# PUERT: Probabilistic Under-sampling and Explicable Reconstruction Network for CS-MRI

arXiv: https://arxiv.org/abs/2204.11189 · HTML: https://mlsanity.com/paper/2204.11189/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 84 references that are in the map.
- First posted on arXiv 24 Apr 2022.

## Builds on (most cited first, 30 of 35)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [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)
- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Forward and Backward Information Retention for Accurate Binary Neural Networks](https://mlsanity.com/paper/1909.10788/) (2019)
- [Convolutional Recurrent Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1712.01751/) (2017)
- [Deep learning for undersampled MRI reconstruction](https://mlsanity.com/paper/1709.02576/) (2017)
- [Compressed Sensing MRI Reconstruction using a Generative Adversarial Network with a Cyclic Loss](https://mlsanity.com/paper/1709.00753/) (2017)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Deep Optics for Single-shot High-dynamic-range Imaging](https://mlsanity.com/paper/1908.00620/) (2019)
- [Reducing Uncertainty in Undersampled MRI Reconstruction with Active Acquisition](https://mlsanity.com/paper/1902.03051/) (2019)
- [ISTA-Net++: Flexible Deep Unfolding Network for Compressive Sensing](https://mlsanity.com/paper/2103.11554/) (2021)
- [Memory-Augmented Deep Unfolding Network for Compressive Sensing](https://mlsanity.com/paper/2110.09766/) (2021)
- [AI-based Reconstruction for Fast MRI -- A Systematic Review and Meta-analysis](https://mlsanity.com/paper/2112.12744/) (2021)
- [Learning-based Optimization of the Under-sampling Pattern in MRI](https://mlsanity.com/paper/1901.01960/) (2019)
- [COAST: COntrollable Arbitrary-Sampling NeTwork for Compressive Sensing](https://mlsanity.com/paper/2107.07225/) (2021)
- [Dense Deep Unfolding Network with 3D-CNN Prior for Snapshot Compressive Imaging](https://mlsanity.com/paper/2109.06548/) (2021)
- [Learning Sensor Multiplexing Design through Back-propagation](https://mlsanity.com/paper/1605.07078/) (2016)
- [Self-Supervised Deep Active Accelerated MRI](https://mlsanity.com/paper/1901.04547/) (2019)
- [PILOT: Physics-Informed Learned Optimized Trajectories for Accelerated MRI](https://mlsanity.com/paper/1909.05773/) (2019)
- [Learning the Sampling Pattern for MRI](https://mlsanity.com/paper/1906.08754/) (2019)
- [Learning-Based Compressive MRI](https://mlsanity.com/paper/1805.01266/) (2018)
- [Convolutional neural networks that teach microscopes how to image](https://mlsanity.com/paper/1709.07223/) (2017)
- [OEDIPUS: An Experiment Design Framework for Sparsity-Constrained MRI](https://mlsanity.com/paper/1805.00524/) (2018)
- [Simulating single-coil MRI from the responses of multiple coils](https://mlsanity.com/paper/1811.08026/) (2018)

## Built on by (most cited first, 5 of 5)

- [Constrained Probabilistic Mask Learning for Task-specific Undersampled MRI Reconstruction](https://mlsanity.com/paper/2305.16376/) (2023)
- [RadioDUN: A Physics-Inspired Deep Unfolding Network for Radio Map Estimation](https://mlsanity.com/paper/2506.08418/) (2025)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [Anatomical Token Uncertainty for Transformer-Guided Active MRI Acquisition](https://mlsanity.com/paper/2603.21806/) (2026)
- [Information-Theoretic Optimization for Task-Adapted Compressed Sensing Magnetic Resonance Imaging](https://mlsanity.com/paper/2604.12709/) (2026)

## Abstract (arXiv)

Compressed Sensing MRI (CS-MRI) aims at reconstructing de-aliased images from sub-Nyquist sampling k-space data to accelerate MR Imaging, thus presenting two basic issues, i.e., where to sample and how to reconstruct. To deal with both problems simultaneously, we propose a novel end-to-end Probabilistic Under-sampling and Explicable Reconstruction neTwork, dubbed PUERT, to jointly optimize the sampling pattern and the reconstruction network. Instead of learning a deterministic mask, the proposed sampling subnet explores an optimal probabilistic sub-sampling pattern, which describes independent Bernoulli random variables at each possible sampling point, thus retaining robustness and stochastics for a more reliable CS reconstruction. A dynamic gradient estimation strategy is further introduced to gradually approximate the binarization function in backward propagation, which efficiently preserves the gradient information and further improves the reconstruction quality. Moreover, in our reconstruction subnet, we adopt a model-based network design scheme with high efficiency and interpretability, which is shown to assist in further exploitation for the sampling subnet. Extensive experiments on two widely used MRI datasets demonstrate that our proposed PUERT not only achieves state-of-the-art results in terms of both quantitative metrics and visual quality but also yields a sub-sampling pattern and a reconstruction model that are both customized to training data.