Learning Fourier-Constrained Diffusion Bridges for MRI Reconstruction

Magnetic Resonance Imaging (MRI) serves as a cornerstone of modern diagnostic medicine, yet the necessity for rapid data acquisition often leads to incomplete k-space sampling. Prior research has shown that this undersampling process generates complex aliasing artifacts that possess distinct spatial structures, which differ fundamentally from random thermal noise. Traditional generative models frequently utilize task-agnostic diffusion priors that rely on a denoising trajectory starting from an asymptotic state of pure Gaussian noise. Because these noise-governed paths do not account for the specific physics of signal degradation in medical scanners, they often produce suboptimal results during artifact suppression. Standard approaches like cold or soft diffusion attempt to mitigate this by using degraded measurements, but they frequently employ start-points that are too severely distorted for practical clinical use. This absence of evidence motivated the creation of a specialized diffusion bridge that specifically targets the transition between undersampled and fully-sampled frequency domains.

This research introduces a Fourier-constrained diffusion bridge specifically engineered to execute high-fidelity dealiasing transformations for accelerated magnetic resonance imaging protocols. The primary objective was to move beyond conventional denoising-based generative models by establishing a direct mapping between realistically undersampled data and the desired fully-sampled output. Investigators aimed to develop a stochastic Fourier-constrained degradation operator that could remove spatial frequencies in a manner consistent with the physical constraints of MRI hardware. The study sought to ensure closer alignment between training and test distributions by utilizing model inputs that reflect actual clinical undersampling patterns rather than theoretical noise. Another central goal was the implementation of a novel sampling algorithm capable of providing progressive dealiasing through continuous k-space correction across all reverse diffusion steps. By evaluating this framework on brain MRI datasets, the researchers intended to establish significant improvements in both within-domain and cross-domain reconstruction accuracy.

The methodology centers on the implementation of a Fourier-constrained diffusion bridge that utilizes a stochastic degradation operator to remove a progressively growing set of spatial frequencies. Unlike existing diffusion bridges that rely on weighted linear averages or the addition of noise, this system employs the binary removal of compact k-space sets. This physics-informed approach allows the forward process to reach a start-point of realistically undersampled data that maintains the structural integrity of the original signal. The researchers integrated a progressive dealiasing sampling algorithm that iteratively refines the image by correcting recovered frequency components during each step of the reverse diffusion trajectory. Quantitative assessment of the reconstruction quality was performed using standardized metrics, specifically the Peak Signal-to-Noise Ratio (PSNR) and the Structural Similarity Index Measure (SSIM). The experimental protocol involved rigorous testing on brain MRI scans to compare the performance of this bridge-based model against established task-agnostic diffusion priors.

The Fourier-constrained diffusion bridge achieved a substantial performance gain, outperforming competing methods by 4.5dB in Peak Signal-to-Noise Ratio for within-domain brain MRI reconstruction. Structural Similarity Index Measure scores for these within-domain tasks reflected an 8.3% improvement, indicating superior preservation of anatomical details and contrast. In cross-domain evaluations, the model exhibited even greater robustness, yielding a 4.7dB increase in Peak Signal-to-Noise Ratio over traditional task-agnostic diffusion priors. The cross-domain Structural Similarity Index Measure improved by 16.4%, highlighting the ability of the bridge framework to generalize across different imaging conditions and populations. These results confirm that the use of a realistically undersampled start-point effectively bridges the gap between training simulations and real-world clinical data distributions. The progressive dealiasing algorithm successfully maintained k-space consistency, which prevented the introduction of secondary artifacts often seen in purely denoising-based generative trajectories.

The findings of this study demonstrate that Fourier-constrained diffusion bridges represent a transformative approach to medical image restoration by aligning generative processes with physical reality. By replacing noise-based denoising with direct dealiasing transformations, this framework provides a more reliable path for reconstructing high-resolution images from undersampled k-space data. The researchers conclude that the binary removal of compact k-space sets is a more effective degradation strategy than the additive noise used in conventional diffusion models. This methodology has the potential to significantly accelerate MRI scan times while maintaining the diagnostic quality required for complex brain imaging and other clinical applications. Future research may expand this Fourier-constrained approach to other medical imaging modalities that rely on structured frequency-domain data for signal acquisition and processing. Ultimately, the study establishes a new benchmark for computational MRI reconstruction that leverages the strengths of diffusion bridges to overcome the limitations of task-agnostic priors.