A Novel Dynamic Light-Section 3D Reconstruction Method for Wide-Range Sensing

Existing galvanometer-based laser-scanning systems are challenging to apply in multi-scale 3D reconstruction because of the difficulty in achieving a balance between a high reconstruction accuracy and a wide reconstruction range. This paper presents a novel method that synchronizes laser scanning by switching the field-of-view (FOV) of a camera using multi-galvanometers. Beyond the advanced hardware setup, we establish a comprehensive geometric model of the system by modeling dynamic camera, dynamic laser, and their combined interaction. Furthermore, since existing calibration methods mainly focus on either dynamic lasers or dynamic cameras and have certain limitations, we propose a novel high-precision and flexible calibration method by constructing an error model and minimizing the objective function. The performance of the proposed method was evaluated by scanning standard components. The results show that the proposed 3D reconstruction system achieves an accuracy of 0.3 mm when the measurement range is extended to 1100 mm × 1300 mm × 650 mm. This demonstrates that for meter-scale reconstruction ranges, a sub-millimeter measurement accuracy is achieved, indicating that the proposed method realizes multi-scale 3D reconstruction and simultaneously allows for high-precision and wide-range 3D reconstruction in industrial applications.


I. INTRODUCTION
L IGHT -section vision systems are widely used in many applications for their adaptability, high accuracy, and effective cost [1]- [3], such as rail traffic monitoring [4], medical imaging [5], robotics [6], and industrial production [7].Such systems typically comprise a camera, laser projector, and mechanical scanning platform.The line laser projects laser stripes onto the surface of the object, whereas the camera captures an image of the object with the laser stripes.The three-dimensional (3D) geometric information of the object is then obtained by triangulation, as extensively reviewed in literature [8].The 3D reconstruction of an object can be completed by passing laser stripes or objects through a mechanical scanning platform.
To overcome these limitations, various scanning mechanisms have been proposed.For instance, Du [11] designed a system that mounts a line laser on the end of a robotic arm to improve scanning flexibility; however, its scanning accuracy was limited by the precision of the robotic arm.Jiang [12] proposed a system that uses gimbals to drive the laser and camera for scanning; however, the system size was significant, and its scanning speed was slow.In recent years, galvanometers have emerged as promising scanning devices because of their small size, fast rotation, and high control accuracy.This galvanometer-based solution provides a better alternative in terms of laser scanning accuracy and speed [13].However, existing galvanometer-based laser-scanning systems are primarily designed to perform laser scanning while leaving the camera fixed.The limited FOV of a fixed camera causes a trade-off between the accuracy and the sensing range of the system, which significantly affects its efficiency.
In this study, we propose a novel dynamic light-section 3D reconstruction system that combines dynamic laser and dynamic camera using multi-galvanometers.Our approach utilizes multiple galvanometers to synchronize laser scanning and the FOV switching of the camera, thereby enabling highprecision and wide-range 3D reconstruction.Calibration is required to achieve this, which includes the system calibration of the galvanometer-based dynamic laser and camera, and their joint calibration.For calibrating galvanometer-based dynamic laser systems, Eisert [14] introduced a mathematical model and calibration procedure; however, the model was complicated, and its optimization was difficult, thus leading to low accuracy.Yu [15] designed a one-mirror galvanometer laser scanner.However, the calibration procedure was complex, and the objective function was difficult to optimize.Similarly, Yang [16] proposed a calibration method based on a precision linear stage.However, this approach relies on a precision instrument and lacks flexibility.For calibrating galvanometer-based dynamic camera systems, Ying et al., [17]- [19] introduced self-calibration methods, which were complex in theory and difficult to implement.Kumar [20] proposed a calibration method based on the look-up table (LUT) using simple linear parameters, which required complex pre-processing.Junejo et al., [21]- [24] proposed feature-based calibration methods, which were time-consuming and had low accuracy.Han [25] introduced a calibration method for galvanometer-based camera using an end-to-end single-hidden layer feed forward neural network model, but it was computationally intensive.
Boi et al., [26] [27] proposed manifold constrained Gaussian process regression methods for galvanometer setup calibration, which relied on data-driven and complex calibration procedures.Hu [28] built a galvanometer mirror-based stereo vision measurement system and established a mirror reflection model, but it still lacked an accurate calibration method.
In conclusion, current light-section 3D reconstruction systems cannot simultaneously have high accuracy and wide range.Moreover, existing calibration methods only focus on calibrating either dynamic lasers or dynamic cameras and still have some shortcomings, as mentioned above.To address these limitations, this study proposes a novel dynamic 3D reconstruction system that overcomes the trade-off between accuracy and measurement range by synchronizing laser scanning and FOV switching of a camera based on multiple galvanometers.Additionally, we propose a complete calibration solution for the proposed system that includes the calibration of the dynamic camera, dynamic laser, and joint calibration.The contributions of this study can be summarized as follows: 1) A novel dynamic light-section 3D reconstruction system is designed based on multiple galvanometers.To the best of our knowledge, the system is the first to synchronize laser scanning and FOV switching of camera, thus enabling high-precision and wide-range 3D reconstruction simultaneously.
2) The geometrical model of the dynamic 3D system is developed by the combined modeling of the dynamic camera and laser, which establishes 3D-information acquisition with multi-galvanometers and laser images.3) A flexible and accurate calibration method of the dynamic 3D system is proposed by constructing error models and objective functions.This method is not only applicable to the proposed system but also to other single galvanometerbased laser or camera systems.4) Experiments are conducted to validate the proposed dynamic 3D reconstruction method and demonstrate its accuracy.To the best of our knowledge, compared to all existing galvanometer-based laser scanning methods, our approach has the highest measurement range while maintaining the same level of measurement accuracy.
The system design and geometric model are described in Section II.The proposed calibration method and error compensation methods are described in Section III.Section IV presents the validation experiments and results.Finally, Section V presents the conclusions.

