# Continuous U-Net: Faster, Greater and Noiseless

arXiv: https://arxiv.org/abs/2302.00626 · HTML: https://mlsanity.com/paper/2302.00626/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 44 references that are in the map.
- First posted on arXiv 1 Feb 2023.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [Pyramid Scene Parsing Network](https://mlsanity.com/paper/1612.01105/) (2016)
- [Rethinking Atrous Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1706.05587/) (2017)
- [TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation](https://mlsanity.com/paper/2102.04306/) (2021)
- [Attention U-Net: Learning Where to Look for the Pancreas](https://mlsanity.com/paper/1804.03999/) (2018)
- [An Overview of Multi-Task Learning in Deep Neural Networks](https://mlsanity.com/paper/1706.05098/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Kvasir-SEG: A Segmented Polyp Dataset](https://mlsanity.com/paper/1911.07069/) (2019)
- [UNet++: Redesigning Skip Connections to Exploit Multiscale Features in Image Segmentation](https://mlsanity.com/paper/1912.05074/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [U2-Net: Going Deeper with Nested U-Structure for Salient Object Detection](https://mlsanity.com/paper/2005.09007/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Road Extraction by Deep Residual U-Net](https://mlsanity.com/paper/1711.10684/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Gland Segmentation in Colon Histology Images: The GlaS Challenge Contest](https://mlsanity.com/paper/1603.00275/) (2016)
- [Skin Lesion Analysis toward Melanoma Detection: A Challenge at the International Symposium on Biomedical Imaging (ISBI) 2016, hosted by the International Skin Imaging Collaboration (ISIC)](https://mlsanity.com/paper/1605.01397/) (2016)
- [UNeXt: MLP-based Rapid Medical Image Segmentation Network](https://mlsanity.com/paper/2203.04967/) (2022)
- [On Robustness of Neural Ordinary Differential Equations](https://mlsanity.com/paper/1910.05513/) (2019)
- [Optimized U-Net for Brain Tumor Segmentation](https://mlsanity.com/paper/2110.03352/) (2021)
- [A Partially Reversible U-Net for Memory-Efficient Volumetric Image Segmentation](https://mlsanity.com/paper/1906.06148/) (2019)
- [Neural Ordinary Differential Equations for Semantic Segmentation of Individual Colon Glands](https://mlsanity.com/paper/1910.10470/) (2019)
- [MONAIfbs: MONAI-based fetal brain MRI deep learning segmentation](https://mlsanity.com/paper/2103.13314/) (2021)
- [Dilated deeply supervised networks for hippocampus segmentation in MRI](https://mlsanity.com/paper/1903.09097/) (2019)

## Built on by (most cited first, 8 of 8)

- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEs](https://mlsanity.com/paper/2410.02113/) (2024)
- [The Missing U for Efficient Diffusion Models](https://mlsanity.com/paper/2310.20092/) (2023)
- [NODE-Adapter: Neural Ordinary Differential Equations for Better Vision-Language Reasoning](https://mlsanity.com/paper/2407.08672/) (2024)
- [Cross-Modal Few-Shot Learning with Second-Order Neural Ordinary Differential Equations](https://mlsanity.com/paper/2412.15813/) (2024)
- [ProSMA-UNet: Decoder Conditioning for Proximal-Sparse Skip Feature Selection](https://mlsanity.com/paper/2603.03187/) (2026)
- [Continuous Learned Primal Dual](https://mlsanity.com/paper/2405.02478/) (2024)
- [LIP-CAR: contrast agent reduction by a deep learned inverse problem](https://mlsanity.com/paper/2407.10559/) (2024)

## Abstract (arXiv)

Image segmentation is a fundamental task in image analysis and clinical practice. The current state-of-the-art techniques are based on U-shape type encoder-decoder networks with skip connections, called U-Net. Despite the powerful performance reported by existing U-Net type networks, they suffer from several major limitations. Issues include the hard coding of the receptive field size, compromising the performance and computational cost, as well as the fact that they do not account for inherent noise in the data. They have problems associated with discrete layers, and do not offer any theoretical underpinning. In this work we introduce continuous U-Net, a novel family of networks for image segmentation. Firstly, continuous U-Net is a continuous deep neural network that introduces new dynamic blocks modelled by second order ordinary differential equations. Secondly, we provide theoretical guarantees for our network demonstrating faster convergence, higher robustness and less sensitivity to noise. Thirdly, we derive qualitative measures to tailor-made segmentation tasks. We demonstrate, through extensive numerical and visual results, that our model outperforms existing U-Net blocks for several medical image segmentation benchmarking datasets.