## 3 Methodology
### 3 DGS-Based Dense Scene Representation  
To optimize the scene representation, we adopt 3D Gaussian Splatting (3 DGS) as the map primitive for high-quality scene reconstruction and novel view synthesis. 3 DGS models the scene as a set of Gaussian primitives distributed in 3D space, leveraging differentiable rendering techniques to achieve image reconstruction and scene optimization for given viewpoints. This representation allows for continuous and smooth blending of 3D structures, making it well suited for dynamic environments where precise reconstruction is critical. Although dynamic 3D points are eliminated during initialization to reduce the system's dependence on segmentation methods, we incorporate dynamic mapping points during the 3 DGS training phase to suppress their impact on reconstruction results. This dynamic adjustment ensures that the final reconstruction is not distorted by moving objects or noisy data. Additionally, we introduce a curling loss function to constrain edge projections and further enhance reconstruction accuracy by preserving fine geometric details, particularly in scenes with complex structures or occlusions. We represent the scene as a collection of Gaussian primitives \(\{G_{i}\}_{i=1}^{n}\). Each Gaussian primitive \(G_{i}\) consists of position, covariance, opacity, and color representation parameters:  
\[G_{i}=(X_{i},\Sigma_{i},\Lambda_{i},S_{i}) \tag{12}\]  
where \(X_{i}\in\mathbb{R}^{3}\) is the point location, i.e., the center of the Gaussian, \(\Sigma_{i}\in\mathbb{R}^{3}\times\mathbb{R}^{3}\) is the symmetric positive definite covariance matrix representing the spatial spread and uncertainty of the Gaussian, \(\Lambda_{i}\in\mathbb{R}^{3}\) is the opacity parameter controlling the transparency of the primitive, and \(S_{i}\in\mathbb{R}^{12}\) represents the spherical harmonics parameters for color features, which allows for the encoding of complex lighting and material properties in the scene. Given the camera pose \(P=\{R_{i}|T_{i}\}\) and intrinsics \(K\), the 3D Gaussians are projected and blended via \(\alpha\)-blending to compute the rendered pixel color \(\hat{C}\) and depth \(\hat{D}\):  
\[\hat{C}=\sum_{i\in\mathcal{N}}c_{i}\prod_{j=1}^{i-1}(1-\alpha_{j}),\quad\hat{D }=\sum_{i\in\mathcal{N}}d_{i}\prod_{j=1}^{i-1}(1-\alpha_{j}) \tag{13}\]  
where \(c_{i}\) and \(d_{i}\) are the respective color and depth contributions of the \(i\)-th Gaussian primitive and \(\mathcal{N}\) is the set of Gaussian indices sorted by depth. This formulation allows for realistic rendering by blending the contributions of multiple Gaussians along the camera's viewing rays, ensuring smooth transitions and accurate depth information in the resulting image. For an image with resolution \(H\times W\), we optimize and update the 3D Gaussian scene representation by minimizing the photometric loss \(\mathcal{L}_{c}\) and geometric depth loss \(\mathcal{L}_{d}\) between the sensor-observed color \(C\) and depth \(D\):  
\[\mathcal{L}_{c}=\sum_{m=1}^{HW}\omega_{m}|C_{m}-\hat{C}_{m}|,\quad\mathcal{L} _{d}=\sum_{m=1}^{HW}\omega_{m}|D_{m}-\hat{D}_{m}|, \tag{14}\]\[\omega_{m}=\exp(-\alpha\cdot(1-M_{j})), \tag{15}\]  
where \(M_{j}\) is a binary mask image that identifies dynamic regions, which is obtained through the method described earlier. Dynamic region identification ensures that the model accounts for moving objects in the scene, helping to maintain accurate static scene representation while minimizing the impact of dynamic disruptions. To further improve reconstruction accuracy, we introduce a curling loss inspired by DIM-SLAM [25]. This novel loss function helps to stabilize the reconstruction process by maintaining consistent geometric features, particularly for fine details such as edges and contours.