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Abstract
The light pen is one of the most useful spatial coordinate measurement methods. However, due to the principle of vision measurement, the traditional light pen has the disadvantages such as limited measurement range, large structure size and complex calibration process. In order to revolutionize the design of light pen, this paper proposes a full space coordinate measurement method with a portable light pen by using hybrid light field imaging. Firstly, a light field imaging system with depth estimation ability has been introduced to simplify the light pen structure to three target points by using P3P method. In order to improve the accuracy of the depth estimation, a hybrid imaging method and EPI method have been used. Secondly, a full spatial measurement method with a rotation platform has been presented in detail. Finally, the experiments have been designed to verify the effectiveness and accuracy of the proposed method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Space coordinate measurement is an essential technology for modern manufacturing and processing industry. With the development of the large-scale equipment manufacturing such as aircrafts ships, and automobiles, there is a great need for large-scale, easy-to-use and efficient methods of coordinate measurement [1].
The traditional coordinates measurement method mainly includes contact measurement and non-contact measurem1ent. Coordinate measuring machining (CMM) is one of the most widely used contact measurement method [2,3]. The measuring accuracy CMM is as high as 1µm with 1∼2 meters measured range. However, due to the contact measurement, mechanical transmission mode and limited measured range, the measurement efficiency and the application field is totally restricted. In order to extend the measured range and obtain non-contact measurement mode, laser tracker is one of the optional measurement methods [4]. The principle of laser tracker system is interference of light, and the measured range can be extended to more than 100 meters with 10µm precision. However, the main disadvantage of laser tracker is that only one special point can be measured at each test process. Meanwhile, the expensive price limits the applications of the laser tracker. Another popular coordinate measurement method is Indoor GPS which is a measurement network with several transmitters and portable receivers. However, for large-scale measurements, iGPS requires many signal transmitters to be installed indoors in order to obtain high accuracy which limits its portability [5]. Other 3D coordinates measurement methods such as laser scanner, laser radar can only obtain the point cloud data of measured piece, rather than single point. Therefore, the light pen technology based on vision measurement with its flexibility, real-time, non-contact measurement mode and excellent accuracy is the most suitable 3D coordinates measurement method in most industrial applications.
Many scholars have studied the light pen using the single CCD camera or binocular stereo vision principle [6]. The light pen with two cameras is based on the principle of binocular intersection, and the main advantage is simple in design. However, it can only measure coordinates within overlapped field-of-views of these two cam-eras and the cameras should be pre-calibrated. Conversely, the light pen with a single camera which doesn’t need to be calibrated each time offers good portability and ease of use. However, due to the principle of PNP which is used to solve the coordinates of the light pen with a single camera, at least six sets of points are required for calculation [7–9]. So, the structure of the light pen is much more complex, and the portability is seriously affected. To simplify the light pen with single camera, the minimum number of the points on the light pen with P3P algorithm is three while the distance information has been obtained. Laser tracking light pen measurement systems combined with vision such as the T-Probe from Leica in Switzerland and the Handy PROBE from Creaform in Canada capture the depth information from laser tracker and provide a suitable solution. However, the addition of the laser tracker makes the light pen system expensive and poorly portable. Another similar measurement method based on the rotating laser scanning without cameras can calculate the special coordinates by using the angular information obtained from the intersection laser beam [10]. Due to the similarity with the principle of monocular vision measurement, the minimum number of the marker points is six and the measurement system is much more complicated. Therefore, the light pen with a single camera by using P3P algorithm is the best solution for portability while the depth information can be obtained simultaneously.
Plenoptic imaging is an emerging technique that allows instantaneous 3D imaging of a scene using a single camera [11]. In contrast to conventional digital images, light field images can separately record light from multiple passing points [12–14]. It greatly simplifies the depth estimation process. Although Tao et al. accomplishing the goal of depth estimation with light-field camera by combining off-focus and corresponding depth cues [15,16], the accuracy of depth estimation is greatly limited by the resolution of the light-field camera. Therefore, the low accuracy of depth obtained directly using the light-field camera limits its application in light pen measurements [17].
