# Hybrid-CSR: Coupling Explicit and Implicit Shape Representation for Cortical Surface Reconstruction

arXiv: https://arxiv.org/abs/2307.12299 · HTML: https://mlsanity.com/paper/2307.12299/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 74 references that are in the map.
- First posted on arXiv 23 Jul 2023.

## Builds on (most cited first, 30 of 34)

- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Swin-Unet: Unet-like Pure Transformer for Medical Image Segmentation](https://mlsanity.com/paper/2105.05537/) (2021)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [UNETR: Transformers for 3D Medical Image Segmentation](https://mlsanity.com/paper/2103.10504/) (2021)
- [Accelerating 3D Deep Learning with PyTorch3D](https://mlsanity.com/paper/2007.08501/) (2020)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Pixel2Mesh: Generating 3D Mesh Models from Single RGB Images](https://mlsanity.com/paper/1804.01654/) (2018)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [PIFuHD: Multi-Level Pixel-Aligned Implicit Function for High-Resolution 3D Human Digitization](https://mlsanity.com/paper/2004.00452/) (2020)
- [Road Extraction by Deep Residual U-Net](https://mlsanity.com/paper/1711.10684/) (2017)
- [nnU-Net: Self-adapting Framework for U-Net-Based Medical Image Segmentation](https://mlsanity.com/paper/1809.10486/) (2018)
- [Shape As Points: A Differentiable Poisson Solver](https://mlsanity.com/paper/2106.03452/) (2021)
- [Unsupervised Learning of Probabilistic Diffeomorphic Registration for Images and Surfaces](https://mlsanity.com/paper/1903.03545/) (2019)
- [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018)
- [FastSurfer -- A fast and accurate deep learning based neuroimaging pipeline](https://mlsanity.com/paper/1910.03866/) (2019)
- [Learning a Probabilistic Model for Diffeomorphic Registration](https://mlsanity.com/paper/1812.07460/) (2018)
- [Neural Mesh Flow: 3D Manifold Mesh Generation via Diffeomorphic Flows](https://mlsanity.com/paper/2007.10973/) (2020)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Voxel2Mesh: 3D Mesh Model Generation from Volumetric Data](https://mlsanity.com/paper/1912.03681/) (2019)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Topology-Preserving Shape Reconstruction and Registration via Neural Diffeomorphic Flow](https://mlsanity.com/paper/2203.08652/) (2022)
- [DeepCSR: A 3D Deep Learning Approach for Cortical Surface Reconstruction](https://mlsanity.com/paper/2010.11423/) (2020)

## Built on by (most cited first, 4 of 4)

- [Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning](https://mlsanity.com/paper/2403.01781/) (2024)
- [SimCortex: Collision-free Simultaneous Cortical Surfaces Reconstruction](https://mlsanity.com/paper/2507.06955/) (2025)
- [GeoDiff: Geometry-Guided Diffusion for Metric Depth Estimation](https://mlsanity.com/paper/2510.18291/) (2025)
- [SimCortex v2: Joint Cortical Surface Reconstruction with Near-Zero Collisions and Self-Intersections](https://mlsanity.com/paper/2610.07378/) (2026)

## Abstract (arXiv)

We present Hybrid-CSR, a geometric deep-learning model that combines explicit and implicit shape representations for cortical surface reconstruction. Specifically, Hybrid-CSR begins with explicit deformations of template meshes to obtain coarsely reconstructed cortical surfaces, based on which the oriented point clouds are estimated for the subsequent differentiable poisson surface reconstruction. By doing so, our method unifies explicit (oriented point clouds) and implicit (indicator function) cortical surface reconstruction. Compared to explicit representation-based methods, our hybrid approach is more friendly to capture detailed structures, and when compared with implicit representation-based methods, our method can be topology aware because of end-to-end training with a mesh-based deformation module. In order to address topology defects, we propose a new topology correction pipeline that relies on optimization-based diffeomorphic surface registration. Experimental results on three brain datasets show that our approach surpasses existing implicit and explicit cortical surface reconstruction methods in numeric metrics in terms of accuracy, regularity, and consistency.