## 3 Characteristics of the Actual Laser Imaging Waveform
### Characteristics of Static Laser Imaging Waveforms  
By controlling the exposure time of the CMOS to obtain a gray-scale image, the characteristics of static imaging waveforms of two different measured object were studied. First, 10 groups of waveforms were obtained for two measured objects, and the change rule of waveform shape under different exposure times was analyzed. Then, the centroids of the waveforms were calculated, and the relationships among the centroid positions of the imaging waveform and the peak intensity were studied. The imaging waveforms corresponding to different exposure times are shown in Figure 3. Although the imaging waveforms of the two different objects had obvious morphological differences (Figure 3a,b), the waveforms had a consistent trend when the peak intensity was in a certain range (350-800). The relationship between the exposure time and the peak intensity was further analyzed, as shown in Figure 3c. For different objects, the relationship between the peak intensity and the CMOS exposure time was different; the CMOS exposure time required to reach the target peak intensity could not be simply calculated from a function. Establishing multiple functions to calculate the CMOS exposure time for different measured objects was costly in terms of both software and hardware. Therefore, we intended to develop an iterative algorithm to automatically adjust the CMOS exposure time for different objects. To confirm the ideal peak intensity, we considered the relationship between the peak intensity and the centroid position. As shown in Figure 4a, there was a certain difference in the centroid positions calculated from the waveforms with different peaks. To reduce this difference and to eliminate the interference caused by different colors, we fit the \"centroid position-peak intensity\" curve equation of black and white objects, respectively, as shown in Equation (3):  
\[X(p)=a_{k}p^{k}+a_{k-1}p^{k-1}+a_{k-2}p^{k-2}+\cdots\cdots+a_{1}p+a_{0} \tag{3}\]  
Figure 2: Imaging waveform experiment platform. (**a**) Hardware platform; (**b**) Measured object. where \(a_{k}\) (\(k\) = 0, 1, 2,... ) is the coefficient term, \(p\) is the peak intensity, and X (\(p\)) is the centroid position when the peak intensity was \(p\). The least square method is used to confirm the coefficient term, and the sum of squares of error \(I(a_{0},a_{1},\)... \(,a_{k})\) is constructed, where \(C_{i}\) is the actual centroid position:  
\[I(a_{0},a_{1},\cdots,a_{k})=\sum_{i=1}^{n}[C_{i}-x(p_{i})]^{2}=\sum_{i=1}^{n} \Bigl{(}y_{i}-a_{0}-a_{1}p_{1}-\cdots-a_{k}p_{i}^{k}\Bigr{)} \tag{4}\]  
Solving the corresponding coefficient \(a_{0}\)*, \(a_{1}\)*,... \(,a_{k}\)* such that:  
\[I(a_{0}\)*, \(a_{1}\)*,..., \(a_{k}\)*)=\min_{a_{0},a_{1},\cdots,a_{k}}I(a_{0},a_{1},\ldots,a_{k}) \tag{5}\]  
According to the least square approximation method described above, the centroid position-peak intensity equation can be written as:  
\[ \begin{array}{ll}X(p)&=\left[\begin{array}{c}X_{w}\\ X_{b}\end{array}\right]=\left[\begin{array}{c}1.216e-19\\ 5.878e-19\end{array}\right]p^{7}+\left[\begin{array}{c}-3.39e-16\\ -1.856e-15\end{array}\right]p^{6}+\left[\begin{array}{c}3.916e-13\\ 2.406e-12\end{array}\right]p^{5}\\ &+\left[\begin{array}{c}-2.574e-10\\ -1.649e-9\end{array}\right]p^{4}+\left[\begin{array}{c}1.178e-7\\ 6.383e-7\end{array}\right]p^{3}+\left[\begin{array}{c}-4.255e-5\\ -0.0001399\end{array}\right]p^{2}+\left[\begin{array}{c}0.01124\\ 0.01872\end{array}\right]p+\left[\begin{array}{c}568\\ 567.3\end{array}\right]\end{array}  \tag{6}\]  
where \(X_{w}\) and \(X_{b}\) are the centroid positions when the measured object is white diffuse reflector and black frosted reflector, and the intersection point of the equation curves is the ideal peak intensity (520 in this paper), as shown in Figure 4b. Figure 3: Imaging waveforms corresponding to different photosensitive imaging device (CMOS) exposure times.