# One-shot Generative Prior in Hankel-k-space for Parallel Imaging Reconstruction

arXiv: https://arxiv.org/abs/2208.07181 · HTML: https://mlsanity.com/paper/2208.07181/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 73 references that are in the map.
- First posted on arXiv 15 Aug 2022.

## Builds on (most cited first, 30 of 40)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [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)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Permutation Invariant Graph Generation via Score-Based Generative Modeling](https://mlsanity.com/paper/2003.00638/) (2020)
- [Learning Gradient Fields for Shape Generation](https://mlsanity.com/paper/2008.06520/) (2020)
- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [Unsupervised Medical Image Translation with Adversarial Diffusion Models](https://mlsanity.com/paper/2207.08208/) (2022)
- [UNIT-DDPM: UNpaired Image Translation with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.05358/) (2021)
- [Convolutional Recurrent Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1712.01751/) (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)
- [Measurement-conditioned Denoising Diffusion Probabilistic Model for Under-sampled Medical Image Reconstruction](https://mlsanity.com/paper/2203.03623/) (2022)
- [Bayesian MRI Reconstruction with Joint Uncertainty Estimation using Diffusion Models](https://mlsanity.com/paper/2202.01479/) (2022)
- [GrappaNet: Combining Parallel Imaging with Deep Learning for Multi-Coil MRI Reconstruction](https://mlsanity.com/paper/1910.12325/) (2019)
- [MR image reconstruction using deep density priors](https://mlsanity.com/paper/1711.11386/) (2017)
- [Source Separation with Deep Generative Priors](https://mlsanity.com/paper/2002.07942/) (2020)

## Built on by (most cited first, 12 of 12)

- [Learning Fourier-Constrained Diffusion Bridges for MRI Reconstruction](https://mlsanity.com/paper/2308.01096/) (2023)
- [Correlated and Multi-frequency Diffusion Modeling for Highly Under-sampled MRI Reconstruction](https://mlsanity.com/paper/2309.00853/) (2023)
- [A Survey of Emerging Applications of Diffusion Probabilistic Models in MRI](https://mlsanity.com/paper/2311.11383/) (2023)
- [Highly Accelerated MRI via Implicit Neural Representation Guided Posterior Sampling of Diffusion Models](https://mlsanity.com/paper/2407.02744/) (2024)
- [DP-MDM: Detail-Preserving MR Reconstruction via Multiple Diffusion Models](https://mlsanity.com/paper/2405.05763/) (2024)
- [RadGazeGen: Radiomics and Gaze-guided Medical Image Generation using Diffusion Models](https://mlsanity.com/paper/2410.00307/) (2024)
- [Partitioned Hankel-based Diffusion Models for Few-shot Low-dose CT Reconstruction](https://mlsanity.com/paper/2405.17167/) (2024)
- [Sub-DM:Subspace Diffusion Model with Orthogonal Decomposition for MRI Reconstruction](https://mlsanity.com/paper/2411.03758/) (2024)
- [JSMoCo: Joint Coil Sensitivity and Motion Correction in Parallel MRI with a Self-Calibrating Score-Based Diffusion Model](https://mlsanity.com/paper/2310.09625/) (2023)
- [Adaptive Mask-guided K-space Diffusion for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2506.18270/) (2025)
- [High-dimensional Embedding Prior for Noisy K-space Domain MRIReconstruction](https://mlsanity.com/paper/2607.01176/) (2026)
- [Hankel Subspace Self-Supervised Learning for Parallel MRI Reconstruction](https://mlsanity.com/paper/2610.11560/) (2026)

## Abstract (arXiv)

Magnetic resonance imaging serves as an essential tool for clinical diagnosis. However, it suffers from a long acquisition time. The utilization of deep learning, especially the deep generative models, offers aggressive acceleration and better reconstruction in magnetic resonance imaging. Nevertheless, learning the data distribution as prior knowledge and reconstructing the image from limited data remains challenging. In this work, we propose a novel Hankel-k-space generative model (HKGM), which can generate samples from a training set of as little as one k-space data. At the prior learning stage, we first construct a large Hankel matrix from k-space data, then extract multiple structured k-space patches from the large Hankel matrix to capture the internal distribution among different patches. Extracting patches from a Hankel matrix enables the generative model to be learned from redundant and low-rank data space. At the iterative reconstruction stage, it is observed that the desired solution obeys the learned prior knowledge. The intermediate reconstruction solution is updated by taking it as the input of the generative model. The updated result is then alternatively operated by imposing low-rank penalty on its Hankel matrix and data consistency con-strain on the measurement data. Experimental results confirmed that the internal statistics of patches within a single k-space data carry enough information for learning a powerful generative model and provide state-of-the-art reconstruction.