Abstract
Camera calibration based on point features leads the main trends in visionbased measurement systems for both fundamental researches and potential applications. However, the calibration results tend to be affected by the precision of the feature point extraction in the camera images. As the point features are noise sensitive, line features are more appropriate to provide a stable calibration due to the noise immunity of line features. We propose a calibration method using the perpendicularity of the lines on a 2D target. The objective function of the camera internal parameters is theoretically constructed by the reverse projections of the image lines on a 2D target in the world coordinate system. We experimentally explore the performances of the perpendicularity method and compare them with the point feature methods at different distances. By the perpendicularity and the noise immunity of the lines, our work achieves a relatively higher calibration precision.
Introduction
Camera is considered as an important instrument in the researches of threedimensional reconstruction and computer vision^{1,2,3,4,5}. The purpose of camera calibration is to obtain the transform parameters between the 2D image and the 3D space, which is crucial for the applications of biology^{6}, materials^{7}, image processing^{8,9,10,11}, photon imaging^{12,13,14,15,16}, physical measurement^{17,18,19,20}, object detection^{21} and sensors^{22,23}. A camera calibration system normally includes a CCD camera to capture the 2D image and a calibration target that is placed in the view filed of the camera in the 3D space. The measurement precision of the camera depends on the calibrated parameters. Therefore, it is significant to calibrate the camera with a precise approach.
The camera calibration is widely studied in recent years. It can be classified in three main categories, 3D cubebased calibration method, 2D planebased calibration method and 1D barbased calibration method. The 3D calibration methods are initially developed by AbdelAziz^{24}. A 3D cubic target is established to provide the coordinates of the 3D points. The direct linear transform is invented to determine the transform matrix of the camera. Xu^{25} donated a threeDOF (degree of freedom) global calibration system to accurately move and rotate the 3D calibration board. A threeDOF global calibration model is constructed to calibrate the binocular systems at different positions. Then, the 2D planebased methods^{26,27,28,29,30} are provided to promote the convenience of the onsite calibration and to simplify the target fabrication. A calibration method is proposed by Ying^{31} based on the geometric invariants. The camera parameters are solved by the projections of two lines and three spheres in the camera calibration. Shishir^{32} presented a method to calibrate a fisheye lens camera. The camera is calibrated by defining a mapping between the points in the world coordinate system and the corresponding point locations in the image plane. Bell^{33} proposed a method to calibrate the camera by using a digital display to generate fringe patterns that encode feature points into the carrier phase. These feature points are accurately recovered, even if the fringe patterns are substantially blurred. Zhang^{34} outlined a camera calibration method that based on a 1D target with feature balls. The camera calibration is solved if one point is fixed. A solution is developed if six or more images of the 1D target are observed. Later, Miyagawa^{35} presented a simple camera calibration method from a single image using five points on two orthogonal 1D targets. The bundle adjustment technique is proposed to optimize the camera parameters. On the whole, although the accurate camera calibration is achieved by the 3D calibration targets, the precise 3D target is always difficult to be fabricated and carried. Moreover, it is difficult to apply in many fields due to the volume of the 3D target. The 2D methods are investigated to provide a convenient calibration compared with the 3D methods. A 2D calibration target contributes sufficient information of geometrical features to accurately calibrate the camera^{36,37,38}. Besides, the 2D calibration target is in moderate size and easy to be fabricated. Although the 1D target takes the simplest model in geometry, the 1D calibration method generates less information compared with the 2D calibration method in one image.
A reliable method is outlined in this paper according to the 2D calibration target. As the point features are normally adopted in the 2D calibration method, the transform matrix is calculated by the relationship between the 2D projective lines in the image and the lines in the 3D space. Then, the reprojection errors of the lines are studied in order to verify the validity of the method. The pointbased calibration methods are compared with the linebased method to evaluate the accuracy and the noise immunity in the calibration process. Finally, the lines on the target are reconstructed by the projective lines and the transform matrix with the camera parameters. The objective function is built and optimized by the perpendicularity of the reconstructed lines. The perpendicularity method is compared with the original methods to evaluate the performances of the approaches.
