# 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
#### 2.1.2 Image Filtering  
Due to the long detection range and wide field of view of LiDAR, the resulting point cloud data are large, with the target points occupying only a small portion of the entire image. Furthermore, most of the point cloud data consists of meaningless background information. However, after converting the 3D point cloud into a cylindrical unfolding projection, the background pixels typically exhibit similar depth characteristics due to the uniformity of distance values. To address this issue, the method proposed in this paper applies a fast Fourier transform (FFT) to the grayscale image, allowing the frequency characteristics of the image to be analyzed in the frequency domain. We selected FFT for its high computational efficiency, which is crucial when processing large-scale LiDAR data. The FFT allows for fast transformation of the data from the time domain to the frequency domain, enabling the efficient identification and removal of periodic background noise. This is particularly important in LiDAR applications, where background noise often exhibits certain frequency patterns, which can be efficiently filtered out using FFT. While alternative filtering methods, such as wavelet transforms or Gaussian filtering, are also commonly used in signal processing, they typically come with higher computational costs. For example, wavelet transforms are highly effective in dealing with non-stationary signals but require more computational resources, which may not be ideal for real-time LiDAR data processing. Gaussian filters, on the other hand, are effective for removing low-frequency noise but are less efficient when processing large datasets in real-time applications. FFT, due to its ability to handle large volumes of data efficiently while maintaining critical signal information, is the most suitable choice for our study. It has been successfully applied in various signal processing applications, such as radar and LiDAR systems, where its effectiveness in noise reduction and signal preservation has been well-documented. For example, [PERSON] et al. [21] demonstrated the use of FFT in LiDAR systems to efficiently reduce noise without compromising target detection performance. Similarly, [PERSON] et al. [22] applied FFT in radar signal processing for noise reduction and signal enhancement, further validating its suitability for our LiDAR application. FFT computes the frequency domain of the discrete process, yielding a frequency-domain representation of the original spherical image. Each point in I\({}^{\prime}\) corresponds to a specific frequency component from the original grayscale image I. The transformation of the grayscale image into the frequency domain is as follows:  
\[\text{I}_{(i,j)}\overset{\mathcal{F}}{\rightarrow}\hat{\text{I}}_{(i,j)} \tag{12}\]  
Based on the frequency domain characteristics, the zero-frequency component is located at the top-left corner of the frequency plot. To better interpret this feature in the frequency domain, the low-frequency components of the frequency plot are moved to the center of the image. By applying a Fourier mask to the frequency domain plot, the low-frequency components corresponding to the majority of background point clouds can be filtered out. The expression for the Fourier mask M\({}_{\mathcal{F}}\) is as follows:  
\[\text{M}_{\mathcal{F}(i,j)}= \begin{cases}0\,,\;\left(\frac{N}{2}-1\right)<j< \left(\frac{N}{2}+1\right);\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;1, \;\text{others}\end{cases}  \tag{13}\]  
Here, \(N\) represents the width of the frequency domain image obtained after the 2D grayscale image is transformed into the frequency domain. The frequency domain matrix \(\hat{\text{I}}\) corresponding to the grayscale image is multiplied element-wise by the mask matrix M\({}_{\mathcal{F}}\), which effectively filters out the low-frequency components of the grayscale image.