An onsite structure parameters calibration of large FOV binocular stereovision based on small-size 2D target

Optik 124 (2013) 5164–5169

Contents lists available at ScienceDirect

Optik journal homepage: www.elsevier.de/ijleo

An onsite structure parameters calibration of large FOV binocular stereovision based on small-size 2D target Zhenxing Wang a , Zhuoqi Wu a , Xijin Zhen a,b , Rundang Yang b , Juntong Xi a,c,d,∗ a

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Shipbuilding Technology Research Institute, Shanghai 200030, China State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China d Shanghai Key Laboratory of Advanced Manufacturing Environment, Shanghai Jiao Tong University, Shanghai 200030, China b c

a r t i c l e

i n f o

Article history: Received 10 October 2012 Accepted 13 March 2013

Keywords: Onsite structure-parameter calibration Binocular stereovision Large field of view Small-size target

a b s t r a c t An onsite structure-parameter calibration method for large field of view (FOV) binocular stereovision based on a small-size 2D target is proposed. Compared with the traditional large-size target, a small-size 2D target is much easier to be manufactured with high accuracy and can be used in onsite calibration. Compared with the 1D target, a 2D target can complete the calibration quickly with fewer calibration images. Additionally, operating and maintaining a small-size 2D target are considerably convenient and flexible. Calibration experiments show that only 5 placements of the calibration target can accomplish the accurate calibration of structure parameters and the average value of 10 measurements for a distance of 1071 mm is 1071.057 mm with a RMS less than 0.2 mm. Moreover, the measurement accuracy of the stereovision calibrated with our method is a little more accurate than that of the one calibrated with traditional method. © 2013 Elsevier GmbH. All rights reserved.

1. Introduction Currently with fast measurement speed, high accuracy and large measurement range, binocular stereovision is broadly applied in the field of large dimension measurement [1–4]. Generally, the calibration of stereovision parameters, which consists of intrinsic parameters and structure parameters of cameras, should be done before measurement. The accuracy of stereovision calibration, which has a direct impact on the 3D reconstruction accuracy, mainly depends on the calibration algorithm, the calibration process and the calibration target. So it is very important to design an appropriate calibration target for the stereovision calibration, especially for the calibration of large FOV binocular stereovision, which is more difficult than that of conventional stereovision due to large measurement range. Recently, the calibration of large FOV binocular stereovision has aroused several scholars’ attention [5–11]. Más et al. [5] and Li et al. [6] calibrated the large FOV stereovision rig respectively in their research papers by using a large-size 2D target. However, as it is not convenient to be operated for large size, a large-size 2D target is not appropriate for onsite calibration in a complex production workshop. Xiao et al. [7] proposed an onsite calibration

∗ Corresponding author at: School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. Tel.: +86 021 34206343. E-mail address: [email protected] (J. Xi). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.077

method of binocular stereovision with large-scale FOV, using a noncoplanar cross target with ring coded points as feature information. The world coordinates of these ring coded points do not need to be known, which benefits the fabrication of the calibration target. But the size of the cross target is still big, which reduces the calibration flexibility. Xu et al. [8] adopted a large-size 3D target to calibrate a binocular stereovision rig. However, a large-size 3D target is also not suitable for onsite calibration of large FOV stereovision because of its low shape accuracy and inconvenience of maintenance and use. As the 2D and 3D calibration targets both had some shortages in the onsite calibration of large FOV, Xu et al. [9] and Sun et al. [10] used a 1D target to calibrate structure parameters of a stereovision rig with large FOV respectively. Compared to the large-size 2D and 3D targets, a 1D target is easier to be manufactured and more adaptable to be operated in complex measurement environment. However, with less feature points on a 1D target, a user should capture many more images of a 1D target during the calibration process, which decreases the calibration efficiency. Sometimes, an on-site assembly of a stereovision rig is often needed for using and maintaining. In addition, calibration parameters of stereovision, structure parameters in particular, are prone to change with an effect of working conditions such as temperature, vibration and so on, resulting in lowering measurement accuracy [11–13]. So the onsite calibration of a stereovision rig is extremely important in such circumstance. In fact, camera intrinsic parameters do not change as the geometry of the two cameras relative to one another, described by structure parameters, varies. Therefore

