# Spherical U-Net on Cortical Surfaces: Methods and Applications

arXiv: https://arxiv.org/abs/1904.00906 · HTML: https://mlsanity.com/paper/1904.00906/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Apr 2019.

## Built on by (most cited first, 30 of 34)

- [Graph-Based Deep Learning for Medical Diagnosis and Analysis: Past, Present and Future](https://mlsanity.com/paper/2105.13137/) (2021)
- [Surface Vision Transformers: Attention-Based Modelling applied to Cortical Analysis](https://mlsanity.com/paper/2203.16414/) (2022)
- [3D Scene Geometry Estimation from 360^circ Imagery: A Survey](https://mlsanity.com/paper/2401.09252/) (2024)
- [A Deep-Discrete Learning Framework for Spherical Surface Registration](https://mlsanity.com/paper/2203.12999/) (2022)
- [The Multiscale Surface Vision Transformer](https://mlsanity.com/paper/2303.11909/) (2023)
- [Automatic Landmark Detection and Registration of Brain Cortical Surfaces via Quasi-Conformal Geometry and Convolutional Neural Networks](https://mlsanity.com/paper/2208.07010/) (2022)
- [SUGAR: Spherical Ultrafast Graph Attention Framework for Cortical Surface Registration](https://mlsanity.com/paper/2307.00511/) (2023)
- [Surface Analysis with Vision Transformers](https://mlsanity.com/paper/2205.15836/) (2022)
- [Spatio-Temporal Encoding of Brain Dynamics with Surface Masked Autoencoders](https://mlsanity.com/paper/2308.05474/) (2023)
- [SurfGNN: A robust surface-based prediction model with interpretability for coactivation maps of spatial and cortical features](https://mlsanity.com/paper/2411.05825/) (2024)
- [Generalized Spherical Neural Operators: Green's Function Formulation](https://mlsanity.com/paper/2512.10723/) (2025)
- [From Flat to Round: Redefining Brain Decoding with Surface-Based fMRI and Cortex Structure](https://mlsanity.com/paper/2507.16389/) (2025)
- [An explainability framework for cortical surface-based deep learning](https://mlsanity.com/paper/2203.08312/) (2022)
- [A Deep Generative Model of Neonatal Cortical Surface Development](https://mlsanity.com/paper/2206.07542/) (2022)
- [Neurodevelopmental Phenotype Prediction: A State-of-the-Art Deep Learning Model](https://mlsanity.com/paper/2211.08831/) (2022)
- [NeuroExplainer: Fine-Grained Attention Decoding to Uncover Cortical Development Patterns of Preterm Infants](https://mlsanity.com/paper/2301.00815/) (2023)
- [Individualized Mapping of Aberrant Cortical Thickness via Stochastic Cortical Self-Reconstruction](https://mlsanity.com/paper/2403.06837/) (2024)
- [Surface Vision Mamba: Leveraging Bidirectional State Space Model for Efficient Spherical Manifold Representation](https://mlsanity.com/paper/2501.14679/) (2025)
- [Community-Level Modeling of Gyral Folding Patterns for Robust and Anatomically Informed Individualized Brain Mapping](https://mlsanity.com/paper/2602.01482/) (2026)
- [Surface Vision Transformers: Flexible Attention-Based Modelling of Biomedical Surfaces](https://mlsanity.com/paper/2204.03408/) (2022)
- [Utility of Equivariant Message Passing in Cortical Mesh Segmentation](https://mlsanity.com/paper/2206.03164/) (2022)
- [Unsupervised Multimodal Surface Registration with Geometric Deep Learning](https://mlsanity.com/paper/2311.13022/) (2023)
- [Controllable Surface Diffusion Generative Model for Neurodevelopmental Trajectories](https://mlsanity.com/paper/2508.03706/) (2025)
- [Spherical Brownian Bridge Diffusion Models for Conditional Cortical Thickness Forecasting](https://mlsanity.com/paper/2509.08442/) (2025)
- [Unified Brain Surface and Volume Registration](https://mlsanity.com/paper/2512.19928/) (2025)
- [JParc: Joint cortical surface parcellation with registration](https://mlsanity.com/paper/2512.22485/) (2025)
- [Neural Operators for Biomedical Spherical Heterogeneity](https://mlsanity.com/paper/2601.03561/) (2026)
- [Two-chart Beltrami Optimization for Distortion-Controlled Spherical Bijection with Application to Brain Surface Registration](https://mlsanity.com/paper/2602.01589/) (2026)
- [SurfAge-Net: A Hierarchical Surface-Based Network for Interpretable Fine-Grained Brain Age Prediction](https://mlsanity.com/paper/2602.06994/) (2026)
- [NeurIPS: Neuro-anatomical Inductive Priors for Sphere-based Brain Decoding](https://mlsanity.com/paper/2605.24993/) (2026)

## Abstract (arXiv)

Convolutional Neural Networks (CNNs) have been providing the state-of-the-art performance for learning-related problems involving 2D/3D images in Euclidean space. However, unlike in the Euclidean space, the shapes of many structures in medical imaging have a spherical topology in a manifold space, e.g., brain cortical or subcortical surfaces represented by triangular meshes, with large inter-subject and intrasubject variations in vertex number and local connectivity. Hence, there is no consistent neighborhood definition and thus no straightforward convolution/transposed convolution operations for cortical/subcortical surface data. In this paper, by leveraging the regular and consistent geometric structure of the resampled cortical surface mapped onto the spherical space, we propose a novel convolution filter analogous to the standard convolution on the image grid. Accordingly, we develop corresponding operations for convolution, pooling, and transposed convolution for spherical surface data and thus construct spherical CNNs. Specifically, we propose the Spherical U-Net architecture by replacing all operations in the standard U-Net with their spherical operation counterparts. We then apply the Spherical U-Net to two challenging and neuroscientifically important tasks in infant brains: cortical surface parcellation and cortical attribute map development prediction. Both applications demonstrate the competitive performance in the accuracy, computational efficiency, and effectiveness of our proposed Spherical U-Net, in comparison with the state-of-the-art methods.