II. SYSTEM DESIGN AND GEOMETRIC MODEL A. System Design
The dynamic light-section 3D reconstruction system consists of a CMOS camera, a line laser, and two galvanometer mirror systems, as shown in Fig. 1.The camera and Galvanometer-1 form a dynamic camera system, whereas the laser and Galvanometer-2 form a dynamic laser system.Based on the mathematical model of the system and precalibration, the 3D information of the target can be calculated from the captured laser image and voltage values of the two galvanometers.The working principle is illustrated in Fig. 2. A spherical object on a flat plane is employed to demonstrate the process of dynamic 3D reconstruction.First, the system utilizes multigalvanometer control to scan the target surface.When the system is activated, a line laser projects a laser stripe onto Galvanometer-2, which reflects the stripe onto the surface of the object.By controlling the voltage of Galvanometer-2, the laser stripe can scan the target.Simultaneously, the dynamic camera system captures laser images from different angles by adjusting the voltage of Galvanometer-1.Next, the lasercenter-pixel coordinates are obtained using the laser stripe extraction algorithm.The 3D reconstruction of the laser stripe is performed by combining the voltage values of multiple galvanometers, laser pixel coordinates, geometric models of the dynamic 3D reconstruction system, and calibrated parameters.The point clouds of all the laser stripes are converted to the same coordinate frame using the transfer matrix of the dynamic camera.The system performs error correction based on the joint calibration to optimize accuracy.Finally, the system generates a point cloud for the target and completes the dynamic 3D reconstruction.Accurate mathematical modeling and calibration methods are essential to ensure the 3D reconstruction accuracy of the system.