Z. Wang et al. / Optik 124 (2013) 5164–5169

we can accurately calibrate camera intrinsic parameters offline, leaving structure parameters to be calibrated onsite [14]. For that reason this paper proposes an onsite structure parameters calibration method of large FOV binocular stereovision based on small-size 2D target. Using a small-size 2D target to calibrate a stereovision rig, on one hand, can solve the problem of the inconvenience of onsite calibration with a large-size target. On the other hand, a small-size 2D target is simple to be manufactured with high accuracy and convenient to be maintained and used with flexibility. More importantly, only a few images of a small-size 2D target can finish the onsite calibration of a stereovision rig with high precision. This paper is structured as follows. Section 2 describes some related mathematical models employed in this paper. Experimental results are provided in Section 3. Finally, Section 4 summaries the paper.

5165

Fig. 1. Structure-parameter calibration model of stereovision.

of the left and right cameras respectively. According to the camera model and the binocular stereovision model, we have

2. Mathematical model

Mol = Rl Mo + Tl

(3)

Mor = Rr Mo + Tr

(4)

Mor = RMol + T

(5)

2.1. Calibration model of camera The pinhole model, an ideal non-distorted linear model, is the simplest and most commonly used camera model. Let Mw (Xw , Yw , Zw ) be a 3D point in the world coordinate frame and Mc (Xc , Yc , Zc ) be the corresponding 3D point in the camera coordinate frame. m(u,v,1) is the homogeneous coordinates of the 2D projection point. If the transformation between the world and camw era coordinate frames can be represented by R w c and T c , we can get: sm = AMc =

A( Rw c Mw

+ Tw c )

(1)





fx 0 u0 is the intrinsic 0 fy v0 0 0 1 parameter matrix of a camera; fx and fy represent the focal length of the camera in terms of pixel dimensions in the x and y direction respectively; (u0 , v0 ) are the coordinates of principle point. In practice, however, no lens is perfect without distortions. Thus, distortion correction is necessary for a lens [15–17]. Assuming that mp (up , vp ) and md (ud , vd ) are the distortion-free and distorted normalized image coordinates, respectively, we have [18]:

where s is a scale factor and A =



up

vp



 2

4

= (1 + k1 r + k2 r )

ud

vd

  +

2p1 ud vd + p2 (r 2 + 2u2d ) p1 (r 2 + 2v2d ) + 2p2 ud vd

 (2)

where r 2 = u2d + v2d ; k1 and k2 are the coefficients of radial distortion; p1 and p2 are the coefficients of tangential distortion. Given an adequate number of visible points whose world coordinates are precisely known, as well as their corresponding image coordinates, we can calibrate a camera by estimating the internal parameters A, w k1 , k2 , p1 , p2 and external parameters Rw c and Tc according to Eqs. (1) and (2). 2.2. Calibration model of structure parameters

with Eqs. (3)–(5), we can get R = Rr RTl

(6)

T = Tr − RTl

(7)

Eqs. (6) and (7) provide a method to compute the structure parameters of stereovision, when extrinsic parameters (Rl , Tl , Rr , Tr ) of the left and right cameras are given. Actually, if intrinsic parameters of a camera have been calibrated, we can figure out extrinsic parameters of the camera given at least three non-collinear points of the target and their corresponding image projections according to Eq. (1). Therefore, each location of the calibration target in the measurement volume gives a new set of extrinsic parameters of the left and right cameras with intrinsic parameters calibrated offline. Then structure parameters can be got again according to Eqs. (6) and (7). In theory, putting the target in the measurement volume for one time can complete the calibration of structure parameters. However, because of image noise and rounding errors, each location of the target results in slightly different values for R and T [19,20]. In order to reduce errors and noise, we can take the median values of the R and T parameters as the initial approximation of the true solution, and then establish a cost function to find the best R and T, which satisfy that the reprojection error of the target feature points for both camera views is minimum. The cost function is expressed by

