# Rapid Target Extraction in LiDAR Sensing and Its Application in Rocket Launch Phase Measurement
## 2 Principle and Methods
### Principle of Rapid Target Point Cloud Extraction Method
#### 2.1.1 Cylindrical Projection
The scatter plot is denoted as \(S\), and each row in the matrix corresponds to the projection information of a 3D data point, as shown in the following equation:  
\[S=\left[\hat{x}_{s};\hat{y}_{s};\hat{\sigma}_{s}\right] \tag{4}\]  
In this matrix, \(S\) represents the matrix of the 2D projection image. \(\hat{x}_{s},\hat{y}_{s}\) represent the coordinates of the original point cloud data \(P\) after the projection transformation into the scatter plot. \(\hat{\sigma}_{s}\) corresponds to the depth value (either distance or reflectance intensity). Since the scatter plot consists of a large number of discrete points, it is less convenient for mathematical operations and image processing tasks compared to a grayscale image. Furthermore, a grayscale image is more suitable for subsequent tasks such as target edge extraction and segmentation. Therefore, the resulting scatter plot \(S_{k}\) is saved as a 2D image through a series of matrix transformations. The specific process is as follows: First, the  
Figure 1: Cylindrical projection schematic. depth values \(\vartheta_{s}\) of each point in the scatter plot are mapped to the range [0, 1]. The points are then grid-quantized based on imaging requirements, and the mapped coordinates are converted into pixel values. This generates a matrix with dimensions \(m\times n\), where the grayscale value of each pixel in matrix \(I\), denoted as \(\delta_{i,j}\), represents the corresponding distance or reflectance intensity value of the point. Here, \(i,j\) denote the row and column indices of the pixel, respectively. \[\delta_{i,j}=\frac{\vartheta_{s}}{\max\{\vartheta_{s}\}}\times 255 \tag{5}\]  
When converting the scatter plot into a grayscale image, the pixels containing the scan data are located in the central region, while the outer edges of the image are filled with meaningless black background pixels. To facilitate subsequent computational processing, the grayscale values of these black background pixels are handled accordingly. The position of each pixel in the grayscale image is denoted as \(I\) (\(x_{I},y_{I}\)), where \(x_{I},y_{I}\) represent the row and column indices of the pixel in matrix \(I\), and the pixel values are determined according to the following equation:  
\[ \begin{cases}x_{I}=i\\ y_{I}=j\end{cases} ;\text{ where},\ \ \delta_{i,j}\
eq 0 \tag{6}\]  
This process involves generating a normalized grid and filling the pixels of the grayscale image. During the conversion from the scatter plot to the 2D pixel image, the primary operation is the processing of scatter point values. The scatter plot \(S\) is multiplied by a value transformation matrix \(K\) where \(K\) is a 3 \(\times\) 3 matrix. The specific values of the matrix \(K\) are determined by the pixel size of the resulting grayscale image. A schematic of this process is shown in Figure 2. The transformation matrix \(K\) can be computed using the following method. Figure 2: Schematic diagram of the grayscale map conversion. First, from the original 3D point cloud data PPP, the scatter plot \(S\) is obtained through cylindrical projection. The four corner points of the scatter plot \(S\) are extracted, denoted as \(A(\hat{x}_{s1},\hat{y}_{s1})\), \(B(\hat{x}_{s2},\hat{y}_{s2})\), \(C(\hat{x}_{s3},\hat{y}_{s3})\), \(D(\hat{x}_{s4},\hat{y}_{s4})\). Similarly, the four corner points of the grayscale image \(I\) are denoted as \(a(x_{1I},y_{1I})\), \(b(x_{I2},y_{I2})\), \(c(x_{I3},y_{I3})\), \(d(x_{I4},y_{I4})\). These corner points are assigned values as follows:  
\[S: \begin{cases}\hat{x}_{s1}=\hat{x}_{s3}=min\{\hat{x}_{s}\}\\ \hat{x}_{s2}=\hat{x}_{s4}=max\{\hat{y}_{s}\}\\ \hat{y}_{s1}=\hat{y}_{s3}=min\{\hat{x}_{s}\}\\ \hat{y}_{s2}=\hat{y}_{s4}=max\{\hat{y}_{s}\}\\ \end{cases} I: \begin{cases}x_{I1}=x_{I3}=min\{x_{I}\}\\ x_{I2}=x_{I4}=max\{y_{I}\}\\ y_{I1}=y_{I3}=min\{x_{I}\}\\ y_{I2}=y_{I4}=max\{y_{I}\}\\ \end{cases}  \tag{7}\]  
By matching the four corner points, the mapping matrix \(K\) between the scatter plot and the grayscale image can be derived using Gaussian elimination.