B. Geometric Model
The mathematical model of the system is shown in Fig. 2(b).Five coordinate frames are created: image pixel coordinate frame {I}, camera coordinate frame {C}, virtual camera coordinate frame {V }, world coordinate frame {W }, and Galvanometer-1 mirror coordinate frame {G}.According to the operating principle of a galvanometer, the rotation angle of the pan-tilt mirror is proportional to the voltage.For Galvanometer-1, the voltages of the pan-tilt mirrors are denoted by U 1−pan and U 1−tilt .Therefore, the rotation angle of the pan mirror is θ 1 = k 1−pan U 1−pan and the rotation angle of the tilt mirror is θ 2 = k 1−tilt U 1−tilt .As the pantilt mirror rotates, {V } reflects the change in U 1−pan and U 1−tilt .When U 1−pan = U 1−tilt = 0, the virtual camera coordinate frame is denoted by {V 0 }.The relationship between the virtual coordinate frame {V } after the motion and the initial coordinate frame {V 0 } is given by Eq. (1).
Line laser

High-speed camera
Galvanometer-1 l-Distance between the optical center of the camera and the center of mirror 1 ; d-Distance between the center of the mirror 1 and the center of the mirror 2 ; -Rotation axis of mirror 4 .
where V T V0 is the transfer matrix between {V 0 } and {V }.
V T G denotes the transfer matrix between {G} and {V }.G T V0 denotes the transfer matrix between {V 0 } and {G}.As shown in the geometric model diagram, {C} is first reflected by a pan mirror and then by a tilt mirror.The geometries of the two reflections are modeled using Eq. ( 2).For {V 0 }, the rotation angle of the pan-tilt mirror is θ 1 = θ 2 = 45 • .The transfer matrix V T V0 is calculated using Eq. ( 3).Thus, the geometric model of the dynamic camera is established.
The coordinates of the pixel points on the laser stripe are denoted by (u, v).The coordinates of the corresponding 3D points in {V } are denoted by (X V , Y V , Z V ).Based on the pinhole model of the camera, the mapping relationship between their coordinates can be obtained using Eq. ( 4).
where (u 0 , v 0 ) is the principal point of the image, f x and f y are the focal length of the camera.For Galvanometer-2, the rotation angle of the pan mirror is denoted by θ 3 = k 2−pan U 2−pan , and the rotation angle of the tilt mirror is denoted by the dynamic laser is in its initial position.The initial laser plane in {V 0 } is denoted by V0 plane 0 and its equation is The plane of the laser after rotation about the rotation axis is denoted as V0 plane, and the equation is The light path of the dynamic laser is reflected by a mirror and rotated along its axis.The rotation angle of the laser plane is twice that of the mirror plane.Therefore, the equation for the dynamic laser plane after rotation in {V 0 } can be solved using Eq. ( 5), where R represents the Rodrigues transformation.Using a point (x n , y n , z n ) on the rotation axis, D V0 can be calculated as Combining this with the transfer matrix V T V 0 in Eq. ( 3), the equation for the dynamic laser plane in {V } can be calculated as The equation for V plane is denoted as A V x+B V y+C V z+ D V = 0.By extracting the pixel points (u, v) from the laser stripe, the corresponding 3D point can be calculated as V P = (X V , Y V , Z V ).Therefore, the relationship between V P and the change in the galvanometer mirror angles can be expressed by Eq. (7).
F represents a parameterized mapping function, where θ 1 , θ 2 , θ 4 are variables and other parameters are constants.Because {V } changes constantly with the scanning angle, it is necessary to convert all V P into a coordinate frame that is fixed with {W }.For ease of calculation, we choose {V 0 } and obtain V0 P = (X V0 , Y V0 , Z V0 ) = V0 T V V P .Thus, we establish the relationship between (u, v) and (X V0 , Y V0 , Z V0 ) to formulate the 3D reconstruction.In the mathematical model of the 3D dynamic system, Eq. (7) shows that f x , f y , u 0 , v 0 can be obtained by calibrating the camera.A 0 , B 0 , C 0 , D 0 can be obtained from the laser plane calibration.l, d, − → n = (n x , n y , n z ), P = (X 0 , Y 0 , Z 0 ) are unknown, and a calibration algorithm must be designed to obtain the unknowns.

III. CALIBRATION METHOD
The proposed system calibration method is divided into three parts: the dynamic camera calibration, the dynamic laser calibration, and the joint calibration of the dynamic camera and laser for error correction.

