# Spline Deformation Field

arXiv: https://arxiv.org/abs/2507.07521 · HTML: https://mlsanity.com/paper/2507.07521/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 34 references that are in the map.
- First posted on arXiv 10 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [4D Gaussian Splatting for Real-Time Dynamic Scene Rendering](https://mlsanity.com/paper/2310.08528/) (2023)
- [Deformable 3D Gaussians for High-Fidelity Monocular Dynamic Scene Reconstruction](https://mlsanity.com/paper/2309.13101/) (2023)
- [K-Planes: Explicit Radiance Fields in Space, Time, and Appearance](https://mlsanity.com/paper/2301.10241/) (2023)
- [HexPlane: A Fast Representation for Dynamic Scenes](https://mlsanity.com/paper/2301.09632/) (2023)
- [SC-GS: Sparse-Controlled Gaussian Splatting for Editable Dynamic Scenes](https://mlsanity.com/paper/2312.14937/) (2023)
- [Gaussian-Flow: 4D Reconstruction with Dynamic 3D Gaussian Particle](https://mlsanity.com/paper/2312.03431/) (2023)
- [4DComplete: Non-Rigid Motion Estimation Beyond the Observable Surface](https://mlsanity.com/paper/2105.01905/) (2021)
- [NeRF-DS: Neural Radiance Fields for Dynamic Specular Objects](https://mlsanity.com/paper/2303.14435/) (2023)
- [SplatFields: Neural Gaussian Splats for Sparse 3D and 4D Reconstruction](https://mlsanity.com/paper/2409.11211/) (2024)
- [ResFields: Residual Neural Fields for Spatiotemporal Signals](https://mlsanity.com/paper/2309.03160/) (2023)
- [Neural Spline Fields for Burst Image Fusion and Layer Separation](https://mlsanity.com/paper/2312.14235/) (2023)
- [Divergence-Free Shape Interpolation and Correspondence](https://mlsanity.com/paper/1806.10417/) (2018)
- [Degrees of Freedom Matter: Inferring Dynamics from Point Trajectories](https://mlsanity.com/paper/2406.03625/) (2024)

## Built on by (most cited first, 3 of 3)

- [BiMotion: B-spline Motion for Text-guided Dynamic 3D Character Generation](https://mlsanity.com/paper/2602.18873/) (2026)
- [Progressive Pose-Guided 4D Animal Reconstruction from Monocular Video](https://mlsanity.com/paper/2607.00157/) (2026)
- [Cepheid: Decoupling Scale from Structure for Cross-Family Motion Completion](https://mlsanity.com/paper/2610.18871/) (2026)

## Abstract (arXiv)

Trajectory modeling of dense points usually employs implicit deformation fields, represented as neural networks that map coordinates to relate canonical spatial positions to temporal offsets. However, the inductive biases inherent in neural networks can hinder spatial coherence in ill-posed scenarios. Current methods focus either on enhancing encoding strategies for deformation fields, often resulting in opaque and less intuitive models, or adopt explicit techniques like linear blend skinning, which rely on heuristic-based node initialization. Additionally, the potential of implicit representations for interpolating sparse temporal signals remains under-explored. To address these challenges, we propose a spline-based trajectory representation, where the number of knots explicitly determines the degrees of freedom. This approach enables efficient analytical derivation of velocities, preserving spatial coherence and accelerations, while mitigating temporal fluctuations. To model knot characteristics in both spatial and temporal domains, we introduce a novel low-rank time-variant spatial encoding, replacing conventional coupled spatiotemporal techniques. Our method demonstrates superior performance in temporal interpolation for fitting continuous fields with sparse inputs. Furthermore, it achieves competitive dynamic scene reconstruction quality compared to state-of-the-art methods while enhancing motion coherence without relying on linear blend skinning or as-rigid-as-possible constraints.