⎡   2 ml,i,j − m ⎣ ˆ l,i,j (Rl,i , Tl,i ) min i

j

⎤  2 mr,i,j − m ˆ r,i,j (Rr,i , Tr,i ) ⎦ +

(8)

j

The calibration model of structure parameters of stereovision is shown in Fig. 1. oo − xo yo zo is the coordinate frame of a calibration target. ol − xl yl zl and or − xr yr zr represent the left and right camera coordinate frames. R and T, the structure parameters, are the rotation matrix and the translation vector that bring the left-camera coordinate frame into the right; Rl and Tl (Rr and Tr , respectively) denote the rotation matrix and the translation vector from the target to the camera for the left (right, respectively) camera. Mo is a 3D point in the target coordinate system; Mol and Mor denote the locations of the 3D point Mo from the coordinate frame

where Rl ,i and Tl ,i (respectively, Rr,i and Tr,i ) denote the rotation matrix and the translation vector from the target in the ith view to the camera for the left (respectively, right) camera; ml,i,j and mr,i,j are real projections of the jth calibration mark in the ith image ˆ l,i,j (Rl,i , Tl,i ) (respecof the left and right cameras, respectively; m ˆ r,i,j (Rr,i , Tr,i )) denotes the calculated reprojection point on tively, m the image corresponding to ml,i,j (respectively, mr,i,j ).Turning to Eqs. (6) and (7), we deduce that Rr,i = RRl,i

(9)

5166

Z. Wang et al. / Optik 124 (2013) 5164–5169

Table 1 Calibration results of two cameras intrinsic parameters. Camera

Focal (pixel) fx

Cam L Cam R

fy

2574.86175 2577.77006 2570.70834 2573.49207

Principal point (pixel)

Distortion coefficients

cx

k1

cy

687.94187 543.58048 698.81932 525.98183

Tr,i = T + RTl,i

−0.06421 0.27867 −0.04531 0.18679

(10)

Following from Eqs. (9) and (10), the Eq. (8) becomes min

  2 ml,i,j − m ˆ l,i,j (Rl,i , Tl,i ) [ i

j

 2 mr,i,j − m ˆ r,i,j (Rl,i , Tl,i , R, T) ] +

k2

(11)

j

At last, the optimal solution of structure parameters can be obtained by running a robust Levenberg–Marquardt iterative algorithm [21]. The calibration process of the structure parameters can be summarized as follows: (1) Put the calibration target in the measurement volume for the ith time. (2) Let the left and right camera capture images of the target and calculate the extrinsic parameters of the two cameras, Rl,i , Tl,i , Rr,i , Rr,i , respectively. Then, determine the structure parameters, Ri , Ti . (3) If i < N, reset the target in a new position, and let i = i+1. Then return to step (1) until i = N. Where N is total number of target placement. (4) Take the median values of the {Ri } and {Ti } as the initial approximation of the structure parameters. (5) Run the Levenberg–Marquardt iterative algorithm to obtain an optimal solution of structure parameters, R and T. 3. Experiments

Reprojection error (pixel) p1

p2

εx

εy

0.00081 0.00014

−0.00068 −0.00200

0.21577 0.21499

0.09202 0.09107

to the calibration model of structure parameters, the calibration target should satisfy that it has at least three non-collinear feature points, while there were few requirements on its size. So we could use small-size 2D target to calibrate the structure parameters. Actually, a 14 LCD displayer, resolution of 1366 pixels × 768 pixels and pixel pitch of 0.227 mm, was chose as calibrate target for its common use. By placing the displayer, showing a chessboard on the screen, in different positions of the measurement volume for 5 times, 10 images were taken by the cameras, which are shown in Fig. 4. First, 5 sets of structure parameters, corresponding to the 5 different positions of the displayer, of the stereovision can be obtained according to calibration model of structure parameters. Then, by taking the median values of the 5 sets of structure parameters, we can estimate the initial value of structure parameters as:

 R0 =

0.7252 −0.0227 0.9997 0.0124 −0.6884 −0.0058

T0 = [ −1822.18662

0.6881 0.0199 0.7253

−2.72396



719.14529 ]

At last, after running the Levenberg–Marquardt iterative algorithm with the initial value of structure parameters, we get:

 R=

0.7273 −0.0224 0.0120 0.9997 −0.6862 −0.0063

T = [ −1821.16799

0.6860 0.0199 0.7274

−4.16747



715.36212 ]

The camera sensor used in this paper is CCD (charge-coupled device) with 1040 pixels × 1392 pixels, and the size of each pixel is 4.65 ␮m × 4.65 ␮m. The focal length of lens is 12 mm. The experimental calibration process is divided into three parts: firstly, calibrate intrinsic parameters of each camera individually; secondly, set up a binocular stereovision with two cameras calibrated in the first step and then calibrate the structure parameters; finally, evaluate the calibration precision of the stereovision. 3.1. Calibrating camera intrinsic parameters We used the method proposed by Zhang [22] to calibrate the camera intrinsic parameters with a working distance of about 2500 mm, considering the second-order radial and tangential distortion coefficients of the lens. The calibration pattern of chessboard was displayed by a 55 LCD with 1920 pixels × 1080 pixels, which had a pixel pitch of 0.63 mm. The calibration target is shown in Fig. 2. Table 1 shows the calibration results of two cameras identified by ‘Cam L’ and ‘Cam R’, respectively.

