Learning Fourier-Constrained Diffusion Bridges for MRI Reconstruction

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.