发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对3D高斯点云构建等变架构的表示瓶颈问题，提出基于表示理论的统一解决方案，通过将光度学与几何张量同构等方法，引入统一矩阵嵌入，构建E3DGS架构，提升了相机帧变化下的鲁棒性和数据效率。
AI 中文摘要
3D高斯点云（3DGS）通过将显式几何（位置、协方差）与视图相关的光度学（球谐函数）相结合来捕捉场景。然而，在这些基元上构建$\mathrm{SE}(3)$等变架构存在基本的表示瓶颈。颜色一直被视为信号而非几何实体，在相机帧变化时统一几何和外观的对称性并非易事。平移由相对坐标处理，而旋转对不同属性的作用方式不同。这种不匹配使严格等变性变得复杂，导致现有方法要么丢弃要么扁平化SH系数，从而破坏对称性。我们提出了一种基于表示理论的统一解决方案：对于SH度数$\ell\leq2$，光度学在代数上与秩为2的几何张量同构。我们证明了对这些SH系数的维格纳-$D$作用可以精确地重新表述为对$3\times3$矩阵的共轭作用。利用这一点，我们引入了统一矩阵嵌入，将所有高斯属性映射到统一的载体空间$\mathfrak{gl}(3)$。基于“颜色即几何”的公式，我们提出了E3DGS，一种无需克莱布施-戈尔丹张量积就能处理3D高斯的刚体（$\mathrm{SE}(3)$）等变架构。在物体视觉和动作条件高斯世界建模上的评估表明，我们的统一方法在相机帧变化下具有很强的鲁棒性，并提高了数据效率。
英文摘要
3D Gaussian Splatting (3DGS) captures scenes by coupling explicit geometry (position, covariance) with view-dependent photometry (Spherical Harmonics). However, building $\mathrm{SE}(3)$-equivariant architectures on these primitives presents a fundamental representation bottleneck. Color has been treated as a signal rather than a geometric entity, making it nontrivial to unify symmetry across geometry and appearance as the camera frame changes. While translations are handled by relative coordinates, rotations act heterogeneously across attributes: $μ\mapsto Rμ$, $Σ\mapsto RΣR^\top$, and $f_\ell\mapsto D^\ell(R)f_\ell$. This mismatch complicates strict equivariance, leading existing methods to either discard or flatten SH coefficients, thereby breaking symmetry. We propose a unified solution rooted in representation theory: for SH degrees $\ell\le2$, photometry is algebraically isomorphic to a rank-2 geometric tensor. We prove that the Wigner-$D$ action on these SH coefficients can be exactly reformulated as the conjugation action on $3\times3$ matrices. Leveraging this, we introduce the Unified Matrix Embedding, a lifting that maps all Gaussian attributes into a unified carrier space, $\mathfrak{gl}(3)$. Building on the "Color-as-Geometry" formulation, we present E3DGS, a rigid-body ($\mathrm{SE}(3)$) equivariant architecture that processes 3D Gaussians without Clebsch-Gordan tensor products. Evaluations on object vision and action-conditioned Gaussian world modeling demonstrate that our unified approach yields strong robustness under camera-frame changes and improved data efficiency.
Comments31 pages, 7 figures, 5 tables
