# Learning Fourier-Constrained Diffusion Bridges for MRI Reconstruction

arXiv: https://arxiv.org/abs/2308.01096 · HTML: https://mlsanity.com/paper/2308.01096/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 60 of its 103 references that are in the map.
- First posted on arXiv 2 Aug 2023.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 30 of 60)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Unsupervised Medical Image Translation with Adversarial Diffusion Models](https://mlsanity.com/paper/2207.08208/) (2022)
- [ResViT: Residual vision transformers for multi-modal medical image synthesis](https://mlsanity.com/paper/2106.16031/) (2021)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Adaptive Diffusion Priors for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2207.05876/) (2022)
- [Soft Diffusion: Score Matching for General Corruptions](https://mlsanity.com/paper/2209.05442/) (2022)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [Deep learning for undersampled MRI reconstruction](https://mlsanity.com/paper/1709.02576/) (2017)
- [Unsupervised MRI Reconstruction via Zero-Shot Learned Adversarial Transformers](https://mlsanity.com/paper/2105.08059/) (2021)
- [A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction](https://mlsanity.com/paper/1703.00555/) (2017)
- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)
- [Measurement-conditioned Denoising Diffusion Probabilistic Model for Under-sampled Medical Image Reconstruction](https://mlsanity.com/paper/2203.03623/) (2022)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Towards performant and reliable undersampled MR reconstruction via diffusion model sampling](https://mlsanity.com/paper/2203.04292/) (2022)
- [Deep Residual Learning for Accelerated MRI using Magnitude and Phase Networks](https://mlsanity.com/paper/1804.00432/) (2018)
- [ReconFormer: Accelerated MRI Reconstruction Using Recurrent Transformer](https://mlsanity.com/paper/2201.09376/) (2022)

## Built on by (most cited first, 14 of 14)

- [Diffusion Models for Image Restoration and Enhancement: A Comprehensive Survey](https://mlsanity.com/paper/2308.09388/) (2023)
- [Self-Consistent Recursive Diffusion Bridge for Medical Image Translation](https://mlsanity.com/paper/2405.06789/) (2024)
- [I2I-Mamba: Multi-modal medical image synthesis via selective state space modeling](https://mlsanity.com/paper/2405.14022/) (2024)
- [MMR-Mamba: Multi-Modal MRI Reconstruction with Mamba and Spatial-Frequency Information Fusion](https://mlsanity.com/paper/2406.18950/) (2024)
- [DenoMamba: A fused state-space model for low-dose CT denoising](https://mlsanity.com/paper/2409.13094/) (2024)
- [Physics-Driven Autoregressive State Space Models for Medical Image Reconstruction](https://mlsanity.com/paper/2412.09331/) (2024)
- [Generative Autoregressive Transformers for Model-Agnostic Federated MRI Reconstruction](https://mlsanity.com/paper/2502.04521/) (2025)
- [Guided MRI Reconstruction via Schr"odinger Bridge](https://mlsanity.com/paper/2411.14269/) (2024)
- [ADOBI: Adaptive Diffusion Bridge For Blind Inverse Problems with Application to MRI Reconstruction](https://mlsanity.com/paper/2411.16535/) (2024)
- [I2I-PR: Deep Iterative Refinement for Phase Retrieval using Image-to-Image Diffusion Models](https://mlsanity.com/paper/2507.09609/) (2025)
- [Adaptive Dual-Mode Distillation with Incentive Schemes for Scalable, Heterogeneous Federated Learning on Non-IID Data](https://mlsanity.com/paper/2509.22507/) (2025)
- [Self-Auditing Residual Drifting for Pathology-Preserving Accelerated Knee MRI](https://mlsanity.com/paper/2607.02428/) (2026)
- [Evaluating the Safety of Deep Learning-Based Brain MRI Reconstruction](https://mlsanity.com/paper/2608.28714/) (2026)
- [Stochastic Finite-Transition Learning and Exploration for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2610.13095/) (2026)

## Abstract (arXiv)

Deep generative models have gained recent traction in accelerated MRI reconstruction. Diffusion priors are particularly promising given their representational fidelity. Instead of the target transformation from undersampled to fully-sampled data required for MRI reconstruction, common diffusion priors are trained to learn a task-agnostic transformation from an asymptotic start-point of Gaussian noise onto the finite end-point of fully-sampled data. During inference, data-consistency projections are injected in between reverse diffusion steps to reach a compromise solution within the span of both the trained diffusion prior and the imaging operator for an accelerated MRI acquisition. Unfortunately, performance losses can occur due to the discrepancy between target and learned transformations given the asymptotic normality assumption in diffusion priors. To address this discrepancy, here we introduce a novel Fourier-constrained diffusion bridge (FDB) for MRI reconstruction that transforms between a finite start-point of moderately undersampled data and an end-point of fully-sampled data. We derive the theoretical formulation of FDB as a generalized diffusion process based on a stochastic degradation operator that performs random spatial-frequency removal. We propose an enhanced sampling algorithm with a learned correction term for soft dealiasing across reverse diffusion steps. Demonstrations on brain MRI indicate that FDB outperforms state-of-the-art methods including non-diffusion and diffusion priors.