# High-Fold 3D Gaussian Splitting Model Pruning Method Assisted by Opacity
## 3 Methods
### Gaussian Quantity Compression  
#### 3.1.1 Opacity Update Module  
When a 3D Gaussian Splitting model performs cloning and splitting during the densification phase, the new Gaussian points inherit the opacity of the original Gaussian points. According to the alpha blending principle of rendering, this direct numerical replication can introduce errors in opacity. Especially when fitting a local area with complex colors, reconstruction of this area often requires a large number of Gaussian points to participate in cloning and splitting, resulting in an excessive density of Gaussian points in the local space. This high-density space will aggravate the opacity error during the training iteration process. Inspired by the explanation of the Gaussian point cloning principle in [39], we introduce the updated opacity formula from a new perspective. As shown in Figure 2 left part, suppose a ray of light passes sequentially through two Gaussian points, \(\mathrm{g}_{1}\) and \(\mathrm{g}_{2}\), during rendering. The opacity of Gaussian point \(\mathrm{g}_{1}\) is \(\alpha_{1}\) and its color is \(c_{1}\), while the opacity of Gaussian point \(\mathrm{g}_{2}\) is \(\alpha_{2}\) and its color is \(c_{2}\). According to the alpha blending logic, the opacity parameter is multiplied by the color parameter to weight the color. The Gaussian point further along the ray is affected not only by its own opacity but also by the product of its opacity and the difference between 1 and the total opacity of the preceding points. Therefore, before cloning the Gaussian point, take the left schematic in Figure 2 as an example. The color of this ray rendered to a pixel is calculated as follows:  
\[C=\alpha_{1}\cdot c_{1}+(1-\alpha_{1})\cdot\alpha_{2}\cdot c_{2} \tag{1}\]  
Suppose a new Gaussian point \(\mathrm{g}_{1}^{\prime}\) is cloned from the Gaussian point \(\mathrm{g}_{1}\) along the direction of the ray, resulting in the ray passing through three Gaussian points. Following the original cloning method, as illustrated in the upper right schematic of Figure 2, except for the position, the new and old Gaussian parameters are the same. At this point, \(\alpha_{1}=\alpha_{1}^{\prime}\), \(c_{1}=c_{1}^{\prime}\). Although the Gaussian point \(\mathrm{g}_{2}\) remains unchanged, its color weight along the  
Figure 1: Illustration of the proposed Gaussian compression framework. Firstly, the 3D Gaussian points are initialized using Structure from Motion (SfM). During the densification process, an opacity-based strategy is applied to guide the cloning of Gaussian points. In the pruning stage, a learnable Gaussian mask is introduced to identify and mask redundant Gaussian points, which are subsequently pruned based on their contribution to image rendering at certain intervals. To address the high memory usage of spherical harmonic coefficients, knowledge distillation and vector quantization are applied, resulting in the final compressed model. ray will decrease due to the increased number of preceding Gaussian points. The color blending formula in this case is:  
\[C=\alpha_{1}\cdot c_{1}+(1-\alpha_{1})\cdot\alpha_{1}^{\prime}\cdot c_{1}^{ \prime}+\left[1-\alpha_{1}-(1-\alpha_{1})\cdot\alpha_{1}^{\prime}\right]\cdot \alpha_{2}\cdot c_{2} \tag{2}\]  
Due to the opacity \(\alpha\in[0,1]\), in this primitive cloning method, the coefficient relationship between the color parameters of the before and after Gaussian points g\({}_{2}\) is:  
\[\left[1-\alpha_{1}-(1-\alpha_{1})\cdot\alpha_{1}^{\prime}\right]\cdot\alpha_{ 2}\leq(1-\alpha_{1})\cdot\alpha_{2} \tag{3}\]  
It can be seen that, although the Gaussian point g\({}_{2}\) is not cloned, it is still affected by other Gaussian points during rendering. After cloning Gaussian point g\({}_{1}\), it has less weight in the calculation of color. From a theoretical perspective, the primary purpose of cloning new Gaussian points g\({}_{1}^{\prime}\) is to correct deviations in scale and position. However, simply copying the opacity can lead to cumulative errors in rendering along the ray.