Although MRI reconstruction requires a dealiasing transformation from undersampled to fully-sampled data, task-agnostic diffusion priors sample images via a denoising-based generative trajectory from an asymptotic start-point of Gaussian noise onto fully-sampled data. Since aliasing artifacts in MR images carry spatial structure deviating from Gaussian noise, this noise-governed trajectory can cause suboptimal artifact suppression. To address this limitation, we introduce the first Fourier-constrained diffusion bridge (FDB) for MRI reconstruction in the literature. Unlike task-agnostic diffusion priors, FDB does not rely on noise in its forward process and instead learns a dealiasing transformation between a start-point of undersampled data and the end-point of fully-sampled data. The start-point is derived via a stochastic Fourier-constrained degradation operator that removes a progressively growing set of spatial frequencies. Unlike cold/soft diffusion priors that use an asymptotic start-point of severely degraded measurements, FDB uses a realistically undersampled start-point to ensure closer alignment of model input between training and test distributions. Unlike existing diffusion bridges that use degradations based on weighted linear averages and noise addition, FDB implements degradations based on binary removal of compact k-space sets to conform to the physics of accelerated MRI. To further improve image quality, FDB leverages a novel sampling algorithm based on progressive dealiasing by continually correcting recovered k-space data across reverse diffusion steps. Demonstrations on brain MRI show that FDB outperforms competing methods by 4.5dB PSNR and 8.3% SSIM in within-domain and by 4.7dB PSNR and 16.4% SSIM in cross-domain reconstructions.

Learning-based translation between MRI contrasts involves supervised deep models trained using high-quality source- and target-contrast images derived from fully-sampled acquisitions, which might be difficult to collect under limitations on scan costs or time. To facilitate curation of training sets, here we introduce the first semi-supervised model for MRI contrast translation (ssGAN) that can be trained directly using undersampled k-space data. To enable semi-supervised learning on undersampled data, ssGAN introduces novel multi-coil losses in image, k-space, and adversarial domains. The multi-coil losses are selectively enforced on acquired k-space samples unlike traditional losses in single-coil synthesis models. Comprehensive experiments on retrospectively undersampled multi-contrast brain MRI datasets are provided. Our results demonstrate that ssGAN yields on par performance to a supervised model, while outperforming single-coil models trained on coil-combined magnitude images. It also outperforms cascaded reconstruction-synthesis models where a supervised synthesis model is trained following self-supervised reconstruction of undersampled data. Thus, ssGAN holds great promise to improve the feasibility of learning-based multi-contrast MRI synthesis.