# A Local Discrete Feature Histogram for Point Cloud Feature Representation
## 3 Methods
### Parameter Settings  
The LDFH descriptor requires the configuration of 8 parameters. These parameters include the support radius \(R\), the number of partitions for the radial distance \(N_{r}\), the number of bins for the three geometric attributes \(\left(N_{\theta},N_{\psi},N_{\phi}\right)\), and the values of the three weights \(\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)\). A critical parameter for local feature descriptors is the support radius. A large value can impact computing performance and enhance sensitivity to clutter and occlusion, while a small value may reduce the descriptiveness. Based on recommendations in references [19; 20], the support radius \(R\) is set to 20 mr.  
The remaining 7 parameters are detailed below. To set the LDFH descriptor parameters, we use a scenario from the B3R dataset (see Section 4.1) that combines 1/4 mesh decimation with 0.3 mr Gaussian noise. The recall vs. 1-precision curve (RPC) serves as a widely adopted approach to evaluate descriptor performance [9; 15]. To enhance compactness, we assessed the performance of LDFH descriptors using the area under the RPC curve (denoted as AUCpr). The current parameter is selected by fixing other parameters during the LDFH testing process, and its value is gradually increased within a specific range. Specifically, the range of \(N_{r},N_{\theta},N_{\psi}\), and \(N_{\phi}\) is 2 to 20, with a step size of 1, and the range of \(\lambda_{1},\lambda_{2}\), and \(\lambda_{3}\) is 0.2 to 2.4, with a step size of 0.1. Referring to the parameter settings in [19], the initial values for \(N_{\theta},N_{\psi}\), and \(N_{\phi}\) are all set to 15, and the initial values for \(\lambda_{1},\lambda_{2}\), and \(\lambda_{3}\) are all set to 1. Once the present parameter has been tested, its value is fixed and used in subsequent parameter tuning. Table 1 displays the specific parameter tuning process for LDFH, and Figure 2 shows the corresponding AUCpr results of the experiments. Based on the test results and a trade-off between the descriptiveness, robustness, and compactness of the descriptor, the values of \(N_{r},N_{\theta},N_{\psi},N_{\phi},\lambda_{1},\lambda_{2}\), and \(\lambda_{3}\) are set to 8, 9, 14, 2, 1.5, 1.2, and 0.7, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(N_{r}\) & \(N_{\theta}\) & \(N_{\psi}\) & \(N_{\phi}\) & \(N_{\phi}\) & \(\lambda_{1}\) & \(\lambda_{2}\) & \(\lambda_{3}\) & **R (mr)** \\ \hline Setting \(N_{r}\) & 2–20 & 15 & 15 & 15 & 1 & 1 & 1 & 20 \\ Setting \(N_{\theta}\) & 8 & 2–20 & 15 & 15 & 1 & 1 & 1 & 20 \\ Setting \(N_{\psi}\) & 8 & 9 & 2–20 & 15 & 1 & 1 & 1 & 20 \\ Setting \(N_{\phi}\) & 8 & 9 & 14 & 2–20 & 1 & 1 & 1 & 20 \\ Setting \(\lambda_{1}\) & 8 & 9 & 14 & 2 & 0.2–2.4 & 1 & 1 & 20 \\ Setting \(\lambda_{2}\) & 8 & 9 & 14 & 2 & 1.5 & 0.2–2.4 & 1 & 20 \\ Setting \(\lambda_{3}\) & 8 & 9 & 14 & 2 & 1.5 & 1.2 & 0.2–2.4 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter settings for the LDFH descriptor.  
Figure 2: The parameter settings for LDFH descriptor. The solid markers present the selected parameter values.