\section{Preliminaries: 3D Gaussian Splatting}
\label{sec:preliminaries}

3DGS~\citep{kerbl2023gaussiansplatting} represents a scene as a set of anisotropic Gaussians $\mathcal{S}=\{\mathcal{G}_i\}_{i=1}^N$, rendered through depth-ordered $\alpha$-compositing. Each Gaussian $\mathcal{G}=(\mathbf{p},\mathbf{o},\mathbf{s},\mathbf{r},\mathbf{k})$ is parameterized by a center $\mathbf{p}\in\mathbb{R}^3$, opacity $\mathbf{o}\in[0,1]$, scale $\mathbf{s}\in\mathbb{R}^3$, quaternion rotation $\mathbf{r}\in\mathbb{R}^4$, and spherical harmonics (SH) coefficients $\mathbf{k}=\{\mathbf{k}^0,\dots,\mathbf{k}^L\}$. The spatial covariance is defined as:
\[
\boldsymbol{\Sigma} = R\,S\,S^\top R^\top,
\]
where $S=\mathrm{diag}(\mathbf{s})$ and $R=\mathrm{quat2rot}(\mathbf{r})$. The opacity at a 3D position $\mathbf{x}$ is given by:
\[
\alpha(\mathbf{x},\mathcal{G}) = \mathbf{o}\cdot \exp\!\Big(-\tfrac{1}{2}(\mathbf{x}-\mathbf{p})^\top \boldsymbol{\Sigma}^{-1}(\mathbf{x}-\mathbf{p})\Big).
\]

Appearance is expressed with SH functions along the viewing direction $\mathbf{d}$:
\[
\mathbf{c}(\mathbf{d},\mathcal{G}) = \mathbf{k}^0 + \sum_{l=1}^L H_l(\mathbf{d})\,\mathbf{k}^l ,
\]
where $H_l$ denotes the SH basis of degree $l$. During rasterization, each 3D Gaussian is projected to a 2D elliptical footprint on the image plane, whose size and orientation are determined by the Jacobian of the projection. The model is initialized with Gaussians from a Structure-from-Motion (SfM) reconstruction~\citep{schonberger2016structure}. It is then optimized with a combination of an $\ell_1$ loss and a DSSIM loss, supplemented by optional depth alignment or opacity regularization terms.