A. Dynamic Camera Calibration
To calibrate the intrinsic parameters, f x , f y , u 0 , v 0 , of the camera, we use the method proposed by Zhang [29].The dynamic camera calibration method described in Section II is used to obtain the constraint relationship between {V } and {G}, as shown in Eq. ( 2); thus, we obtain the parameters l and d.
The proposed calibration method uses a large calibration board as shown in Fig. 2(b).The calibration board measures 740 × 740 mm and comprises a total of 35 × 35 circular markers.These markers are constructed from 25 individual 7 × 7 sub-patterns.Each sub-pattern features a central larger circular marker with a diameter of 15 mm, while the remaining smaller circular markers have a diameter of 10 mm, with a center-to-center spacing of 20 mm.The purpose of the larger circular markers is to establish the relationships between calibration points across different FOVs.The calibration board is scanned by varying the galvanometer voltage to obtain numerous images at different rotation angles.Each image corresponds to a virtual coordinate frame.The number of images is denoted as n.The conversion matrix between {V } and {W } can be obtained by extrinsic parameter calibration as V0 T W , V1 T W , V2 T W , . . ., Vn T W .As the relative positions between the calibration points in these images are known, the transformation matrix between virtual coordinate frames is calculated as V1 T V0 , V2 T V0 , . . ., Vn T V0 .These values are taken as observations.Multiple sets of observations are used to solve for the parameters to be calibrated.The initial pan-tilt angles of Galvanometer-1 are denoted by θ where k ij and g ij represent the function of {V 0 } and {V n }. l and d are parameters to be calibrated using Eq.(7).The transformation matrix between {V 0 } and {V n } is a 4×4 matrix, which can be expressed as: Simultaneously, {V n } can be calculated using {V 0 } from Eq. ( 8), and Vn T V0 .For ease of representation, this is denoted as h ij .
{G} is the base coordinate frame and {V } = V T G • {G}; therefore, Eq. ( 2) is the mathematical model of {V }.Eq. ( 10) is the result of {V } obtained from multiple observations.Eq. ( 8) shows the results calculated using the mathematical model of the dynamic camera.For all the measured coordinate frames ({V 0 }, {V 1 }, {V 2 }, . . ., {V n }), the sum of the errors between the theoretical and measured values must be minimized.Therefore, the objective function is formulated using Eq.(11).
In Eq. ( 2), the parameters l and d exist only in translation vector.Based on the objective function, Eq. ( 12) can be obtained.Finally, the parameters l and d are obtained by solving Eq. ( 12) using the least-square method.

B. Dynamic Laser Calibration
Based on the mathematical modeling of the dynamic laser presented in Section II, the equation of V0 plane 0 and the rotation axis of the dynamic laser − → n must be calibrated when Galvanometer-1 is at the initial position.The calibration of V0 plane 0 is conducted utilizing the methodology outlined in [30].This process involve the acquisition of laser images at various positions using a checkerboard calibration plate.Through this approach, the laser plane is accurately defined by fitting multiple laser lines.For the extraction of the laser center, the technique described in [31] is employed, offering the advantage of sub-pixel precision in the extraction process.Following these procedures, we successfully derive the equation . ., U m are used to move the laser and obtain multiple laser planes.Next, the respective equations are calibrated in the same manner as V0 plane 0 and denoted as V0 plane 1 , V0 plane 2 , . . ., V0 plane m .The unit normal vectors of these planes are calculated as → n m (n xm , n ym , n zm ).In the absence of errors, the laser planes intersect along the same straight line.This line is the laser rotation axis − → n and is also the rotation axis of the tilt mirror in Galvanometer-2.For a normal vector in any laser plane, we obtain − → n • − → n i = 0(i = 0, 1, 2, . . ., m).
However, − → n • − → n i do not exactly equal zero because of various errors.Therefore, for all laser planes, the objective function is formulated as shown in Eq. (13).
The direction vector − → n of the rotation axis is obtained by minimizing the objective function.P = (X 0 , Y 0 , Z 0 ) is a point on the rotation axis in all laser planes and can be obtained using the least-square method.All parameters in Eq. ( 7) are obtained by calibrating the camera, laser plane, dynamic camera, and dynamic laser.Thus, we complete the calibration of the proposed dynamic 3D system.

