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

We introduce Grassmannian splatting, a dynamic scene representation whose primitives are Gaussians supported on 3-planes in spacetime $\R^4$: generically, spatial 2-planes in uniform translation along their normals. Each primitive carries a unit normal $n \in \mathbb S^3/\{\pm 1\} \cong \mathrm{Gr}(3,4)$ and an unconstrained factor $L \in \mathbb R^{4 \times 3}$, with covariance \[ Σ_{4\mathrm{D}} = (P_n L)(P_n L)^T, \qquad P_n = I - n n^T. \] For generic $L$ and $n \neq \pm e_0$, 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 [\it{Yang 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