# HNOSeg-XS: Extremely Small Hartley Neural Operator for Efficient and Resolution-Robust 3D Image Segmentation

arXiv: https://arxiv.org/abs/2507.08205 · HTML: https://mlsanity.com/paper/2507.08205/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 51 references that are in the map.
- First posted on arXiv 10 Jul 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [UNet++: A Nested U-Net Architecture for Medical Image Segmentation](https://mlsanity.com/paper/1807.10165/) (2018)
- [Swin-Unet: Unet-like Pure Transformer for Medical Image Segmentation](https://mlsanity.com/paper/2105.05537/) (2021)
- [UNETR: Transformers for 3D Medical Image Segmentation](https://mlsanity.com/paper/2103.10504/) (2021)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [FDA: Fourier Domain Adaptation for Semantic Segmentation](https://mlsanity.com/paper/2004.05498/) (2020)
- [The RSNA-ASNR-MICCAI BraTS 2021 Benchmark on Brain Tumor Segmentation and Radiogenomic Classification](https://mlsanity.com/paper/2107.02314/) (2021)
- [Global Filter Networks for Image Classification](https://mlsanity.com/paper/2107.00645/) (2021)
- [CoTr: Efficiently Bridging CNN and Transformer for 3D Medical Image Segmentation](https://mlsanity.com/paper/2103.03024/) (2021)
- [nnFormer: Interleaved Transformer for Volumetric Segmentation](https://mlsanity.com/paper/2109.03201/) (2021)
- [Medical Image Segmentation Using Deep Learning: A Survey](https://mlsanity.com/paper/2009.13120/) (2020)
- [UTNet: A Hybrid Transformer Architecture for Medical Image Segmentation](https://mlsanity.com/paper/2107.00781/) (2021)
- [The KiTS21 Challenge: Automatic segmentation of kidneys, renal tumors, and renal cysts in corticomedullary-phase CT](https://mlsanity.com/paper/2307.01984/) (2023)
- [Implicit Neural Representation in Medical Imaging: A Comparative Survey](https://mlsanity.com/paper/2307.16142/) (2023)
- [3D Segmentation with Exponential Logarithmic Loss for Highly Unbalanced Object Sizes](https://mlsanity.com/paper/1809.00076/) (2018)
- [NISF: Neural Implicit Segmentation Functions](https://mlsanity.com/paper/2309.08643/) (2023)
- [Implicit Anatomical Rendering for Medical Image Segmentation with Stochastic Experts](https://mlsanity.com/paper/2304.03209/) (2023)
- [FreMIM: Fourier Transform Meets Masked Image Modeling for Medical Image Segmentation](https://mlsanity.com/paper/2304.10864/) (2023)
- [3D Segmentation with Fully Trainable Gabor Kernels and Pearson's Correlation Coefficient](https://mlsanity.com/paper/2201.03644/) (2022)
- [FNOSeg3D: Resolution-Robust 3D Image Segmentation with Fourier Neural Operator](https://mlsanity.com/paper/2310.03872/) (2023)

## Built on by (most cited first, 2 of 2)

- [Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment](https://mlsanity.com/paper/2606.24851/) (2026)
- [When Fusion Fails: Corruption-Aware Rebalanced Fusion for Multi-Modal Medical Image Segmentation](https://mlsanity.com/paper/2609.10261/) (2026)

## Abstract (arXiv)

In medical image segmentation, convolutional neural networks (CNNs) and transformers are dominant. For CNNs, given the local receptive fields of convolutional layers, long-range spatial correlations are captured through consecutive convolutions and pooling. However, as the computational cost and memory footprint can be prohibitively large, 3D models can only afford fewer layers than 2D models with reduced receptive fields and abstract levels. For transformers, although long-range correlations can be captured by multi-head attention, its quadratic complexity with respect to input size is computationally demanding. Therefore, either model may require input size reduction to allow more filters and layers for better segmentation. Nevertheless, given their discrete nature, models trained with patch-wise training or image downsampling may produce suboptimal results when applied on higher resolutions. To address this issue, here we propose the resolution-robust HNOSeg-XS architecture. We model image segmentation by learnable partial differential equations through the Fourier neural operator which has the zero-shot super-resolution property. By replacing the Fourier transform by the Hartley transform and reformulating the problem in the frequency domain, we created the HNOSeg-XS model, which is resolution robust, fast, memory efficient, and extremely parameter efficient. When tested on the BraTS'23, KiTS'23, and MVSeg'23 datasets with a Tesla V100 GPU, HNOSeg-XS showed its superior resolution robustness with fewer than 34.7k model parameters. It also achieved the overall best inference time (< 0.24 s) and memory efficiency (< 1.8 GiB) compared to the tested CNN and transformer models.