In order to extend the application field of the light pen based on vision measurement principle with only three points, this paper presents a full-space three-point light pen measurement system based on hybrid imaging [18]. The system consists of a light field camera and a conventional industrial camera, a motorised turntable and a three-point light pen. Then the depth measurement accuracy of the light pen can be improved significantly by using the hybrid imaging system and the full space dimensional measurement can be achieved with the turntable rotation method. Therefore, the portable light pen with the advantages of full space, high accuracy, large scale measurement, multi-tasking will be accomplished.
This paper is organized as follows. Section 2 describes the principles of the proposed method, including hybrid light field images, Epipolar Surface Image (EPI) depth estimation [19–21], spinning parallelogram operator (SPO) measurements [22–24] and the P3P method. In Section 3, experiments are conducted based on a platform composed of cameras and get the accuracy of the method is analysed according to the experimental results. Finally, in section 4, we present concluding remarks and a brief overview of future improvements.
2 Method
2.1 Light field image fusion
The light field camera is consisted of a main lens, an image sensor and a microlens array which is mounted front the image sensor [25]. It is possible to obtain the depth information directly from the light field image by using the refocus method. However, the accuracy is limited by the size of the microlens array [26].
In order to improve the depth estimation accuracy, this paper proposes a hybrid light field imaging method. With Homography correspondence, it is possible to fuse light field images with images obtained by other methods of measurement to achieve hybrid imaging. As shown in Fig. 1, the overall hybrid light field imaging measurement system consists of these components. The nib of the light pen is used as the position point to be measured, and the traditional CCD and the light field camera are fixed side-by-side on a rotating stage. The traditional CCD is fixed parallel to the light field camera. The plane mirror and 50-50 beam splitter are parallel, and located directly in front of the two cameras at an angle of 45° to ensure that the imaging centers of the two cameras are in the same position. These two imaging systems shoot the light pen through a 50-50 beam splitter and a plane mirror to achieve the measurement of the nib position. The 50-50 beam splitter enables the transmitted and reflected light to be of the same intensity, so that the images captured by both cameras can be considered at the same angle. In this way, two sets of images can be obtained from these two cameras.
To ensure the accuracy of the hybrid light field imaging, the correspondence between the large pixels in the light field image and the pixels in the traditional CCD image will be obtained according to the homography principle. Firstly, by photographing the same checkerboard grid, the best (relatively similar) set of features of the two camera images can be obtained. Then the matching points can be used to calculate the Homography matrix that combinations the two images together. The relationship between the coordinates of the checkerboard grid in CCD image and light field image can be represented by Eq. (1). Finally, according to the known parameters, the correspondence of the other points can be calculated.
Figure 2 gives an illustration of the pixel fusion process between the light field images and the high-resolution traditional CCD images. The light field image contains $m \times n$ macro pixels which contains the pixel points covered under a single micro lens. Meanwhile, the high precision traditional CCD contains $p \times q$ effective pixels. Then the macro pixels can be mapped to the corresponding CCD imaging pixels by the Homography matrix, while the non-directly mapped pixel points can be calculated from the macro pixel information of the adjacent area. Finally, the high positional and angular resolution can be achieved simultaneously by the fusion algorithm from the hardware level. For each macro pixel in the light field image, a correspond pixel in high spatial resolution image can be found. Then these two images can be fused as one high resolution image in which some points have light field information and traditional CCD information while some points have only CCD information.
As illustrated in Fig. 2, when a traditional CCD image with $p \times q$ pixels and a light field image with $m \times n$ macro pixels have been captured, the light field information can be calculated from the neighboring yellow macropixels for each blue pixel $(i,j)$ as formula (2):
2.2 EPI depth estimation
The light field image contains multi-view information from the scene, which makes depth estimation possible. In order to obtain high precision depth estimation, the EPI (Epipolar plane image) method has been introduced in this paper. By fixing the rows or columns of the subaperture array of a light field camera, a certain row or column of the light field subaperture image can be arranged in the corresponding position to obtain an EPI.
