# CaDeX: Learning Canonical Deformation Coordinate Space for Dynamic Surface Representation via Neural Homeomorphism

arXiv: https://arxiv.org/abs/2203.16529 · HTML: https://mlsanity.com/paper/2203.16529/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 61 references that are in the map.
- First posted on arXiv 30 Mar 2022.

## Builds on (most cited first, 30 of 48)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Implicit Functions in Feature Space for 3D Shape Reconstruction and Completion](https://mlsanity.com/paper/2003.01456/) (2020)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [SNARF: Differentiable Forward Skinning for Animating Non-Rigid Neural Implicit Shapes](https://mlsanity.com/paper/2104.03953/) (2021)
- [SCANimate: Weakly Supervised Learning of Skinned Clothed Avatar Networks](https://mlsanity.com/paper/2104.03313/) (2021)
- [Embodied Hands: Modeling and Capturing Hands and Bodies Together](https://mlsanity.com/paper/2201.02610/) (2022)
- [NASA: Neural Articulated Shape Approximation](https://mlsanity.com/paper/1912.03207/) (2019)
- [4DComplete: Non-Rigid Motion Estimation Beyond the Observable Surface](https://mlsanity.com/paper/2105.01905/) (2021)
- [Shape2Motion: Joint Analysis of Motion Parts and Attributes from 3D Shapes](https://mlsanity.com/paper/1903.03911/) (2019)
- [Deformed Implicit Field: Modeling 3D Shapes with Learned Dense Correspondence](https://mlsanity.com/paper/2011.13650/) (2020)
- [A-SDF: Learning Disentangled Signed Distance Functions for Articulated Shape Representation](https://mlsanity.com/paper/2104.07645/) (2021)
- [i3DMM: Deep Implicit 3D Morphable Model of Human Heads](https://mlsanity.com/paper/2011.14143/) (2020)

## Built on by (most cited first, 30 of 57)

- [Tracking Everything Everywhere All at Once](https://mlsanity.com/paper/2306.05422/) (2023)
- [SpatialTracker: Tracking Any 2D Pixels in 3D Space](https://mlsanity.com/paper/2404.04319/) (2024)
- [CARTO: Category and Joint Agnostic Reconstruction of ARTiculated Objects](https://mlsanity.com/paper/2303.15782/) (2023)
- [Neural Surface Reconstruction of Dynamic Scenes with Monocular RGB-D Camera](https://mlsanity.com/paper/2206.15258/) (2022)
- [MoDGS: Dynamic Gaussian Splatting from Casually-captured Monocular Videos with Depth Priors](https://mlsanity.com/paper/2406.00434/) (2024)
- [Reconstructing Animatable Categories from Videos](https://mlsanity.com/paper/2305.06351/) (2023)
- [REACTO: Reconstructing Articulated Objects from a Single Video](https://mlsanity.com/paper/2404.11151/) (2024)
- [Flow supervision for Deformable NeRF](https://mlsanity.com/paper/2303.16333/) (2023)
- [PhysCtrl: Generative Physics for Controllable and Physics-Grounded Video Generation](https://mlsanity.com/paper/2509.20358/) (2025)
- [Dynamic Point Fields](https://mlsanity.com/paper/2304.02626/) (2023)
- [CASA: Category-agnostic Skeletal Animal Reconstruction](https://mlsanity.com/paper/2211.03568/) (2022)
- [MonoNPHM: Dynamic Head Reconstruction from Monocular Videos](https://mlsanity.com/paper/2312.06740/) (2023)
- [Motion2VecSets: 4D Latent Vector Set Diffusion for Non-rigid Shape Reconstruction and Tracking](https://mlsanity.com/paper/2401.06614/) (2024)
- [K-VIL: Keypoints-based Visual Imitation Learning](https://mlsanity.com/paper/2209.03277/) (2022)
- [NAP: Neural 3D Articulation Prior](https://mlsanity.com/paper/2305.16315/) (2023)
- [Banana: Banach Fixed-Point Network for Pointcloud Segmentation with Inter-Part Equivariance](https://mlsanity.com/paper/2305.16314/) (2023)
- [EFEM: Equivariant Neural Field Expectation Maximization for 3D Object Segmentation Without Scene Supervision](https://mlsanity.com/paper/2303.15440/) (2023)
- [Diffusion Priors for Dynamic View Synthesis from Monocular Videos](https://mlsanity.com/paper/2401.05583/) (2024)
- [LEIA: Latent View-invariant Embeddings for Implicit 3D Articulation](https://mlsanity.com/paper/2409.06703/) (2024)
- [Track Everything Everywhere Fast and Robustly](https://mlsanity.com/paper/2403.17931/) (2024)
- [Zero-shot Pose Transfer for Unrigged Stylized 3D Characters](https://mlsanity.com/paper/2306.00200/) (2023)
- [Relightable and Animatable Neural Avatars from Videos](https://mlsanity.com/paper/2312.12877/) (2023)
- [Neural Vector Fields for Implicit Surface Representation and Inference](https://mlsanity.com/paper/2204.06552/) (2022)
- [DPHMs: Diffusion Parametric Head Models for Depth-based Tracking](https://mlsanity.com/paper/2312.01068/) (2023)
- [textttNePhi: Neural Deformation Fields for Approximately Diffeomorphic Medical Image Registration](https://mlsanity.com/paper/2309.07322/) (2023)
- [DynoSurf: Neural Deformation-based Temporally Consistent Dynamic Surface Reconstruction](https://mlsanity.com/paper/2403.11586/) (2024)
- [OP-Align: Object-level and Part-level Alignment for Self-supervised Category-level Articulated Object Pose Estimation](https://mlsanity.com/paper/2408.16547/) (2024)
- [NAISR: A 3D Neural Additive Model for Interpretable Shape Representation](https://mlsanity.com/paper/2303.09234/) (2023)
- [Fluid Dynamics Network: Topology-Agnostic 4D Reconstruction via Fluid Dynamics Priors](https://mlsanity.com/paper/2303.09871/) (2023)
- [Dynamic 3D Point Cloud Sequences as 2D Videos](https://mlsanity.com/paper/2403.01129/) (2024)

## Abstract (arXiv)

While neural representations for static 3D shapes are widely studied, representations for deformable surfaces are limited to be template-dependent or lack efficiency. We introduce Canonical Deformation Coordinate Space (CaDeX), a unified representation of both shape and nonrigid motion. Our key insight is the factorization of the deformation between frames by continuous bijective canonical maps (homeomorphisms) and their inverses that go through a learned canonical shape. Our novel deformation representation and its implementation are simple, efficient, and guarantee cycle consistency, topology preservation, and, if needed, volume conservation. Our modelling of the learned canonical shapes provides a flexible and stable space for shape prior learning. We demonstrate state-of-the-art performance in modelling a wide range of deformable geometries: human bodies, animal bodies, and articulated objects.