Results
In the process of the camera calibration, it is important to provide the relationship between the coordinates of the 2D image and the ones of the 3D space. A method is proposed for the camera calibration according to the perpendicularity of 2D lines in Fig. 1. A 2D target with the checkerboard pattern is adopted as the calibration object to construct the perpendicularity of 2D lines. The world coordinate system O_{w}X_{w}Y_{w}Z_{w} is attached on the 2D calibration board and the original point of the world coordinate system O_{w} is located at the upper left corner of the 2D calibration board. The axes O_{w}X_{w} and O_{w}Y_{w} are defined along two vertical sides of the target. The 2D calibration board is arbitrarily placed in the view filed of the camera. O_{c}X_{c}Y_{c}Z_{c} indicates the camera coordinate system. The original point of the camera coordinate system O_{c} is located at the optical center of the camera. O_{c}X_{c}, O_{c}Y_{c} are respectively parallel to O_{p}X_{p}, O_{p}Y_{p} of the image coordinate system. O_{c}Z_{c} is the optical axis that is perpendicular to the image plane. a_{i} and b_{i} are two perpendicular lines generated from the checkerboard pattern on the 2D target. As the world coordinate system is attached to the 2D target, a_{i} and b_{i} are defined in the world coordinate system. and are the lines in the image coordinate system corresponding to the two perpendicular lines a_{i} and b_{i} in the world coordinate system. The transformation from the world coordinate system to the image coordinate system is a typical projective transformation; therefore, the line is not usually perpendicular to the line in the image. The vectors of the two lines and can be solved from the observations of the camera.
According to the calibration method, the transform matrix H_{a} is generated from the 2D lines in the world coordinate system and the 2D projective lines in the image coordinate system, firstly. Then, the initial solutions of the intrinsic parameters of the camera are contributed by the transform matrices of the observations. Finally, the optimal solutions are provided by minimizing the objective function. The following experiments are divided to two aspects, the initial solution experiments and the optimal solution experiments, in order to verify the validity of the method based on the perpendicularity of 2D lines. The pointbased methods proposed by Zhang^{37} and Tsai^{39} are adopted as the comparative methods. The two methods are 2D planebased calibrations. An A4 paper with the checkerboard pattern is covered on the 2D target. The size of each square is 10 mm × 10 mm. Four capture distances, 400 mm, 500 mm, 600 mm and 800 mm, are chosen to study the effect of distance in the experiments, respectively. For each distance, ten images are captured to calibrate the camera. The resolution of the images is 1024 × 768. In the process of camera calibration, 32 lines are defined by the checkerboard paper in the world coordinate system. Figure 2(a–d) show the original images observed at the distances of 400 mm, 500 mm, 600 mm and 800 mm in the first group of images, respectively. Similarly, Fig. 2(e–h) are the second group of images at the distances of 400 mm, 500 mm, 600 mm and 800 mm, respectively.
The coordinates of the 2D projective lines are extracted by the Hough transform^{40,41,42} in the images. Hough transform finds the straight line in the parameter space that is less affected by noises. So the result of the line extraction is more stable than the result of the point extraction. Figure 3 shows the results of the Hough transform in the polar coordinate system. A sinusoidal curve corresponds to a 2D point in the Cartesian coordinate system. The blue crosses symbolize the radial coordinates ρ and the angular coordinates θ of 32 lines in the polar coordinate system. The extraction results of the lines are illustrated in Fig. 4(a–h). It is obviously that the Hough transform accurately detects the lines on the calibration target.
The linebased method is compared to the pointbased methods to verify the calibration validity and noise immunity. The transform matrix H_{a} of the line projection is experimentally obtained from the coordinates of the lines in the image coordinate system and the coordinates of the lines in the world coordinate system. Then we have^{43}
where a_{ai} is the line in the world coordinate system, is the reprojection of the line a_{ai} by the transform matrix H_{a}. The transform matrices of the Zhang’s method and Tsai’s method are denoted by H_{m}, H_{t}. We have^{43}
where M_{i} is the point in the world coordinate system, m_{i} is the point in the image coordinate system. The image points m_{i} are fitted to a straight line by the least square method. The coordinates of these lines are denoted by , . The errors of three methods are defined by
where Δa_{ai} is the error of the linebased method, Δa_{mi} is the error of the Zhang’s pointbased method with 2D plane target, Δa_{ti} is the error of the Tsai’s pointbased method with 2D plane target, is the image line generated by the Hough transform, is the reprojection of the line a_{ai} by the transform matrix H_{a}, is the fitted line derived from the transform matrix H_{m} and the reprojections of points, is the fitted line derived from the transform matrix H_{t} and the reprojections of points.
