# A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise

arXiv: https://arxiv.org/abs/2205.10278 · HTML: https://mlsanity.com/paper/2205.10278/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 68 references that are in the map.
- First posted on arXiv 20 May 2022.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images](https://mlsanity.com/paper/2106.07009/) (2021)
- [A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction](https://mlsanity.com/paper/1610.09736/) (2016)
- [Compressed Sensing MRI Reconstruction using a Generative Adversarial Network with a Cyclic Loss](https://mlsanity.com/paper/1709.00753/) (2017)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [Time-Dependent Deep Image Prior for Dynamic MRI](https://mlsanity.com/paper/1910.01684/) (2019)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Noise2Same: Optimizing A Self-Supervised Bound for Image Denoising](https://mlsanity.com/paper/2010.11971/) (2020)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [Physics-Driven Deep Learning for Computational Magnetic Resonance Imaging](https://mlsanity.com/paper/2203.12215/) (2022)
- [AI-based Reconstruction for Fast MRI -- A Systematic Review and Meta-analysis](https://mlsanity.com/paper/2112.12744/) (2021)
- [DIMENSION: Dynamic MR Imaging with Both K-space and Spatial Prior Knowledge Obtained via Multi-Supervised Network Training](https://mlsanity.com/paper/1810.00302/) (2018)
- [Self-Supervised Learning for MRI Reconstruction with a Parallel Network Training Framework](https://mlsanity.com/paper/2109.12502/) (2021)
- [Unsupervised MRI Reconstruction with Generative Adversarial Networks](https://mlsanity.com/paper/2008.13065/) (2020)
- [Multi-Mask Self-Supervised Learning for Physics-Guided Neural Networks in Highly Accelerated MRI](https://mlsanity.com/paper/2008.06029/) (2020)
- [Deformation-Compensated Learning for Image Reconstruction without Ground Truth](https://mlsanity.com/paper/2107.05533/) (2021)
- [PARCEL: Physics-based Unsupervised Contrastive Representation Learning for Multi-coil MR Imaging](https://mlsanity.com/paper/2202.01494/) (2022)
- [Deep Plug-and-Play Prior for Parallel MRI Reconstruction](https://mlsanity.com/paper/1909.00089/) (2019)
- [Improved Supervised Training of Physics-Guided Deep Learning Image Reconstruction with Multi-Masking](https://mlsanity.com/paper/2010.13868/) (2020)

## Built on by (most cited first, 30 of 30)

- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [Analyzing the Sample Complexity of Self-Supervised Image Reconstruction Methods](https://mlsanity.com/paper/2305.19079/) (2023)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Measurement Score-Based Diffusion Model](https://mlsanity.com/paper/2505.11853/) (2025)
- [A Deep Learning Method for Simultaneous Denoising and Missing Wedge Reconstruction in Cryogenic Electron Tomography](https://mlsanity.com/paper/2311.05539/) (2023)
- [K-band: Self-supervised MRI Reconstruction via Stochastic Gradient Descent over K-space Subsets](https://mlsanity.com/paper/2308.02958/) (2023)
- [Noisier2Inverse: Self-Supervised Learning for Image Reconstruction with Correlated Noise](https://mlsanity.com/paper/2503.19468/) (2025)
- [Self-Supervised Learning from Noisy and Incomplete Data](https://mlsanity.com/paper/2601.03244/) (2026)
- [Joint Supervised and Self-supervised Learning for MRI Reconstruction](https://mlsanity.com/paper/2311.15856/) (2023)
- [Robust multi-coil MRI reconstruction via self-supervised denoising](https://mlsanity.com/paper/2411.12919/) (2024)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Sparsity-Driven Parallel Imaging Consistency for Improved Self-Supervised MRI Reconstruction](https://mlsanity.com/paper/2505.24136/) (2025)
- [Deep Learning-based Accelerated MR Cholangiopancreatography without Fully-sampled Data](https://mlsanity.com/paper/2405.03732/) (2024)
- [MotionTTT: 2D Test-Time-Training Motion Estimation for 3D Motion Corrected MRI](https://mlsanity.com/paper/2409.09370/) (2024)
- [Fast MRI for All: Bridging Access Gaps by Training without Raw Data](https://mlsanity.com/paper/2411.13022/) (2024)
- [Re-Visible Dual-Domain Self-Supervised Deep Unfolding Network for MRI Reconstruction](https://mlsanity.com/paper/2501.03737/) (2025)
- [Equivariant Splitting: Self-supervised learning from incomplete data](https://mlsanity.com/paper/2510.00929/) (2025)
- [UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matching](https://mlsanity.com/paper/2512.17493/) (2025)
- [An Adaptive, Disentangled Representation for Multidimensional MRI Reconstruction](https://mlsanity.com/paper/2512.24674/) (2025)
- [Extendable Generalization Self-Supervised Diffusion for Low-Dose CT Reconstruction](https://mlsanity.com/paper/2509.23885/) (2025)
- [MID: A Self-supervised Multimodal Iterative Denoising Framework](https://mlsanity.com/paper/2511.00997/) (2025)
- [Locally-Supervised Global Image Restoration](https://mlsanity.com/paper/2511.01998/) (2025)
- [HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training](https://mlsanity.com/paper/2601.01228/) (2026)
- [Accelerating Stroke MRI with Diffusion Probabilistic Models through Large-Scale Pre-training and Target-Specific Fine-Tuning](https://mlsanity.com/paper/2603.13007/) (2026)
- [Self-Supervised Uncertainty Estimation For Super-Resolution of Satellite Images](https://mlsanity.com/paper/2603.14074/) (2026)
- [Self-Supervised Angular Deblurring in Photoacoustic Reconstruction via Noisier2Inverse](https://mlsanity.com/paper/2604.15681/) (2026)
- [CoilDrop-MRI: Self-supervised physics-guided MRI reconstruction with coil dropout](https://mlsanity.com/paper/2606.00100/) (2026)
- [Enhancing Implicit Neural Representations with Image Feature Embedding for Unsupervised Cardiac Cine MRI Reconstruction](https://mlsanity.com/paper/2607.04069/) (2026)
- [Hankel Subspace Self-Supervised Learning for Parallel MRI Reconstruction](https://mlsanity.com/paper/2610.11560/) (2026)

## Abstract (arXiv)

In recent years, there has been attention on leveraging the statistical modeling capabilities of neural networks for reconstructing sub-sampled Magnetic Resonance Imaging (MRI) data. Most proposed methods assume the existence of a representative fully-sampled dataset and use fully-supervised training. However, for many applications, fully sampled training data is not available, and may be highly impractical to acquire. The development and understanding of self-supervised methods, which use only sub-sampled data for training, are therefore highly desirable. This work extends the Noisier2Noise framework, which was originally constructed for self-supervised denoising tasks, to variable density sub-sampled MRI data. We use the Noisier2Noise framework to analytically explain the performance of Self-Supervised Learning via Data Undersampling (SSDU), a recently proposed method that performs well in practice but until now lacked theoretical justification. Further, we propose two modifications of SSDU that arise as a consequence of the theoretical developments. Firstly, we propose partitioning the sampling set so that the subsets have the same type of distribution as the original sampling mask. Secondly, we propose a loss weighting that compensates for the sampling and partitioning densities. On the fastMRI dataset we show that these changes significantly improve SSDU's image restoration quality and robustness to the partitioning parameters.