C. Joint Calibration for Error Correction
For a well-calibrated dynamic light-section 3D reconstruction system, there are two sources of error, dynamic camera and dynamic laser, as listed in Table I.
This study proposes an error correction method based on the joint calibration of a dynamic camera and dynamic laser.After the calibration is completed, error correction is performed based on the 3D reconstructed results.Theoretically, when Galvanometer-1 is scanning and Galvanometer-2 is fixed, the reconstructed laser point cloud coincides perfectly.However, as explained in the error source analysis, there is some deviation between the multiple laser point clouds owing to these errors.We correct these errors using point-cloud registration to obtain the accuracy conversion matrix.
The calibration process is designed based on the following principle.If Galvanometer-2 remains stationary, the laser will be out of the FOV after Galvanometer-1 has scanned a certain range.Therefore, multiple calibration positions must be set in advance to maintain the laser in the FOV.These are set in advance as p 1 , p 2 , . . ., p n and the corresponding voltages of Galvanometer-2 at these positions are (s 1 , t 1 ), (s 2 , t 2 ), . . ., (s n , t n ), respectively.The laser plane equations plane p1 , plane p2 , . . ., plane pn at these positions are calibrated using the laser plane calibration method described in Part B. The voltage of Galvanometer-1 is denoted by , respectively.The error-correction flow is presented in Algorithm 1.

Algorithm 1 Joint Calibration for Error Correction
, M ← Unit matrix.4: Using Eq. ( 7) to obtain the laser point cloud V0 P 1 .5: while i < m do 6: Laser Image capture and center curve extraction;

IV. EXPERIMENT
The proposed dynamic 3D reconstruction system is built as shown in Fig. 3.The camera model is MV-CA004-10UC, with a pixel size of 6.9 µm × 6.9 µm, resolution of 720 pixels × 540 pixels, and frame rate of 500 fps.The exposure time of the camera to capture the dark image of the laser is 500 µs.The laser model is LXL65050-16 and the laser wavelength is 650 nm.The models of both Galvanometer-1 and Galvanometer-2 are TSH8310.The galvanometer is used to scan a range of ±20 • using a control voltage range from −10V to +10V .The maximum scan frequency is 1 kHz, with an angular resolution of 0.0008, thus, the system has the potential for high accuracy and resolution.
A. Calibration Accuracy Verification 1) Dynamic camera calibration accuracy: Twenty-five images of the calibration board are collected when U 1−pan = U 1−tilt = 0.The camera is calibrated using Zhang's [29] calibration method in OpenCV.After calibration, the intrinsic parameters are f x = 7801.38,f y = 7798.24,u 0 = 359.51,and v 0 = 269.54.The focal length is 53.83 mm.According to the calibration method for the dynamic cameras presented in Section III, the system parameters are solved as l = 83.45mm and d = 22.14 mm.
Based on these calibration results, the mathematical model proposed in Section II can be used to calculate the theoretical transfer matrix for the pan-tilt mirror of Galvanometer-1 at different angles.The transfer matrices corresponding to these angles are directly measured using a calibration board.The matrix 2-norm is calculated according to Eq. ( 14) to compare the theoretical matrices A and measured transfer matrices B for calibration accuracy verification.The pan-tilt voltages of Galvanometer-1 are varied from −10V to 10V , at intervals of 4V .Thirty-six positions are measured.The error between the theoretical and measured transfer matrices is obtained, and the error curves are shown in Fig. 4(a).The results show that the RMSE (Root Mean Square Error) is 1.231 mm between the theoretical and measured values.
This confirms the accuracy of the dynamic camera calibration.The observed errors originate from the geometric model and the calibration process, as explained in the error analysis section.It is important to note that the measured values obtained for the virtual camera using the calibration board may also exhibit slight deviations.
Consequently, these findings serve as a validation of the accuracy of the dynamic camera calibration; however, these cannot be solely relied upon to assess the accuracy of the calibration.A more detailed accuracy verification can be conducted based on the outcomes of the 3D reconstruction analysis.
2) Dynamic laser calibration accuracy: With Galvanometer-1 fixed, the calibration board is positioned within the FOV of the virtual camera.The tilt mirror of Galvanometer-2 is rotated 30 times with a step size of 0.1V , allowing the system to scan the calibration board, whose position is randomly changed five times (ensuring clear imaging in the virtual camera); the same 30 scans are repeated for each position.The laser rotation axis is solved as Notably, the calibrated rotation axis align with the intersection of the laser planes, providing evidence for the accuracy of the dynamic laser calibration.A detailed accuracy assessment is subsequently performed by analyzing the results of the 3D reconstruction.
3) Joint calibration accuracy: A calibration sphere is selected as the 3D reconstruction target for error correction.Galvanometer-2 is controlled to project the laser stripe onto the sphere, while Galvanometer-1 is fixed.The virtual camera, controlled by Galvanometer-1, captures images of the laser stripe from different views.The 3D reconstruction of these laser stripe images is performed based on the calibration results and mathematical models of the 3D dynamic system.The reconstructed point clouds, which are indicated as white, are shown in Fig. 2(a), 'Error Correction'.Notably, white point clouds exhibit non-overlapping regions owing to errors.The correction method described in Algorithm 1 is employed to register these white point clouds.The registration results are shown as colored point clouds in Fig. 2(a).The distance between the point clouds before and after the correction is calculated to evaluate the error.The calculation formula is as follows: Here, p s represents the point cloud of the laser stripe captured in the first virtual camera view, and p t represents the point cloud of the laser stripe captured from another view.The error between p t and p s is determined by performing a nearest-neighbor search, denoted as Error1.After the pointcloud registration, the error between p t and p s is calculated as Error2.In addition, the error before correction is computed as Error3 using the matched points from the point-cloud registration result.The error curves are shown in Fig. 4(c).The RMSE of Error1 and Error3 before correction are calculated as 4.928 and 5.475 mm, respectively.However, after error correction, the RMSE of Error2 is significantly reduced to 0.197 mm.These results evidently indicate a substantial improvement in the accuracy following the error-correction process.analyze its reconstruction accuracy at different angles.The stair block has a distance of 30 mm between its two planes, with machining errors within 1 µm.The 3D dynamic system proposed in the paper is used to reconstruct a stair block.Scanning is performed by synchronously controlling the tilt mirrors of both Galvanometer-1 and Galvanometer-2, rotating each by 0.1 Â°.