As shown in Fig. 3, the Epipolar plane view shows that there are “boundaries” in the image. Taking into account that each point on the sloping line in the EPI image is obtained from a different view of the same point in the real world. Considering this particularity of EPI, the spinning parallelogram operator (SPO) can be used to find the angle of a sloping line. A parallelogram operator is created in the Epipolar plane with a direction equal to the possible slope $\theta $. In this EPI image, the tilt angle of the segmentation line is $\theta $. Due to the limitation of computer computing power and the limitation of the spatial resolution of the real light field camera, 32 labels are chosen in this paper. The method measures the slopes by maximizing distribution distances between two parts of the parallelogram window to extract depth information.
This correct matching of points as well as the line will split the parallelogram into two different parts, so the difference between these two parts can be used to predict the direction of the line. The center point of the parallelogram is set as the reference point and the orientation of the center line through the center point is the difference to be estimated. The center lines with different orientations, divide the window into two parts of the same size. The correct line, indicating the disparity information, can be figured out by finding the maximum distance between the distributions of pixel values on either side of the lines.
Firstly, the size of parallelogram operator will be defined by using a weighted function. The weights in the window are used to measure each pixel’s contribution to the distance. The weighting function $w(i,j)$ will be created by using the derivative of a Gaussian, as illustrated in Fig. 4. Specifically, for the reference point $P({x_r},{u_r})$ in the EPI ${I_{y,v}}(x,u)$ the corresponding pixels in the defined window are weighted by formula (3):
Then the weight of the point has been set according to its horizontal distance from the hypothetical matching line so that the point which has the same distance from the line will contribute the same amount of information to the distance measurement. Therefore, this method separates the window based on the pixel’s distance to the hypothetical line. After dealing with the large disparities which contains stairstep structures due to the sparse sampling, the weighted window shows the similar stairstep structure to estimate the slope. Compared with the other slope estimation methods which need the densely sampling of the large slope, the SPO can be applied immediately.
2.3 Distribution rule of the P3P method’s multiple solutions
When the relatively accuracy distance between the measurement system and the light pen has been estimated by using the above method, the coordinates of the nib of the light pen can be calculated through PNP method (Perspective N points). The PNP method is an algorithm for solving camera poses based on 3D to 2D point pairs. While a set of points have been known to correspond to projections onto an image plane and given the intrinsic camera parameters, the transformation matrix (three rotation parameters and three translation parameters) between the object frame to the camera frame can be calculated. In the light pen measurement systems, while the world and image coordinates of the characteristic laser points on the light pen have been captured, the camera’s poses can be solved through the PNP method.
It has been proved that when the number of point pairs is greater than 3, the PNP problem has unique solution while the points on the light pen are not distributed on the same plane, and the commercially available light pen always has more than 6 feature points. However, as the number of the feature points increases, the portability deteriorates. Therefore, 3 feature points light pen has been designed in our method. When the number of point pairs is reduced to 3, the PNP algorithm is simplified to the P3P algorithm. Only 3 pairs of 3D-2D matched points are required to solve the external reference matrix of the camera to find the spatial coordinates of the measured point.
As shown in Fig. 5, these three-imaging points a, b and c corresponds the three spatial points A,B and C on the light pen. The center of the camera optical axis is the point O.
In the spatial triangle, Eq. (4) can be obtained through the cosine theorem:
The unknown elements x,y can be solved by Eq. (5) and then the known quantities $\cos \left\langle {a,b} \right\rangle$, $\cos \left\langle {a,c} \right\rangle$, $\cos \left\langle {b,c} \right\rangle$, v and w can be calculated. The P3P calculation is a system of binary quadratic Equations which have up to 4 sets of solutions. Moreover, there are two sets of imaginary roots, which do not conform to the length proportional relationship. Therefore, two sets of RT matrices corresponding to two special planes can be calculated through P3P problem.
The RT matrix allows the calculation of the coordinates of any point in the plane in this camera coordinate system. In order to eliminate the wrong solution, the depth of these points relative to the measurement system will be used.
