# Grassmannian Splatting I: Moving rank-2 Spacetime Surfels for Dynamic Scene Rendering

arXiv: https://arxiv.org/abs/2607.10489 · HTML: https://mlsanity.com/paper/2607.10489/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 14 references that are in the map.
- First posted on arXiv 11 Jul 2026.

## Builds on (most cited first, 12 of 12)

- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [4D Gaussian Splatting for Real-Time Dynamic Scene Rendering](https://mlsanity.com/paper/2310.08528/) (2023)
- [2D Gaussian Splatting for Geometrically Accurate Radiance Fields](https://mlsanity.com/paper/2403.17888/) (2024)
- [Deformable 3D Gaussians for High-Fidelity Monocular Dynamic Scene Reconstruction](https://mlsanity.com/paper/2309.13101/) (2023)
- [Real-time Photorealistic Dynamic Scene Representation and Rendering with 4D Gaussian Splatting](https://mlsanity.com/paper/2310.10642/) (2023)
- [Spacetime Gaussian Feature Splatting for Real-Time Dynamic View Synthesis](https://mlsanity.com/paper/2312.16812/) (2023)
- [4D-Rotor Gaussian Splatting: Towards Efficient Novel View Synthesis for Dynamic Scenes](https://mlsanity.com/paper/2402.03307/) (2024)
- [A Compact Dynamic 3D Gaussian Representation for Real-Time Dynamic View Synthesis](https://mlsanity.com/paper/2311.12897/) (2023)
- [Orientation-anchored Hyper-Gaussian for 4D Reconstruction from Casual Videos](https://mlsanity.com/paper/2509.23492/) (2025)
- [Space-time 2D Gaussian Splatting for Accurate Surface Reconstruction under Complex Dynamic Scenes](https://mlsanity.com/paper/2409.18852/) (2024)
- [SpeeDe3DGS: Speedy Deformable 3D Gaussian Splatting with Temporal Pruning and Motion Grouping](https://mlsanity.com/paper/2506.07917/) (2025)

## Abstract (arXiv)

We introduce Grassmannian splatting, a dynamic scene representation whose primitives are Gaussians supported on 3-planes in spacetime R4: generically, spatial 2-planes in uniform translation along their normals. Each primitive carries a unit normal n ∈ mathbb S3/± 1 cong Gr(3,4) and an unconstrained factor L ∈ mathbb R4 × 3, with covariance [ Sigma4D = (Pn L)(Pn L)T, qquad Pn = I - n nT. ] For generic L and n neq ± e0, conditioning on time returns a rank-2 surfel at every frame. The normal of the disk and its velocity along that normal are read off from n; the disk shape and the tangential drift of its center are set by L. Existing native 4D Gaussian splatting methods [itYang et. al. 2023,Duan et. al. 2024] slice full-rank spacetime covariances, so their per-frame primitive is a volumetric ellipsoid; since conditioning lowers rank by exactly one, a rank-2 surfel in the slice requires a rank-3 spacetime covariance, and the parameterization above realizes exactly these. The motion model is closed form, i.e. no deformation field is learned, and no custom CUDA is required: the conditioned disk feeds a standard 3DGS rasterizer through its precomputed-covariance interface. A soft clamp in the Schur denominator regularizes the static orientation and continuously bridges rank-3 static and rank-2 dynamic behavior, so static and moving primitives form a single continuous family. On the 17 HyperNeRF scenes of MonoDyGauBench, training is fastest among all compared methods (4.9 to 5.6 times faster than the strongest quality baselines), while ranking second in PSNR, MS-SSIM, and LPIPS. Code: https://github.com/PaulCelanCoding/grassmannian-splatting