Fig. 2. Calibration target of a single camera.

3.2. Calibrating structure parameters of stereovision Now, with the two cameras calibrated in Section 3.1, a stereovision rig was set up with a working distance of about 2500 mm and a baseline distance of about 2000 mm, the distance between the two cameras’ projection centers. The setup is shown in Fig. 3. The measurement volume was about 1500 mm × 1000 mm. According

Fig. 3. Binocular stereovision rig.

Z. Wang et al. / Optik 124 (2013) 5164–5169

5167

The reprojection error of the chessboard corners for both camera views is 0.064 pixel after optimization. In addition, the calibration process mentioned above demonstrates that the method proposed in this paper can finish the accurate calibration of structure parameters with fewer times of placing the target in the measurement volume, while the method proposed by Sun [11] calibrated the structure parameters with 22 times of placing the 1D target in the measurement area. So our method is more efficient.

compute the distance between each pair of corners in the world coordinate frame, represented by the left camera coordinate frame. Afterwards, compare the difference between the calculated distance and theoretical distance for each pair of corners and the results are shown in Table 2. Table 2 shows that the average value of the calculated distance is 1071.057 mm and the RMS error is less than 0.2 mm. Moreover, we discover that the error decreases when the pair of corners is close to the middle of image and increases as the pair of corners is away from the image middle. The main reason for that is as follows: first, it is caused by the residual lens distortions, which affect the pixel near the periphery more than the one near the center; second, it is caused by the tiny deformation of the LCD displayer. As there is a support in the middle part of the LCD displayer, the deformation is small in the middle region but large in the upper and lower parts. Therefore, the object to be measured should be placed near the center of the FOV. Furthermore, we compared our calibration method with the traditional method, which calibrated the camera intrinsic parameters and structure parameters simultaneously with the large-size 2D LCD displayer. Table 3 lists all the measurement results. Table 3 shows that RMS error of traditional method is a little more than that of our method. In addition, we can find that the average deviation of our method is smaller than that of traditional method. Reasons for that may be as follows. First, when the camera intrinsic-parameter is calibrated by the traditional method, the orientation of the calibration target is subject to the different viewpoint of each camera because the calibration target must be seen by the two cameras simultaneously. This situation is shown in Fig. 6(a), where the angle ˛ between the optical axis of camera C1 and calibration plate is limited within (0, ␲–), which leads to an incomprehensive calibration of each camera. However, the range of angle ˛ can extend to (0, ␲) when the camera is calibrated alone shown in Fig. 6(b). Therefore, with our method, the camera intrinsic-parameter calibration is more comprehensive for the separate calibration of each camera. Second, in terms of structureparameter calibration, the calibration target used in our method is a small-size LCD with a physical resolution of 0.227 mm/pixel, which has higher accuracy than the large-size LCD with a physical resolution of 0.63 mm/pixel used in traditional method. Third, the flatness error of the 14 LCD is smaller than that of the 55 LCD for small size.

3.3. Calibration evaluation

4. Summary

Fig. 5 shows that 10 pairs of corners are set on the chessboard displayed by the large-size LCD mentioned above and the pixel distance of each pair of corners is 1700 pixels. With the pixel pitch of 0.63 mm of the LCD displayer, we can calculate that the physical distance of each pair of corners is 1071 mm. Then use the stereovision to measure the 3D coordinates of each pair of corners and

In this paper, we described an onsite structure-parameter calibration method for large FOV binocular stereovision based on a small-size 2D target. By utilizing a small-size 2D target, the method has some advantages. First, compared with the

Fig. 4. Calibration process of structure parameters: (a) Cam L 1; (b) Cam R 1; (c) Cam L 2; (d) Cam R 2; (e) Cam L 3; (f) Cam R 3; (g) Cam L 4; (h) Cam R 4; (i) Cam L 5; (j) Cam R 5; Cam L and Cam R denote the left and right cameras, respectively.

