# Dynamic Downsampling Algorithm for 3D Point Cloud Map Based on Voxel Filtering
## 3 Preliminaries  
Voxel downsampling is a method for reducing the density of point clouds while preserving their structural information. In voxel downsampling, \(p_{i}\) denotes the \(i^{th}\) point in a point cloud of \(n\) points, with coordinates \((x_{i},y_{i},z_{i})\). The process divides the cloud into cubic voxels of edge length \(l\), assigning each point to a voxel with center coordinates \((x_{c},y_{c},z_{c})\). The floor function, \(\lfloor\cdot\rfloor\), ensures that points are accurately grouped by rounding down their coordinates to the nearest voxel center, facilitating efficient downsampling while retaining essential spatial information. It involves dividing the point cloud into equally sized cubic voxel blocks with a side length of \(l\). Overall, the process includes three steps:  
1. Divide the point cloud into cubic voxel blocks with a side length of \(l\), allowing for the calculation of the center coordinates of each voxel block, denoted by \((x_{c},y_{c},z_{c})\), as follows: \[ \begin{split} x_{c}&=l\cdot\left\lfloor\frac{x_{i}} {l}\right\rfloor+\frac{l}{2}\\ y_{c}&=l\cdot\left\lfloor\frac{y_{i}}{l}\right\rfloor+ \frac{l}{2}\\ z_{c}&=l\cdot\left\lfloor\frac{z_{i}}{l}\right\rfloor+ \frac{l}{2}\end{split} \] (1)
2. For each voxel block, select a representative point based on a chosen strategy, such as the point closest to the center or the one closest to the plane. 3. Combine all the selected representative points to create a new point cloud, which represents the downsampled result. The process of voxel downsampling is illustrated in Figure 4. By adjusting the side length (\(l\)) of the voxel blocks, we can control the density of the downsampled point cloud. A smaller \(l\) value retains more representative points and results in higher point cloud density, while a larger \(l\) value retains fewer representative points and leads to lower point cloud density. Therefore, in practical applications, the appropriate \(l\) value should be selected based on particular requirements. Classical voxel filtering is a commonly used point cloud denoising method and is based on the idea of segmenting the point cloud into regular voxel grids and averaging the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Advantages** & **Disadvantages** \\ \hline \multirow{4}{*}{Cluster downsampling} & (1) It can effectively eliminate noise and redundant information while preserving local details. (2) It performs better on point clouds with non-uniform distributions. & It performs poorly in processing high-density point clouds and is high-density point clouds and is prone to creating voids. \\ \hline \multirow{4}{*}{Voxel downsampling} & (1) Simple and easy to implement, its sampling density can be controlled by adjusting the voxel size. (2) It can preserve the overall shape and structural characteristics of the original point cloud. & It cannot handle local detail information and may introduce noisy points. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._points within each voxel. While this method effectively reduces point cloud noise, it has the following limitations:  
* Loss of point cloud map information:Voxel filtering essentially involves the sampling and downsampling of point clouds. However, due to retaining only a single average value within each grid cell, it suffers from information loss. In applications where precise reconstruction and analysis of point cloud data are required, this information loss can lead to significant errors. * Point cloud map blurring: Voxel filtering applies averaging operations to the points within each grid cell, potentially blurring fine details on the surfaces of objects in the point cloud map. This may pose issues in applications that require the preservation of detailed surface information.