## 2 Proposed Method  
### Principles of Laser Triangulation  
The method of laser triangulation measurement is to form a triangular mathematical model by the position relation of the optical system, the laser emitter, and the surface spot of target objects, which uses the displacement change received by the optical system to calculate the speckle height information. The measurement schematic is shown in Figure 1. The laser emitter emits a target beam projecting onto the object measured to form a laser line. The sensor in the optical system receives the laser line information from the target by receiving the lens, and the laser line information received by the base plane (no measured material) is displaced \(x\) in relation to the laser line information. Based on this, a triangular function relationship is formed. The relative height relationship between the surface of the object under test and the base plane is calculated by the relative displacement \(x\) of the laser line on the sensor, the angle of the laser and the receiving light \(\beta\), and so on. Figure 1: The principle of the laser triangulation measuring method. Laser triangulation can be divided into vertical and oblique, based on the incident position of the laser. The characteristic of the vertical incident type is to direct the light emitted by the line laser perpendicular to the base platform, which makes it easy to construct the mathematical model of the triangulation method, and the environment requirement is low. The inclined incident is the light emitted by the line laser into the benchmark platform at a certain angle. There is information on several angular parameters, meaning it is not easy to construct the mathematical model and needs higher environment requirements. As this study is about information on the lithium battery film surface height, the detection environment is relatively stable, and the vertical incident model is relatively simple; thus, the measurement speed is fast with a higher measurement accuracy and stability, so the vertical incident method is chosen in this work. As shown in Figure 1, the laser beam is projected vertically at the \(O^{\prime}\) point in the base plane. The angle of the light received with the camera is \(\beta\). Through the \(O\) point of the photocenter, the laser beam is imaged at the \(M^{\prime}\) point on the optical sensor. The distance between the \(O^{\prime}\) and the optical imaging system is \(b\), the distance from the imaging point \(M^{\prime}\) to the center of the light \(O\) is \(a\); when acting on a high object, the \(M^{\prime}\) point on the optical sensor is displaced, and the information of the \(N\) point on the surface of the object measured is projected on the \(N^{\prime}\) point of the sensor. The distance between the \(M^{\prime}\) and \(N^{\prime}\) is \(x\), the height difference between the point of the measured object and the base plane is \(h\). The \(N\) point is passed to create the vertical line of the receiving light, and the Pendant is the \(M\) point. Shown in the geometric \(\triangle OMN\sim\triangle OM^{\prime}N^{\prime}\) consisting of the laser light, according to the triangular similarity principle:  
\[\frac{OM}{OM^{\prime}}=\frac{MN}{M^{\prime}N^{\prime}} \tag{1}\]  
where \(O^{\prime}N=h\), \(\angle MO^{\prime}N=\beta\), \(OM^{\prime}=a\), \(OO^{\prime}=b\), \(M^{\prime}N^{\prime}=x\). In \(\triangle O^{\prime}MN\), according to the geometric relation:  
\[O^{\prime}M=hcos\beta \tag{2}\]  
\[MN=hsin\beta \tag{3}\]  
and then:  
\[OM=b-hcos\beta \tag{4}\]  
The Formulas (1) and (4) are united and sorted out:  
\[x(b-hcos\beta)=ahsin\beta \tag{5}\]  
\[h=\frac{xb}{asin\beta+xcos\beta} \tag{6}\]  
As can be seen from Figure 2, the 3D reconstruction system is that the camera captures images containing laser lines at \(\beta\) angles based on laser triangulation. The computer terminal calculates the surface height h at different times, and the measured height information is recorded in the slice model.