Assuming that the optical center of the camera coordinate system and the origin of the light pen coordinate system are both on the optical axis line. As shown in Fig. 6, the Euclidean distance is d, and the light pen system is rotated around its $x$-axis by $\alpha $. The rotation matrix $\textrm{R}$ of the light pen coordinate system with respect to the camera coordinate system can be calculated through Eq. (6):
The translation vector T is $T = {[0\textrm{,}0\textrm{,}d]^T}$, and sets d to 1000 mm, 1500 mm, 2000 mm and 2500 mm. The rotation angle α is at the range of $[ - 30^\circ ,30^\circ ]$ and the step size is set to $1^\circ $. The light pen coordinate system is established with the central light point as the origin, and the center of the light pen tip used in this paper is located 270 mm from the origin. Figure 6 shows that there are two results for P3P to solve the tilted surface, and the simulation is used to understand the difference between these two answers. The simulation results of the difference between the two sets of solutions caused by d difference and the tilt angles in the camera coordinate system are shown in Fig. 7.
From the analysis of the simulation results, the greater the tilt angle, the greater the difference in depth of the P3P multiple solutions within a reasonable range. Hence, in the subsequent light field data collection, in order to increase the robustness of the algorithm and the accuracy of selecting the correct solution, the light pen should be tilted at a certain angle with respect to the camera coordinate system to increase the depth difference of the P3P multi-solution. Then the wrong solution can be excluded and the robustness of the algorithm can be improved.
2.4 Full spatial measurement method with the light pen
For the camera measurement method, the measurement range is limited by the shooting range of the camera. So a rotation stage for measurement is used to solve this problem. As shown in Fig. 8, the camera coordinate system cannot be guaranteed to match the rotation plane exactly due to the assembly. The actual optical center of the camera is located near the rotation axis, and there will be an assembly angle between the rotation axis and the camera coordinate axis, and the camera motion relationship on the rotation axis can be obtained by calibrating the relationship between them.
Since the camera is fixed on the turntable, the rotation axis of the camera around the turntable is uncertain. Therefore, the external parameters of the measurement system before and after the change of rotation cannot be obtained directly. According to the external reference of the camera at different positions with respect to the world coordinate system, the measurement data at different angles can be unified in the same camera coordinate system.
When the camera rotates around a fixed axis the captured checkerboard grid image can be seen as a checkerboard grid rotating around a fixed rotation axis while the camera optical center position remains unchanged. As illustrated in Fig. 9, O is the rotation center, N is the rotation axis, and ${\textbf{R}_1}{\textbf{T}_1},{\textbf{R}_2}{\textbf{T}_2}, \cdot{\cdot} \cdot ,{\textbf{R}_n}{\textbf{T}_n}$ are the pose parameters of the camera at different angles in the coordinate system of the calibration plate, where $\textbf{R}$ is the rotation matrix and $\textbf{T}$ is the translation vector.
The monoclinic relationship between the world and pixel coordinates of the corner points on the calibration plate is obtained by combining the camera internal parameters according to the Eq. (7):
By converting the camera coordinate systems of different angles into the first camera coordinate system, the pose of the $i$-th camera with respect to the first camera can be calculated by Eq. (8):
By reading the coordinates of the checkerboard grid vertices at different angles, it is actually the position of the camera's optical center in a certain checkerboard grid coordinate system. As shown in Fig. 10, the optical center of the camera will form a circle which is distributed on a plane in three-dimensional space, and the camera's axis of rotation is a vector through the center of the circle and perpendicular to the plane. Therefore, the direction of rotation axis can be fitted using these coordinates. Due to manufacturing and assembly errors between the camera and the turntable, the axis of the turntable is not precisely vertical for the measurement work, and the camera's optical center does not pass through the axis of rotation. The actual structure of the camera's optical and rotational axes mounted on the turntable is shown in Fig. 10.
The circle center of the camera rotation is at the intersection of the rotation plane and the rotation axis. Then a three-dimensional spatial circle fitting method is used to obtain the coordinates of the center of the circle. Assume that the fitted plane equation is $ux + vy + wz + d = 0$, where $N = (u,v,w)$ is the directional quantity of the plane. The projection points ${O_i} = ({x_i},{y_i},{z_i})$ are obtained by projecting the 3D coordinates $( {x^{\prime}_i},{y^{\prime}_i},{z^{\prime}_i})$ of the camera optical center at different positions onto the plane through Eq. (9).