Fig. 5. Image of calibration evaluation.

Fig. 6. Effect of calibration style on orientation of calibration plate’s placement: (a) traditional method; (b) our method.  denotes the angle between the two optical axes of camera C1 and camera C2 . The two optical axes intersect at point O.

5168

Z. Wang et al. / Optik 124 (2013) 5164–5169

Table 2 Measurement data and error of the stereovision rig. NO.

1# 2# 3# 4# 5# 6# 7# 8# 9# 10#

Image coordinates of left camera

Image coordinates of right camera

Reconstructed 3D coordinates

u (pixel)

v (pixel)

u (pixel)

v (pixel)

192.089539 1148.323120 193.946747 1152.554810 195.791153 1156.783569 197.758728 1161.210571 199.782455 1165.563721 201.881989 1169.918335 204.225662 1174.303101 206.512085 1178.645630 208.728607 1183.214233 211.061981 1187.669434

332.644531 366.643158 399.394928 422.983826 466.580719 479.487427 534.197083 536.441162 602.219055 593.492615 670.513733 650.909546 739.261047 708.618286 808.446472 766.728882 877.934265 824.888367 947.854858 883.581604

225.627151 379.069855 1230.543213 381.517029 220.934235 436.817566 1228.102905 447.076843 215.999390 494.666687 1225.653687 512.996094 211.182358 552.951965 1223.126587 579.520264 206.278793 611.519287 1220.443970 646.330322 201.196426 670.438171 1217.855957 713.593140 196.001160 729.604248 1215.229858 781.352539 190.809372 789.118713 1212.438110 849.498901 185.582535 848.873901 1209.773071 918.109924 180.459213 909.171814 1207.037231 987.209473 Average (mm)

X (mm)

Y (mm)

Z (mm)

−82.567021 −122.752323 −19.734928 −59.841023 43.024864 2.945365 105.843272 65.906376 168.651691 128.677464 231.425295 191.47592 294.169459 254.297045 356.935903 317.154134 419.615012 379.866277 482.422789 442.733766

−463.437109 605.741383 −460.812318 608.300552 −458.291186 610.876584 −455.666748 613.490517 −453.052234 615.993116 −450.459596 618.599649 −447.753426 621.219028 −445.104013 623.732806 −442.530982 626.422232 −439.842887 629.051358

162.228472 107.673684 166.281187 112.4606 170.627901 117.209579 174.864901 122.300687 179.228836 127.416847 183.877203 132.370024 188.893497 137.363389 193.764747 142.421055 198.547091 147.655976 203.219032 152.723017

RMS error(mm)

Calculated distance (mm)

Error (mm)

1071.323355

0.323355

1071.217758

0.217758

1071.251419

0.251419

1071.193357

0.193357

1071.046395

0.046395

1071.044635

0.044635

1070.95624

−0.04376

1070.808531

−0.191469

1070.901883

−0.098117

1070.822101

−0.177899

1071.0565674

0.0565674

0.18396

Table 3 Measurement data and error of the distance of 1071 mm. Method

Maximum positive and negative deviation (mm)

Average deviation (mm)

RMS error (mm)

Traditional method Our method

1071.363167 1071.323355

1071.08455 1071.0565674

0.200477 0.18396

1070.829938 1070.808531

large-size 2D and 3D targets, a small-size 2D target is much easier to be manufactured with high accuracy and can be used in onsite calibration with a complex environment; second, compared with the 1D target, a 2D target can complete the calibration quickly with fewer calibration images; third, operating and maintaining a small-size 2D target are considerably convenient and flexible. Calibration experiment shows that only 5 placements of the calibration target can accomplish the accurate calibration of structure parameters and the average value of 10 measurements for a distance of 1071 mm is 1071.057 mm with a RMS less than 0.2 mm. Moreover, the measurement accuracy of the stereovision calibrated with our method is a little more accurate than that of the one calibrated with traditional method. So the method proposed in this paper can be used in practical onsite structure-parameter calibration of large FOV binocular stereovision. Acknowledgments This work was supported by Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Project no. 51121063), National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Project no. 2012BAF12B01) and Shanghai Postdoctoral Sustentation Fund (12R21420400). References [1] T. Pinto, C. Kohler, A. Albertazzi, Regular mesh measurement of large free form surfaces using stereo vision and fringe projection, Opt. Laser Eng. 50 (2012) 910–916. [2] J. Sun, G. Zhang, Z. Wei, F. Zhou, Large 3D free surface measurement using a mobile coded light-based stereo vision system, Sens. Actuators A: Phys. 132 (2006) 460–471.

