# Degrees of Freedom Matter: Inferring Dynamics from Point Trajectories

arXiv: https://arxiv.org/abs/2406.03625 · HTML: https://mlsanity.com/paper/2406.03625/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 75 references that are in the map.
- First posted on arXiv 5 Jun 2024.

## Builds on (most cited first, 30 of 46)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Expressive Body Capture: 3D Hands, Face, and Body from a Single Image](https://mlsanity.com/paper/1904.05866/) (2019)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [AMASS: Archive of Motion Capture as Surface Shapes](https://mlsanity.com/paper/1904.03278/) (2019)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [Nerfies: Deformable Neural Radiance Fields](https://mlsanity.com/paper/2011.12948/) (2020)
- [FlowNet: Learning Optical Flow with Convolutional Networks](https://mlsanity.com/paper/1504.06852/) (2015)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [End-to-end Recovery of Human Shape and Pose](https://mlsanity.com/paper/1712.06584/) (2017)
- [CoTracker: It is Better to Track Together](https://mlsanity.com/paper/2307.07635/) (2023)
- [HumanNeRF: Free-viewpoint Rendering of Moving People from Monocular Video](https://mlsanity.com/paper/2201.04127/) (2022)
- [Advances in Neural Rendering](https://mlsanity.com/paper/2111.05849/) (2021)
- [HuMoR: 3D Human Motion Model for Robust Pose Estimation](https://mlsanity.com/paper/2105.04668/) (2021)
- [Learning Category-Specific Mesh Reconstruction from Image Collections](https://mlsanity.com/paper/1803.07549/) (2018)
- [3D Menagerie: Modeling the 3D shape and pose of animals](https://mlsanity.com/paper/1611.07700/) (2016)
- [DynIBaR: Neural Dynamic Image-Based Rendering](https://mlsanity.com/paper/2211.11082/) (2022)
- [SNARF: Differentiable Forward Skinning for Animating Non-Rigid Neural Implicit Shapes](https://mlsanity.com/paper/2104.03953/) (2021)
- [BANMo: Building Animatable 3D Neural Models from Many Casual Videos](https://mlsanity.com/paper/2112.12761/) (2021)
- [3DGS-Avatar: Animatable Avatars via Deformable 3D Gaussian Splatting](https://mlsanity.com/paper/2312.09228/) (2023)
- [Tracking Everything Everywhere All at Once](https://mlsanity.com/paper/2306.05422/) (2023)
- [SCANimate: Weakly Supervised Learning of Skinned Clothed Avatar Networks](https://mlsanity.com/paper/2104.03313/) (2021)
- [3D-CODED : 3D Correspondences by Deep Deformation](https://mlsanity.com/paper/1806.05228/) (2018)
- [BundleSDF: Neural 6-DoF Tracking and 3D Reconstruction of Unknown Objects](https://mlsanity.com/paper/2303.14158/) (2023)
- [ARAH: Animatable Volume Rendering of Articulated Human SDFs](https://mlsanity.com/paper/2210.10036/) (2022)
- [4DComplete: Non-Rigid Motion Estimation Beyond the Observable Surface](https://mlsanity.com/paper/2105.01905/) (2021)
- [We are More than Our Joints: Predicting how 3D Bodies Move](https://mlsanity.com/paper/2012.00619/) (2020)
- [SimPoE: Simulated Character Control for 3D Human Pose Estimation](https://mlsanity.com/paper/2104.00683/) (2021)

## Built on by (most cited first, 5 of 5)

- [SplatFields: Neural Gaussian Splats for Sparse 3D and 4D Reconstruction](https://mlsanity.com/paper/2409.11211/) (2024)
- [Spline Deformation Field](https://mlsanity.com/paper/2507.07521/) (2025)
- [SmallGS: Gaussian Splatting-based Camera Pose Estimation for Small-Baseline Videos](https://mlsanity.com/paper/2504.17810/) (2025)
- [GrowFields: Compositional 4D Neural Fields for Topology-Changing Plant Growth](https://mlsanity.com/paper/2607.03330/) (2026)
- [Cepheid: Decoupling Scale from Structure for Cross-Family Motion Completion](https://mlsanity.com/paper/2610.18871/) (2026)

## Abstract (arXiv)

Understanding the dynamics of generic 3D scenes is fundamentally challenging in computer vision, essential in enhancing applications related to scene reconstruction, motion tracking, and avatar creation. In this work, we address the task as the problem of inferring dense, long-range motion of 3D points. By observing a set of point trajectories, we aim to learn an implicit motion field parameterized by a neural network to predict the movement of novel points within the same domain, without relying on any data-driven or scene-specific priors. To achieve this, our approach builds upon the recently introduced dynamic point field model that learns smooth deformation fields between the canonical frame and individual observation frames. However, temporal consistency between consecutive frames is neglected, and the number of required parameters increases linearly with the sequence length due to per-frame modeling. To address these shortcomings, we exploit the intrinsic regularization provided by SIREN, and modify the input layer to produce a spatiotemporally smooth motion field. Additionally, we analyze the motion field Jacobian matrix, and discover that the motion degrees of freedom (DOFs) in an infinitesimal area around a point and the network hidden variables have different behaviors to affect the model's representational power. This enables us to improve the model representation capability while retaining the model compactness. Furthermore, to reduce the risk of overfitting, we introduce a regularization term based on the assumption of piece-wise motion smoothness. Our experiments assess the model's performance in predicting unseen point trajectories and its application in temporal mesh alignment with guidance. The results demonstrate its superiority and effectiveness. The code and data for the project are publicly available: urlhttps://yz-cnsdqz.github.io/eigenmotion/DOMA/