用四维平面高斯表示动态场景，实现高效精确的时变表面渲染。
Grassmannian Splatting I: Moving rank-2 Spacetime Surfels for Dynamic Scene Rendering
- 以时空四维3平面为基本单元，用法向量和自由参数控制动态表面形状与运动。
- 在17个HyperNeRF场景上训练速度比最强基线快4.9~5.6倍，质量仅次于顶尖方法。
- 无需学习形变场或自定义CUDA，静态与动态对象统一建模，兼容标准3DGS渲染器。
我们提出格拉斯曼点阵（Grassmannian splatting），一种动态场景表示方法，其基本单元是定义在时空ℝ⁴中的高斯分布，支撑于3-平面：通常为沿法线匀速平移的空间2-平面。每个原语携带单位法向量 $n \in \mathbb S^3/\{\pm 1\} \cong \mathrm{Gr}(3,4)$ 与无约束因子 $L \in \mathbb R^{4 \times 3}$，协方差为 $Σ_{4\mathrm{D}} = (P_n L)(P_n L)^T$，其中 $P_n = I - n n^T$。对任意非零 $L$ 且 $n \neq \pm e_0$，时间条件化后每帧返回一个秩-2表面元（surfel）。法向量决定圆盘方向，其沿法线的运动速度由 $n$ 决定；圆盘形状及中心切向漂移则由 $L$ 控制。现有4D高斯点阵方法对全秩时空协方差进行切片，导致每帧为体素椭球；由于条件化降秩1，秩-2表面元需秩-3时空协方差，而上述参数化恰好实现此结构。运动模型为闭式表达，无需学习形变场，也无需定制CUDA：条件化后的圆盘通过预计算协方差接口接入标准3DGS光栅化器。通过舒尔分母中的软夹紧项正则化静态方向，并连续衔接秩-3静态与秩-2动态行为，使静态与动态原语构成单一连续族。在MonoDyGauBench的17个HyperNeRF场景上，训练速度为所有对比方法中最快（比最强质量基线快4.9至5.6倍），在PSNR、MS-SSIM和LPIPS指标上排名第二。
原文摘要 · Abstract (English)
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
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