## 1 Introduction
### Extraction of Discontinuity Sets  
#### 2.3.1 Construction of the Improved Naive Bayes Classifier  
The Naive Bayes classifier, a widely employed supervised learning algorithm, is based on the fundamental [PERSON] theorem, expressed as Equation (2) [51]:  
\[P(A|B)=\frac{P(A)P(B|A)}{P(B)} \tag{2}\]  
where \(P(A)\) and \(P(B)\) represent the probability of events \(A\) and \(B\) occurring independently, respectively, with \(P(A)>0\) and \(P(B)>0\). \(P(B|A)\) signifies the probability of event \(B\) occurring given the occurrence of event \(A\), and \(P(A|B)\) denotes the probability of event \(A\) occurring given the occurrence of event \(B\). For each sample with \(d\) features \(X=\{x_{1},x_{2},\ldots,x_{d}\}\) and all discontinuity set categories \(y_{1}\), \(y_{2}\), \(\ldots\), \(y_{m}\), the Naive Bayes classifier is under the assumption that the \(d\) features are mutually independent. First of all, the probability \(P(y_{i}|X)\) (\(j=1,\ldots,m\)) that the point to be classified belongs to category \(y_{j}\) is calculated by Naive Bayes theorem:  
\[P(y_{j}|X)=\frac{P(y_{j})P(X|y_{j})}{P(X)} \tag{3}\]  
where \(P(y_{j})\) is the proportion of training samples belonging to category \(y_{j}\) in the training samples. The value of \(P(X)\) independent of the category is the evidence factor used for normalization, which can be obtained by the calculation of Equation (4):  
\[P(X)=\sum_{j=1}^{m}P(y_{j})P(X|y_{j}) \tag{4}\]  
Due to the features of the sample \(X\) to be categorized being independent of each other, \(P(X|y_{j})\) can be expressed as Equation (5):  
\[P(X|y_{j})=\prod_{i=1}^{d}P(x_{i}|y_{j}) \tag{5}\]  
Then, substitute Equation (5) into Equation (3):  
\[P(y_{j}|X)=\frac{P(y_{j})}{P(X)}\prod_{i=1}^{d}P(x_{i}|y_{j}) \tag{6}\]  
Figure 5: Point cloud data of case A: assigning RGB colors based on normal vectors. One color per discontinuity set. Considering that \(P(X)\) is independent of the category and its value will not affect the classification result, this paper introduces \(P_{j}\) to denote the probability that the sample \(X\) belongs to each category:  
\[P_{j}=P(y_{j})\prod_{i=1}^{d}P(x_{i}|y_{j}) \tag{7}\]  
After calculating the probability that the sample \(X\) belongs to each category using Equation (7), the category with the highest probability is selected as the classification result:  
\[h_{nb}(X)=\arg\max P_{j}=\arg\max P(y_{j})\prod_{i=1}^{d}P(x_{i}|y_{j}) \tag{8}\]  
The calculation method of \(P(x_{i}|y_{j})\) in Equation (8) depends on the distribution of the features of the samples. In this paper, the components of the normal vectors of the points to be classified in the direction of the \(x\), \(y\), \(z\) coordinate axes are used as the three features used for classification. Due to its numerical changes being continuous and approximately normally distributed, \(P(x_{i}|y_{j})\) in Equation (8) needs to be estimated using the following equation:  
\[P(x_{i}|y_{j})=\frac{1}{\sqrt{2\pi}\sigma_{y_{j},x_{i}}}\exp\left(-\frac{(x_{ i}-\mu_{y_{j},x_{i}})^{2}}{2\sigma_{y_{j},x_{i}}^{2}}\right) \tag{9}\]  
where \(\mu_{y_{j},x_{i}}\), \(\sigma_{y_{j},x_{i}}\), and \(\sigma_{y_{j},x_{i}}^{2}\) is the mean, standard deviation, and variance of feature \(x_{i}\) of the samples belonging to the \(y_{j}\) in the training sample, respectively. The above Naive Bayes classifier takes the category with the highest probability of belonging as the classification result. However, in the point cloud classification process, there are usually some noise points with a very small probability of belonging to any category that need to be removed.