\begin{table}[H]
\centering
\caption{Summary of main notations used in this paper.}
\label{tab:notation}
\resizebox{0.75\textwidth}{!}{
\begin{tabular}{cll}
\toprule
Symbol & Type & Description \\
\midrule
$\mathcal{S}=\{\mathcal{G}_i\}_{i=1}^N$ & Set & Learnable set of Gaussians \\
$\mathbf{x}_i \in \mathbb{R}^3$ & Vector & Center (3D position) of Gaussian $\mathcal{G}_i$ \\
$\Sigma_i \in \mathbb{R}^{3\times 3}$ & Matrix & Covariance matrix (from scale and rotation) \\
$o_i \in [0,1]$ & Scalar & Opacity of Gaussian $\mathcal{G}_i$ \\
$\mathbf{k}_i$ & Vector & Spherical Harmonic (SH) coefficients \\
$\mathcal{N}_k(i)$ & Set & $k$ nearest neighbors of node $i$ \\
$d_M(i,j)$ & Scalar & Directional Mahalanobis distance between $i$ and $j$ \\
$A$ & Matrix & Adjacency matrix of symmetrized graph \\
$C_m$ & Set & $m$-th connected component \\
$\mathcal{R}_{\text{prune}}$ & Set & Indices of pruned small components \\
$\alpha_c(u)$ & Scalar & Alpha value at pixel $u$ in camera $c$ \\
$\mathcal{D}(\cdot)$ & Operator & Morphological dilation (binary mask) \\
$\mathbf{1}[\cdot]$ & Operator & Indicator function (1 if true, else 0) \\
$\#$ & Operator & Cardinality (number of elements in set) \\
$\mathcal{I}_\partial$ & Set & Boundary Gaussian indices for a hole \\
$\mathcal{R}_{\text{rare}}$ & Set & Rare semantic class indices \\
$S_{\text{seg}}:\Omega\to\{0,\dots,C-1\}$ & Mapping & Semantic segmentation map over pixel domain $\Omega$ \\
$\mathcal{B}$ & Set & Semantic boundary pixels (4-neighborhood changes) \\
$\Vert$ & Operator & Concatenation of feature vectors \\
$\|\cdot\|_2$ & Operator & Euclidean $\ell_2$ norm \\
$\langle \cdot,\cdot \rangle$ & Operator & Inner product \\
$\mathrm{sg}(\cdot)$ & Operator & Stop-gradient operation \\
$\arg\min^{(k)}$ & Operator & Indices of $k$ smallest elements \\
\bottomrule
\end{tabular}}
\end{table}
