# Point Cloud Upsampling Algorithm: A Systematic Review
## 2 Benchmark
### Evaluation Metrics  
The current evaluation metrics mainly evaluate the quality of the upsampled point cloud from two aspects: the deviation between the ground truth and the generated point cloud and the uniformity of the generated point cloud. **Surface Deviation (SD)**[5]. This is defined to find the closest point y on the mesh for each predicted point x; the distance between them is then calculated, and the mean and standard deviation are finally computed over all the points. **Chamfer Distance (CD)**[16]. This is the sum of positive distances, which is defined as an unsigned distance function, usually the distance between two curves or two 2D images. Applying CD to 3D space, it is defined as follows:  
\[d_{CD}(S_{1},S_{2})=\frac{1}{S_{1}}\sum_{x\in S_{1}}\min_{y\in S_{2}}\lVert x -y\rVert_{2}^{2}+\frac{1}{S_{2}}\sum_{y\in S_{2}}\min_{x\in S_{1}}\lVert y-x \rVert_{2}^{2} \tag{1}\]  
where \(S_{1}\), \(S_{2}\) represent two sets of 3D point clouds respectively. The formula represents the sum of the minimum distance from any point x to \(S_{2}\) in \(S_{1}\), plus the sum of the minimum distance from any point y to \(S_{1}\) in \(S_{2}\). **Hausdorff Distance (HD)**[17]. HD measures the distance between proper subsets in the metric space. A proper subset is defined as a finite (possibly infinite) set of numbers of elements (points). HD distance can be viewed as the maximum value of the shortest distance from a point set to another, and is defined as follows:  
\[d_{H}(S_{1},S_{2})=\max[\sup_{x\in S_{1}y\in S_{2}}d(x,y),\sup_{y\in S_{2}x \in S_{1}}d(x,y)] \tag{2}\]  
where sup and inf define the supremum and infimum calculations. **Earth Mover's Distance (EMD)**[18]. EMD is the histogram similarity measure based on the efficiency of transportation problems. It is a normalized minimum cost of changing from one distribution to another. It measures, in a certain feature space, the difference between two multi-dimensional distributions. It is defined as follows:  
\[d_{EMD}(S_{1},S_{2})=\min_{\phi\in S_{1}\to S_{2}}\sum_{x\in S_{1}} \lVert x-\phi(x)\,\lvert\lvert_{2} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Samples** & **Training** & **Test** & **Type** & **Representation** \\ \hline \hline \multicolumn{1}{c}{ModelNet10 [7]} & 4899 & 3991 & 605 & Synthetic & Mesh \\ \hline \multicolumn{1}{c}{ModelNet40 [7]} & 12,311 & 9843 & 2468 & Synthetic & Mesh \\ \hline \multicolumn{1}{c}{ShapeNet [8]} & 51,190 & - & - & Synthetic & Mesh \\ \hline \multicolumn{1}{c}{SHREC15 [9]} & 1200 & - & - & Synthetic & Mesh \\ \hline \multicolumn{1}{c}{FAUST [10]} & 300 & 100 & 200 & Real-world & Mesh \\ \hline \multicolumn{1}{c}{ScanObjectNN [11]} & 2902 & 2321 & 581 & Real-world & Point Clouds \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical point cloud datasets for upsampling. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Samples** & **Training** & **Test** & **Type** & **Representation** \\ \hline \hline \multicolumn{1}{c}{PU-Net [5]} & 60 & 40 & 20 & Synthetic & Mesh \\ \hline \hline \multicolumn{1}{c}{EC-Net [12]} & 36 & 36 & - & Synthetic & CAD \\ \hline \hline \multicolumn{1}{c}{PU-GAN [13]} & 147 & 120 & 27 & Synthetic & Mesh \\ \hline \multicolumn{1}{c}{PU1K [14]} & 1147 & 1020 & 127 & Synthetic & Mesh \\ \hline \multicolumn{1}{c}{PUGeo-Net [15]} & 103 & 90 & 13 & Synthetic & Mesh \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Datasets constructed by researchers for upsampling. where \(\phi:S_{1}\to S_{2}\) is the bijection mapping. The formula finds a bijection \(\phi\) between the point sets \(S_{1}\) and \(S_{2}\), which are one-to-one corresponding, so that the sum of Euclidean distances calculated by them is the smallest. **Point to Surface (P2F)**. Evaluation indicators such as CD and HD evaluate the deviation from point to point, whereas P2F measures the distance of the generated point from the surface of the original point cloud. It is the distance between each point and its closest plane.