The first point in the sequence of the optical center is taken as the origin of the plane coordinate system while ${X_0} = {x^{\prime}_0},{Y_0} = {y^{\prime}_0},{Z_0} = {z^{\prime}_0}$.The positive direction of the X-axis is defined as the direction from the first camera photocenter point to the second point, and the positive direction of the Z-axis as the normal vector of the fitted plane, so that the direction of the Y-axis is also obtained. Finally, the three sets of the directions for the new axis can be solved through Eqs. (10)–(12):
The unit vectors of the three axes are $({{n_{1x}},{n_{1y}},{n_{1z}}} )$, $({{n_{2x}},{n_{2y}},{n_{2z}}} )$ and $({{n_{3x}},{n_{3y}},{n_{3z}}} )$, and the conversion matrix from 2D to 3D coordinates can be obtained through Eq. (13):
In this way the three-dimensional photocentric coordinates can be transformed to the same plane as Eq. (14):
Then the objective function is obtained as Eq. (15) using the least squares method for plane circle fitting.
By solving ${a_0},{b_0},r$, the final fitted 3D circle center coordinates $O = (a,b,c)$ can be obtained by Eq. (14).
The normal vector of the axis of rotation is $N = (u,v,w)$ and the position of the axis of rotation is $O = (a,b,c)$. When the camera is rotated by an angle $\theta $ around the rotation axis, the rotation matrix of the camera can be obtained as Eq. (16):
The center of light of the camera is not over the axis of rotation, so there is also some translation. The translation matrix is shown as Eq. (17):
Therefore, once the camera is fixed to the rotation stage, the images taken by the rotated camera can be unified under the same coordinate system by simply calibrating the camera in relation to the spindle.
3. Experimental validation and analysis of results
In order to verify the effectiveness of the proposed method, a set of experiments has been designed. As shown in Fig. 11, the light pen was fixed to the three-dimensional translation stage at a titled angle. The light field camera and traditional CCD were fixed on both sides of the 50-50 beam splitter on the electric rotation stage, ensuring that both cameras were in the same view as far as possible.
The light field camera was Lytro illum, with a microlens array of $625 \times 434$ and $15 \times 15$ pixels behind each microlens. The angular resolution is $15 \times 15$ which means 225 different directions of light emanating from a point can be recorded. This is equivalent to taking a picture of the object at the same time from 225 nearby locations behind the object plane. The reconstructed light field image is 3 channels $9375 \times 6510$, with a fixed focal length of 35 mm, an equivalent focal length of 70 mm, an equivalent conversion factor of 3.19, which means the nominal focal length of f = 21.944 mm, and a sensor image element size of 17.312 um.
This experiment used the HY-LW18-01A Lightweight Network Head, which uses a lightweight worm gear to allow continuous rotation in full horizontal space with a positioning accuracy of 0.1°.
The 5-dimensional packets captured by the light field camera were decoded to obtain the subaperture images and calibrated to obtain the camera internal reference matrix. The Homography matrix of the corresponding subaperture was obtained by calibrating the single-strain relationship captured by the traditional CCD, and the fusion algorithm was used to interpolate the traditional CCD image into the enlarged subaperture image using the single-strain principle to obtain a collection of subaperture images with higher resolution and reconstruct the light field image. Then the depth information of the light pen was obtained based on the light field EPI depth estimation. Two different sets of RT transformation matrices were obtained by the P3P algorithm. The depth values calculated by using EPI were compared with the depth values obtained from the two sets of coordinate transformations respectively, so as to exclude an erroneous solution and finally obtained the positional matrix of the light pen relative to the camera and the exact coordinates of the light pen tip.
The accuracy of the measurement was influenced by the precision of the electric rotation stage and the camera. Firstly, measurements were taken on the tiny moving light pen without rotating cameras. Secondly, the light pen was moved over a wide area and cameras are rotated through the rotation stage to take the measurement.
A checkerboard was fixed on the translation stage and the protation relationship between electric rotation stage and camera by rotating the camera could be obtained. Figure 12 shows the images of a fixed checkerboard captured by the camera at different angles.
