# Deep Learning-Based Estimation of Muckpile Fragmentation Using Simulated 3D Point Cloud Data
## 5 Verification of Particle Size Distribution Estimation  
### Particle Size Distribution Estimation Network  
The particle size distribution estimation network used in this study is a modified version of PointNet++ [37], which is an advanced version of PointNet [35]. PointNet++ [37] is proposed as a method to capture local features by hierarchically applying PointNet to each group of divided point clouds. In this experiment, we used a modified version of PointNet++ for particle size distribution estimation. The network structure is shown in Figure 14. The network consists of a Local Module and a Global Module. The Local Module divides the point cloud into multiple groups using a sphere with radius r. It captures  
Figure 13: Steps in Generating and Processing Multi-View Point Clouds.  
features, learns from all points in each group as a single PointSet, and calculates a score. The Global Module captures global shape features from the sampled point cloud and multiplies them with the final score of the Local Module as weights.  
The Local Module takes the XYZ coordinates of the 3D point cloud as input and selects centroids for group division using FPS (FPS: Farthest Point Sampling). In FPS, a point is selected randomly, and this point becomes the first centroid. The second centroid is the point farthest from the first, and the third centroid is the point farthest from the first and second. This is repeated to select centroids up to then'th. By using FPS, it is possible to sample points spatially evenly. In this study, the points selected as centroids are limited to the range from the \"center of the point cloud\" to \"r inside the edge\" so that local shape patterns can be learned appropriately. Furthermore, by setting the radius r so that the largest rock in the data fits into a sphere, appropriate learning can be achieved. After generating n' groups, feature extraction is performed for each group as shown in Figure 5. After feature extraction, a Shered MLP is applied to each feature. This MLP is shared within the group, learned to output a score from the shape of the group, and finally normalized by the Softmax function to sum up the scores.  
The Global Module takes the XYZ coordinates of the 3D point cloud as input and samples _n_s points using FPS. Feature extraction is performed on the sampled point cloud as shown in Figure 4, and the overall feature is output. Then, an MLP is applied to generate weights. The score output from the Local Module is multiplied by the weight obtained from the Global Module and normalized by the Softmax function to obtain the final output. When the particle size distribution is at the d-stage, the output is a d-dimensional discrete probability distribution.  
In addition, the Kullback-Leibler divergence was used as the loss function for the difference between the output d-dimensional discrete probability distribution and the correct label, which is the particle size distribution. Let the output vector be \(p(x)\) and the correct particle size distribution be \(q(x)\). The loss function is represented by Equation (2).  
\[D_{KL}(Q\|P)=\sum_{x\in X}q(x)\log\left(\frac{q(x)}{p(x)}\right) \tag{2}\]  
To estimate the particle size distribution, the Bhattacharyya Distance (\(BD\): Bhattacharyya Distance) is used as the evaluation function for the difference between the correct label particle size distribution and the estimated particle size distribution. Let the two distributions be \(p(x)\) and \(q(x)\). It is expressed as in Equation (3).  
\[BD(p,q)=\sum_{x\in X}\sqrt{p(x)q(x)} \tag{3}\]  
Figure 14: Architecture of Fragmentation Estimation Network.