# Dual-Dimensional Gaussian Splatting Integrating 2D and 3D Gaussians for Surface Reconstruction
## 3 Methods
### Optimization  
We adopt the depth distortion and normal consistency losses from 2 DGS, and incorporate new regularization terms into our method. In Gaussian splatting, when Gaussians at different depths contribute to the same pixel during blending, the estimated surface may deviate from the true geometry. The depth distortion loss encourages Gaussians to concentrate around the true surface, thereby reducing depth inconsistencies and improving geometric alignment.  
\[\mathcal{L}_{d}=\sum_{i,j}\omega_{i}\omega_{j}\big{|}z_{i}-z_{j}\big{|} \tag{8}\]  
Here, \(\omega_{i}\) and \(\omega_{j}\) denote the blending weights at the \(i\)-th and \(j\)-th intersections, respectively, and \(z_{i}\), \(z_{j}\) represent their corresponding depths.  
The normal consistency loss is effective in aligning Gaussians with the underlying surface geometry. It encourages the normal of each Gaussian to align with the surface normal estimated from the depth map gradient. However, estimating surface normals from nearby depth points can lead to errors in regions with high depth variation. These errors can result in excessive smoothing of the reconstructed surface. To address this issue, we introduce a gradient-aware normal consistency weight \(\omega_{grad}\) to modulate the normal consistency loss:  
\[\begin{split}\mathcal{L}_{n}^{*}=\sum_{i}\omega_{i}\omega_{ grad}\Big{(}1-\mathbf{n}_{i}^{T}\mathbf{N}\Big{)},\quad\text{where}\;\omega_{ grad}=\exp(-\|\
abla I\|),\\ \mathbf{n}_{i}=\mathbf{R}\cdot\text{OneHot}(\arg\min(s_{u},s_{v },s_{w})).\end{split} \tag{9}\]  
Here, \(\mathbf{n}_{i}\) denotes the normal direction of the \(i\)-th Gaussian, computed by identifying the axis with the smallest scaling factor and applying the corresponding rotation, following the approach used in prior work [10; 27; 28]. In the case of 2D Gaussians, the normal direction is inherently determined by the axis associated with the zero-valued scaling factor. \(\mathbf{N}\) is the surface normal estimated from the depth map, and \(\
abla I\) represents the image gradient. \(\omega_{grad}\) assigns lower weights to regions with strong image gradients, facilitating more accurate reconstruction of edge regions.  
To enhance the representation of edge regions, we introduce both first-order and second-order gradient losses. The first-order loss penalizes image gradient differences, while the second-order loss captures curvature variations.  
\[\begin{split}\mathcal{L}_{g1}&=\big{\|}\
abla I- \
abla I\big{\|}_{1},\\ \mathcal{L}_{g2}&=\big{\|}\
abla^{2}I-\
abla^{2}I \big{\|}_{1}\end{split} \tag{10}\]  
Here, \(I\) and \(I\) denote the ground-truth and predicted images, respectively. These losses optimize learnable Gaussian parameters, leading to improved reconstruction of both edges and curvature, and better alignment with underlying surfaces.  
Finally, we optimize the following loss function:  
\[\mathcal{L}=\mathcal{L}_{c}+\lambda_{d}\mathcal{L}_{d}+\lambda_{n}\mathcal{L} _{n}^{*}+\lambda_{g1}\mathcal{L}_{g1}+\lambda_{g2}\mathcal{L}_{g2}. \tag{11}\]  
Here, \(\mathcal{L}_{c}\) is the appearance loss defined in 3 DGS, consisting of a weighted sum of the PSNR and SSIM losses, with weights of 0.8 and 0.2, respectively.