## 3 Methods
### Prune Method for Redundancy of Point Clouds  
In submanifold convolutions, the output coordinates \(P_{out}\) are the same as the input coordinates \(P_{in}\) when the input stride \(s\) is 1, as follows. \[P_{out}=P_{in}. \tag{2}\]  
Employing sparse convolution techniques can effectively preserve the data's inherent sparsity, ensuring efficient computation and resource utilization. But when the input stride \(s\) is 2, or has other values, to execute a downsampling operation or max-pooling,  
Figure 5: **(a)** Scalar operation and **(b)** SIMD operation. The computation traditionally requires four cycles to complete; leveraging SIMD reduces this to a single cycle. both background and foreground points dilate simultaneously. According to Equation \(p_{j}=s\cdot q_{k}+\delta\), the output coordinates need to be changed. As shown in Figure 6, the input coordinates must first be offset to derive the output coordinates. Subsequently, potential output points within the expanded neighborhood of the input coordinates are identified. The final output coordinates are then selected according to the stride of the input coordinates, ensuring precise and accurate placement. These can be expressed as follows. \[P_{dilate}=(P_{in}-offset), \tag{3}\]  
\[P_{out}=P_{dilate}\quad\text{if}\quad P_{dilate}\%s=0. \tag{4}\]  
where % represents mod. This process could disrupt the sparsity of the input points because the background points tend to occupy a significant portion of the points, which indicates that background points will dilate more quickly than foreground points. As illustrated in Figure 3, sparse convolution relies on a hash table that establishes the relationship between input and output coordinates, thereby facilitating computations. The faster the expansion of background points, the more significant the increase in the length of the hash table, leading to a rapid growth in the computational load of the entire sparse convolution. This escalation ultimately undermines the sparsity of the point cloud data, thereby negating the advantages of sparse convolution, as shown in Figure 7.  
Figure 6: Downsampling or maxpooling operation. Here, the input stride is 2. Figure 7: Comparison of pruned and unpruned models. If foreground and background are equally treated, the sparsity will be compromised, and the computational complexity will increase significantly. In the domain of 2D image analysis, one approach to enhancing computational speed and efficiency involves generating masks to flag features that should be ignored [22; 23; 24; 25]. This motivates us to investigate analogous mask application techniques to 3D point clouds. Inspired by the unequal diffusion rate, we propose a pruning method based on the difference of coordinates in adjacent keyframes during the downsampling operation. The difference in coordinates is defined as follows. \[\Deltacoords_{t,m}=\ coords_{t,m}\ -\ coords_{t-1,m}. \tag{5}\]  
where \(coords_{t,m}=(x_{t,m},\ y_{t,m},\ z_{t,m}),\ t>1,\ t\) represents the coordinates at the \(t\)-timeslot, and \(m\) denotes the index of the point. The difference between background points tends to be small, and foreground points are the opposite. So, we define a mask based on the coordinate difference to decide which point is more likely to be the background point, as follows:  
\[Mask=\left\{ \begin{array}{ll}True,&\ \ \text{if}\ |\Deltacoords_{t,m}|>T,\\ False,&\ \ \text{otherwise}.\end{array} \right. \tag{6}\]  
\(|\Deltacoords_{t,m}|\) represents the sum of squared coordinate differences and is expressed as follows:  
\[|\Deltacoords_{t,m}|= (x_{t,m}\ -\ x_{t-1,m})^{2}+(y_{t,m}\ -\ y_{t-1,m})^{2} \tag{7}\] \[+ (z_{t,m}\ -\ z_{t-1,m})^{2}.\]  
This implies that for any given point \(m\) at \(t\)-timeslot, the coordinate displacement is measured as \(|\Deltacoords_{t,m}|\), and a smaller displacement corresponds to a smaller \(|\Deltacoords_{t,m}|\).