B. 3D Reconstruction
Once the scanning and reconstruction processes are completed, a point cloud of the stair block is generated.Two planes (Plane-1 and Plane-2) of the stairs are fitted, and the distance between them is calculated.Point clouds belonging to Plane-2 are used to fit a plane equation using the least-square method.Next, 500 points belonging to Plane-1 are randomly selected and the average distance between these points and Plane-2 is calculated as the distance of the fitted plane.The difference between the calculated and actual distances is considered as the error, which serves as a measure of the reconstruction accuracy achieved by the system.The reconstruction distance is 650 mm.The measurement range of the system is determined by the overlapping FOV of the dynamic camera and dynamic laser, which measures 1100 mm × 1300 mm.Thirty different positions are selected to analyze the reconstruction accuracy at different angles.The dynamic camera and laser simultaneously scan the target from these positions to complete the 3D reconstruction process.Fig. 5(a) shows the example reconstructions obtained from four different positions, providing a visual representation of the reconstructed 3D models.The thickness error, which is related to the rotation angles of Galvanometer-1 and Galvanometer-2, is analyzed, as shown in Fig. 5(b).It is evident from the graph that the error in 3D reconstruction increases as the rotation angles of the galvanometers deviate from their initial positions (calibration position).This is because larger errors occur as the system moves farther away from the calibration position.The RMSE for these thirty positions is calculated as 0.165 mm.These values provide evidence of the high precision achieved by the proposed system for 3D reconstruction.
To compare the performance of the proposed method with that of existing methods [13], [15], [16], [32]- [35], we conduct comparative experiments using the standard component scanning method.The accuracy of the dynamic light-section 3D system depends primarily on the working distance.To perform a fair comparison, we repeat the standard component scanning procedure at various reconstruction distances, namely 100, 200, 350, 400, and 1000 mm, which are consistent with the working distances employed in existing methods.
The 3D reconstruction accuracy and measurement ranges achieved using each method are shown in Fig. 6, with each color oval representing the same working distance.From the obtained results, it can be concluded that the proposed method exhibits smaller errors and larger measurement ranges than the existing methods at the corresponding working distances.This demonstrates the superior performance of our method in terms of accuracy and range compared with existing methods.
2) Large object scanning test: A high-precision machined large flat plate is also utilized to test the 3D reconstruction accuracy.The proposed 3D dynamic system is employed to scan the target and obtain its point clouds.The scanning distance is set at 650 mm.The measurement range is 1100 mm × 1300 mm.The size of the target is 740 mm × 740 mm.The position and angle of the target are changed arbitrarily within the depth of the field, and the reconstruction is repeated three times.After obtaining the reconstructed point cloud, a plane equation is fitted to the data using the RANSAC algorithm.NOM-LSS method [16] (d=100mm) 3DM-LSS method [17] (d=250mm) EAC-LSS method [14] (d=250mm) IS-LSS method [30] (d=350mm) FLR-LSS method [31] (d=400mm) FFV-LSS method [32] (d=400mm) U3D-LSS method [33] (d=1000mm) Our method at corresponding distance Fig. 6.Comparison of 3D reconstruction accuracy and measurement ranges between existing method and proposed method.
The distances between all the points and the fitted plane are calculated, and the average value of these distances is considered as the error of the dynamic 3D system reconstruction.The RMSE for the three measurements is 0.281 mm.These results demonstrate that the proposed system achieves a high level of accuracy in 3D reconstruction measurements.

