# Gaussian-UDSR: Real-Time Unbounded Dynamic Scene Reconstruction with 3D Gaussian Splitting
## 3 Method
### Gaussian Model  
The basic principle of Gaussian splatting is shown in Figure 2, where we initialize the SfM point cloud and the LiDAR point cloud as a cluster of Gaussian ellipsoids in 3D space and classify them into static background models and dynamic object models. All the Gaussian ellipsoids jointly affect the pixels on the corresponding raster, and a large number of ellipsoids perform a throwing process directly to the view plane, which leads to fast and high-quality rendering and building of complex scenes. Static background model. The static background model is represented as a set of three-dimensional Gaussian points in the world coordinate system, and the covariance matrix \(\Sigma_{s}\in R^{3\times 3}\), the positional mean \(\mu_{s}\in R^{3}\), the opacity \(\alpha_{s}\), and the color \(c_{s}\) jointly influence its shape expression:  
\[G_{s}(x)=e^{-\frac{1}{2}(x-\mu_{s})^{T}\Sigma_{s}-1}(x-\mu_{s}) \tag{1}\]  
Meanwhile, the covariance matrix \(\Sigma_{s}\) can be decomposed into scaling matrix \(S_{s}\) and rotation matrix \(R_{s}\), where \(S_{s}\) is represented by diagonal elements and \(R_{s}\) is represented by unit quaternion. Quaternions are a compact and numerically stable representation of 3D rotations, defined by a four-dimensional vector (\(q_{w}\), \(q_{x}\), \(q_{y}\), \(q_{z}\)) subject to the unit norm constraint. Compared to Euler angles, quaternions avoid gimbal lock and provide smooth interpolation (e.g., via SLERP), which is crucial for continuous pose tracking in dynamic scenes. Moreover, quaternions are more efficient and numerically stable than rotation matrices, as they require fewer parameters and avoid the need for orthonormalization. These advantages make quaternions particularly suitable for representing and optimizing camera and object orientations in our dynamic scene reconstruction framework. As shown in Figure 3, the unit sphere in the left figure is transformed into the ellipsoid in the right figure, and Equation (2) describes this transformation process. The rotation matrix \(R_{S}\) changes the orientation of the unit sphere, while the scaling matrix \(S_{S}\) scales it. Through rotation and scaling operations, the originally isotropic unit sphere is transformed into  
Figure 2: 3D Gaussians splatting schematic. an anisotropic ellipsoid, which can more flexibly describe the multivariate correlations in complex environments. The covariance matrix \(\Sigma_{s}\) can be expressed as  
\[\Sigma_{s}=R_{s}S_{s}S_{s}^{T}R_{s}^{T} \tag{2}\]  
In addition to this, the rendering needs to project each Gaussian ellipsoid from the 3D space to the 2D view plane in order to obtain the image in the specified view direction, and the process of changing the projection is a nonlinear approximation, which can be expressed as a local linear approximation of the multivariate function at a point by the Jacobian matrix \(J\). As shown in Figure 4, the Jacobian matrix \(J\) (Equation (3)) in the projection model characterizes the local linear transformation relationship from 3D spatial coordinates to a 2D projection plane, with its elements consisting of the partial derivatives of the projection coordinates with respect to the spatial coordinates. Specifically, the elements \(\frac{df_{1}}{dx}=\frac{n_{1}}{z}\) and \(\frac{df_{2}}{dy}=\frac{n_{2}}{z}\) in the matrix describe the reciprocal relationship between the scaling factors in the x and y directions and the depth z, reflecting the linear response of the projection coordinates to the spatial positions. In contrast, \(\frac{df_{1}}{dz}=-\frac{n_{1}x}{z^{2}}\) and \(\frac{df_{2}}{dz}=-\frac{n_{2}y}{z^{2}}\) capture the nonlinear perspective contraction effect of depth changes on the projections in the x and y directions, whose absolute values increase as the spatial points move away from the optical center (i.e., as z decreases). Additionally, \(\frac{df_{2}}{dz}=-\frac{nf}{z^{2}}\) quantifies the attenuation of the scaling factor by depth, revealing the nonlinear degradation of depth information during the projection process.