[3] N. Ahuja, A.L. Abbott, Active stereo: integrating disparity, vergence, focus, aperture and calibration for surface estimation, IEEE Trans. Pattern Anal. Mach. Intell. 15 (1993) 1007–1029. [4] J. Aguilar, F. Torres, M. Lope, Stereo vision for 3D measurement: accuracy analysis, calibration and industrial applications, Measurement 18 (1996) 193–200. [5] F. Rovira-Más, Q. Wang, Q. Zhang, Design parameters for adjusting the visual field of binocular stereo cameras, Biosystems Eng. 105 (2010) 59–70. [6] Y. Li, Y. Ruichek, C. Cappelle, 3D triangulation based extrinsic calibration between a stereo vision system and a LIDAR, in: Proceedings of 14th International IEEE Conference on Intelligent Transportation Systems, Washington, USA, 2008, pp. 797–802. [7] Z. Xiao, L. Jin, D. Yu, Z. Tang, A cross-target-based accurate calibration method of binocular stereo systems with large-scale field-of-view, Measurement 43 (2010) 747–754. [8] G. Xu, X. Li, J. Su, H. Pan, G. Tian, Precision evaluation of three-dimensional feature points measurement by binocular vision, J. Opt. Soc. Korea 15 (2011) 30–37. [9] Q. Xu, D. Ye, R. Che, Q. Fan, Online extrinsic parameter calibration of stereo vision coordinate measurement system, Proc. SPIE (2008) 672360. [10] J. Sun, Q. Liu, Z. Liu, G. Zhang, A calibration method for stereo vision sensor with large FOV based on 1D targets, Opt. Laser Eng. 49 (2011) 1245–1250. [11] J.M. Collado, C. Hilario, A. de la Escalera, J.M. Armingol, Self-calibration of an on-board stereo-vision system for driver assistance systems, in: Proceedings of IEEE Conference on Intelligent Vehicles Symposium, Tokyo, Japan, 2006, pp. 156–162. [12] N. Pettersson, L. Petersson, Online stereo calibration using FPGAs, in: Proceedings of IEEE Conference on Intelligent Vehicles Symposium, 2005, pp. 55–60. [13] Z. Zhang, A stereovision system for a planetary rover: calibration, correlation, registration, and fusion, Mach. Vis. Appl. 10 (1997) 27–34. [14] F. Zhou, G. Zhang, Z. Wei, J. Jiang, Calibrating binocular vision sensor with onedimensional target of unknown motion, Chin. J. Mech. Eng. 42 (2006) 92–96 (in Chinese). [15] R. Tsai, A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses, IEEE J. Robot. Automat. 3 (1987) 323–344. [16] J. Weng, P. Cohen, M. Herniou, Camera calibration with distortion models and accuracy evaluation, IEEE Trans. Pattern Anal. Mach. Intell. 14 (1992) 965–980. [17] X. Chen, J. Xi, Y. Jin, J. Sun, Accurate calibration for a camera-projector measurement system based on structured light projection, Opt. Laser Eng. 47 (2009) 310–319. [18] D.C. Brown, Close-range camera calibration, Photogram. Eng. 37 (1971) 855–866.

Z. Wang et al. / Optik 124 (2013) 5164–5169 [19] G. Bradski, A. Kaehler, Learning OpenCV: Computer Vision with the OpenCV Library, O’Reilly Media, Sebastopol, CA, 2008. [20] J.Y. Bouguet, Camera calibration toolbox for matlab, http://www.vision. caltech.edu/bouguetj/calib doc, 2004.

5169

[21] J. More, The Levenberg–Marquardt algorithm, implementation and theory, Numer. Anal. (1978) 105–116. [22] Z. Zhang, A flexible new technique for camera calibration, IEEE Trans. Pattern Anal. Mach. Intell. 22 (2000) 1330–1334.