# Dual-Dimensional Gaussian Splatting Integrating 2D and 3D Gaussians for Surface Reconstruction
## 4 Experiments and Discussion  
### Implementation Details  
We increase the number of Gaussians through adaptive densification, following the 3 DGS strategy, until reaching 15,000 iterations out of the total 30,000 iterations. During the first 12,000 iterations, we determine the dimensionality of each Gaussian every 100 iterations by adaptively selecting between 2D and 3D Gaussians. For the proposed gradient-based regularization terms, we compute image gradients from both the ground-truth and rendered images at every iteration. The rasterization process follows the 2 DGS approach and is performed at every iteration via volumetric alpha blending. We compare DDGS with existing explicit methods, including 3 DGS [9], SuGaR [25], and 2 DGS [10]. To comprehensively evaluate our method, we assess its performance on three datasets across different environments. For surface reconstruction evaluation, we use the DTU dataset [35], which contains multi-view scans of small objects, and the Tanks and Temples (TnT) dataset [36], which provides large-scale 360-degree scenes. For image rendering quality, we use the Mip-NeRF 360 dataset [37], which includes a variety of indoor and outdoor environments. The indoor scenes in Mip-NeRF 360 include bonsai, counter, kitchen, and room, while the outdoor scenes consist of bicycle, flowers, garden, stump, and treehill. We evaluate surface reconstruction using Chamfer Distance and F1 score, and 2D image appearance quality with PSNR, SSIM, and LPIPS. Chamfer Distance measures the average closest-point distance between the reconstructed and ground-truth surfaces. \[\text{CD}(P,G)=\frac{1}{2}\Bigg{(}\frac{1}{|P|}\sum_{p\in P}\min_{g\in G}\|p-g \|^{2}+\frac{1}{|G|}\sum_{g\in G}\min_{p\in P}\|g-p\|^{2}\Bigg{)} \tag{12}\]  
Here, \(P\) is the predicted point set sampled from the reconstructed mesh, and \(G\) is the ground truth point set. The F1 score is the harmonic mean of precision and recall for the reconstructed surfaces. Both metrics are computed using a distance threshold \(d\) between predicted and ground truth points. \[\text{Precision}(d) =\frac{|\{p\in P:\min_{g\in G}\|p-g\|<d\}|}{|P|},\] \[\text{Recall}(d) =\frac{|\{g\in G:\min_{p\in P}\|g-p\|<d\}|}{|G|}, \tag{13}\] \[\text{F1-score}(d) =\frac{2\cdot\text{Precision}(d)\cdot\text{Recall}(d)}{\text{ Precision}(d)+\text{Recall}(d)}\]  
The Peak Signal-to-Noise Ratio (PSNR) quantifies image similarity based on pixel-wise mean squared error. \[\text{PSNR}=10\cdot\log_{10}\Bigg{(}\frac{\text{MAX}_{I}^{2}}{\text{MSE}} \Bigg{)} \tag{14}\]  
Here, \(\text{MAX}_{I}\) denotes the maximum pixel intensity value, and MSE represents the mean squared error between the predicted and ground truth images. The Structural Similarity Index (SSIM) compares images based on luminance, contrast, and structural similarity. \[\text{SSIM}(x,y)=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^ {2}+\mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})} \tag{15}\]  
Here, \(\mu_{x}\) and \(\mu_{y}\) are local means, \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) are variances, and \(\sigma_{xy}\) is the covariance between images \(x\) and \(y\). \(C_{1}\) and \(C_{2}\) are small constants for numerical stability. The Learned Perceptual Image Patch Similarity (LPIPS) measures perceptual similarity using deep features from neural networks. \[\text{LPIPS}(x,x_{0})=\sum_{l}\frac{1}{H_{l}W_{l}}\sum_{h,w}\left\lVert w_{l} \odot\left(\hat{y}_{hw}^{l}-\hat{y}_{0 thr}^{l}\right)\right\rVert_{2}^{2} \tag{16}\]  
Here, \(\hat{y}_{hm}^{l}\) and \(\hat{y}_{0 thr}^{l}\) are the normalized deep features at spatial location \((h,w)\) from layer \(l\) for images \(x\) and \(x_{0}\), respectively, and \(w_{l}\) denotes learned channel-wise weights. \(H_{l}\) and \(W_{l}\) are the spatial dimensions of layer \(l\). Following 2 DGS, we set \(\lambda_{d}\) to 1000 for unbounded scenes, 100 for bounded 360-degree scenes, and 0 for large-scale scenes such as Mip-NeRF 360. Similarly, \(\lambda_{n}\) is fixed to 0.05 across all environments.