estimating the underlying k-space distribution from incomplete and noise-corrupted measurements. While diffusion models have recently shown strong potential as generative prior for inverse problems,existingapproachesstruggletohandlenoisyreconstruction settings, especially when operating directly in k-space domain. In this work, we propose a unified high-dimensional k-space reconstruction framework tailored for noisy inverse problems, whichenhancesdiffusion-based solversthroughrepresentation lifting.Ratherthanmodifyingthe underlying optimization procedures, the proposed framework augments the data representation space, enabling existing diffusion-based solvers to operate on enriched k-space embeddings with improved expressiveness. Extensive experiments on both in-house and public datasets across varying noise levels and undersampled factors demonstrate that the proposed frame work consistently improves reconstruction quality for multiple diffusion-based inverse solvers. Notably, the largest gains are observed in high-noise regimes, which is consistent with our theoretical analysis of error propagation under high-dimensional representation. These results suggest that high-dimensional representation provides a general and model-agnostic mechanism for improving diffusion-based MRI reconstruction in noisy settings, offering a new perspective on robust k-space generative modeling for practical inverse problems. The code will be available at https://github.com/yqx7150/HEP-MRIRec.

Accelerated magnetic resonance imaging (MRI) enabled by the training of deep learning (DL)-based image recon. models requires large and diverse raw k-space datasets. In most clinical MRI applications, due to storage and patient privacy concerns, raw k-space data is discarded and magnitude-only images are the only component saved. Consequently, a large portion of the DL-based MRI recon. literature has either relied on small training datasets or has used one of the few available open-source k-space datasets. At the same time, the growing number of anonymized magnitude-only image registries/databases motivates the development of techniques that can use them as training datasets for generalizable DL-based recon. models. Here we propose to address this challenge by employing a generative approach based on conditional score-based diffusion models (SBDMs): given a magnitude-only MR image, it synthesizes a phase map (in the image domain) that realistically corresponds to the magnitude-only image. We evaluate its generative capabilities in a downstream DL-based recon. task whereby a large k-space dataset is generated by combining the SBDM-synthesized phase-maps and the corresponding magnitude-only images, and this k-space dataset is then used to train a DL model for accelerated MRI recon. We compare the performance of the resulting DL model versus those trained according to (a) a naive approach that uses smooth phase, (b) a k-space training dataset generated using synthesized phase maps derived from a generative adversarial network, and (c) the ground truth k-space data. Our results suggest that the DL model trained from SBDM-synthesized k-space data outperforms the other approaches in terms of quantitative metrics as well as qualitatively observed recon. fidelity, i.e., whether the reconstructed images include erroneous or hallucinated features that could adversely impact diagnostic accuracy.

Conventional MRI reconstruction methods treat images and coil sensitivities as discrete objects, leading to high memory demands and limited structural awareness that hamper effective regularization. These limitations hinder accurate reconstruction in highly undersampled scenarios, such as dynamic 3D cardiac magnetic resonance (CMR). We introduce a discretization-free, memory-efficient, model-based framework for dynamic 2D and 3D MRI reconstruction from highly undersampled data. We represent magnetization and coil sensitivities as continuous objects -- differentiable functions -- using tensor products of univariate neural fields. This tensor product structure enables scalable optimization in high-dimensional spatiotemporal settings. Our method outperforms state-of-the-art model-based reconstructions in dynamic 2D and 3D MR settings, preserving structure and motion even under aggressive undersampling (e.g., acceleration factor 16).