The conversion matrix between the camera coordinate system of different angles and the world coordinate system could be calculated by using the monoclinic relationship between the phase plane and the tessellation plane in Eq. (7), and then the conversion matrix $\textbf{R}, \textbf{T}$ of the $i$-th camera relative to the camera coordinate system at the first view could be obtained by using Eq. (8). Taking the camera coordinate system with a 2° change as an example, the transformation matrix of the pose in the camera coordinate system at the first shot was shown in Eq. (18):
The translation vector at the first camera coordinate system after the camera optical center rotation angle is shown in Table 1.
Table 1. Camera translation
After obtaining different angular camera translations, the rotation parameters to the camera calculated using the plane fitting algorithm and the spatial circle fitting algorithm are shown in Table 2.
Table 2. Rotor parameters
The errors in the rotation axis calibration algorithm of this method can be obtained from the rotation axis parameters in Table 2. Where the calculated points are fitted to the same plane, the root mean square error from the point to the fitted result plane is 0.0027mm, and then the circle is fitted on this plane, the root mean square error of the radius is 0.3124mm.
The effect of fitting the rotating axis in space is shown in Fig. 13.
The images obtained from the light field camera and the traditional CCD were fused using Eq. (2) to obtain a high-resolution light-field image. Figure 14 compares the hybrid light field images obtained with this method. The EPI plane is selected and the depth of the synthesized light field image by rotating the parallelogram operator (SPO) was estimated. Figure 15 shows the depth values by SPO operator at each point on a horizontal line. This value indicates the depth difference between the point and the adjacent point.
Fig. 14. image of light field and hybrid light field (a) original light field image (b) hybrid light field image
According to Eq. (5), the calculation results of the two groups could be obtained. The relationship between the position of the light pen placement can be obtained through Fig. 15, and the wrong solution can be excluded from the calculation results. When the camera was not rotating and the small dimensions were measured from 0-200mm, the results can be obtained as shown in the Table 3.
Table 3. Experimental results of light pen translational measurements (The first one is correct) (unit: mm)
When the turntable is introduced, large scale measurements can be implemented. When the moving distance exceeds 2000mm, the corresponding matrix could be obtained according to the rotation angle using Eqs. (16)–(17). The measurement results are shown in Table 4.
Table 4. Large scale measurement experiment (unit: mm)
The above table records the measurement results of the traditional method and this paper's method for large-size equipment, where Method 1 is the traditional PNP measurement method and Method 2 is the proposed measurement method. From the table, it shows that the traditional light pen measurement system cannot measure beyond a certain range due to the small field of view.
It can be seen from Table 3 and Table 4, the measurement accuracy is mainly depended on the P3P algorithm when the rotation angle is during a proper range. Therefore, according to the simulation and Refs. [27], the measurement accuracy of the light pen is mainly affected by the measured distance. As demonstrated in Fig. 16, the measurement system was perpendicular to the movement path of the light pen and the vertical distance is about 3m. So the distance from the measurement system to the light pen would change from 3m to 4m. Therefore, there was no significant change in measurement error which is illustrated in Table 4. In the comparison experiment, the minimum measurement error of the traditional light pen measurement system is 0.16mm, the maximum is 0.23mm, and the average error is 0.18mm. The minimum error of the proposed measurement system is 0.12mm, the maximum error is 0.34mm, the average error is 0.22mm, and the root mean square error is 0.24mm.
4. Conclusion
A full-space hybrid optical field CMM system based on a precision rotary table to address the problems of traditional three-point optical pen vision measurement system has been proposed in this paper. This measurement method compensates for the limited resolution of the plenoptic camera and has higher accuracy when using the P3P method. From the experimental results, the measurement accuracy is mainly depended on the P3P algorithm when the rotation angle is during a proper range. The measurement error of the system is about 0.3 mm in the range of 5 m. This method allows the use of three marker points in a limited field of view to exclude incorrectly calculated values from traditional CCD by mixing light field images. Compared to binocular depth estimation, the proposed system does not require a large spacing and complex calibration process. The introduced rotation stage can effectively expand the measurement range of the three-point light pen measurement system. Finally, the full space coordinate measurement method with portable light pen has been accomplish.
Funding
National Natural Science Foundation of China (52005500); The Basic Science-research Funds of National University (3122019088); Foundation of Tianjin educational committee (2018KJ242); Tianjin Graduate Scientific Research Innovation Project (2020YJS006).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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