V. CONCLUSION
A dynamic light-section 3D reconstruction system is proposed in this study, which overcomes the trade-off between accuracy and measurement range by using multiple galvanometers.A mathematical model of the system is established, and a flexible and accurate calibration method is developed.The experimental results demonstrate that the proposed system performs well in terms of measurements, indicating its potential for industrial applications where high-precision and widerange 3D reconstruction is required.Furthermore, the proposed method can be used in conjunction with the tracking algorithm for 3D reconstruction of moving targets.

Fig. 2 .
Fig. 2. Framework of dynamic 3D reconstruction method.(a) The flowchart of dynamic light-section 3D reconstruction.(b) Geometric modeling and calibration of dynamic 3D reconstruction system.

2 .
are the corresponding angles of {V 0 } and the first calibration image.The pan-tilt angles of Galvanometer-1 corresponding to {V n } and nth calibration images are denoted as θ Therefore, {V 0 } and {V n } are defined as follows:

Fig. 4 .
Fig. 4. Calibration accuracy verification.(a) Error of dynamic camera transfer matrix.(b) Visualization of laser planes and the calibrated rotation axis.(c) Error curve before and after correction.

Fig. 5 .
Fig. 5. Standard blocks reconstruction test at different angles.(a) The point clouds of the stair at different angle.(b) 3D reconstruction error distribution at different angles.
Accuracy Verification 1) Standard blocks reconstruction test: A standard stair block is employed to test the stability of the system and Rotation angle of galvanometer mirror (θ1, θ2) Non-linear deviations exists between voltage and rotation angle of Galvanometer-1. 2 Dynamic camera geometric model The spectacular reflection geometric model has deviations with mechanical structure.3 Calibration of parameter (l, d) This error has been optimized by proposed error model and objective function in this paper.4 Camera intrinsic parameters calibration This non-linear error is optimized using the Zhang's [29] calibration method.5 Rotation angle of galvanometer mirror (θ3, θ4) Non-linear deviations exists between voltage and rotation angle of Galvanometer-2.6 Dynamic laser geometric model This error depends on the accuracy of the laser mechanical installation.7 Calibration of laser rotation axis This error is optimized by the proposed objective function.8 Laser center curve extraction Center extraction algorithm is based on the Hessian Matrix and ensures a high extraction accuracy.