The errors of the perpendicularity method, Zhang’s method and Tsai’s method are shown in Fig. 5(a–h). The errors of the linebased method are less than the pointbased methods. The first group of experiments corresponds to Fig. 5(a–d). The mean errors and the variances are listed in Table 1. The second group of experiments corresponds to Fig. 5(e–h). The mean errors and the variances are listed in Table 2. According to the error data above, the errors in the X direction, Y direction and the rootmeansquare errors of the linebased method are all far less than the errors of the pointbased methods in the two groups of experiments. The errors of the linebased method vary indistinctively with the increasing distance. However, the errors of the pointbased methods show the increasing trend when the distance is on the rise. The error variances of the linebased method have been compared with the error variances of the pointbased methods. It is indicated that the variation range of the errors using the linebased method is smaller than the variation range of the pointbased methods. The variances of the errors adopting the linebased method fluctuate in a small range with the increase of the distance. However, the variances of the errors using the pointbased methods provide a significant jump with the increasing distance. The results reveal that the linebased method is less affected by the capture distance compared to the pointbased methods. According to the results and analyses above, the linebased method contributes higher accuracy in the camera calibration process.
The calibration accuracy of the three methods is further analyzed by adding different levels of Gaussian noises to the original images. The variances of the added noises are 0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02 and 0.05, respectively. The average errors are identified by the rootmeansquare errors of the lines. The linebased method and Zhang’s and Tsai’s pointbased methods are compared in Fig. 6. The values of the Gaussian noises are shown by the denary logarithms and the values of the average errors are presented by the natural logarithms for the purpose of direct observation. Figure 6(a–d) show the relationship between the average errors and the values of the noises at the distances of 400 mm, 500 mm, 600 mm and 800 mm, respectively. The errors of the three methods are all on the increase with the increasing noises. The average rootmeansquare errors of the ten images using the linebased method increase from 9.06 × 10^{−5} to 4.10 × 10^{−2} as the noises vary from 0.0001 to 0.05 at the distance of 400 mm. Nevertheless, the average rootmeansquare errors of the ten images based on the Zhang’s and Tsai’s methods are from 3.02 × 10^{−2} to 2.62 × 10^{−1} and from 5.14×10^{−2} to 3.15×10^{−1}, as the noises vary from 0.0001 to 0.05 at the distance of 400 mm. Moreover, when the distance is 500 mm, the average rootmeansquare errors of the ten images grow from 8.58 × 10^{−5} to 3.80 × 10^{−2}. The average errors of Zhang’s and Tsai’s methods are from 2.81 × 10^{−2} to 2.52 × 10^{−1} and from 5.53×10^{−2} to 3.84×10^{−1}, respectively. At the distance of 600 mm, the rootmeansquare errors of the ten images using the linebased method increase from 9.88 × 10^{−5} to 4.08 × 10^{−2}, respectively. The average errors of Zhang’s and Tsai’s methods are from 3.09 × 10^{−2} to 2.61 × 10^{−1} and from 5.54×10^{−2} to 4.20×10^{−1}, respectively. At the distance of 800 mm, the rootmeansquare errors of the ten images using the linebased method grow from 9.13 × 10^{−5} to 3.93 × 10^{−2}, respectively. The rootmeansquare errors of Zhang’s and Tsai’s methods are from 3.21 × 10^{−2} to 2.57 × 10^{−1} and from 6.07×10^{−2} to 3.85×10^{−1}, respectively. It is evident that the average errors of the linebased method steadily increase as the noises are on the rise. However, the average errors of the linebased method grow more slowly than the errors of the pointbased methods under the nine levels of noises. It proves that the linebased method provides a better noise immunity compared with the pointbased calibration methods.
The calibration results are affected by noises in the camera calibration process. Therefore, the initial solution of the intrinsic parameters should be optimized to approach the real values of the parameters. The perpendicular method is proposed to solve the optimal value of the intrinsic parameter matrix. The elements of u_{0} and v_{0} indicate the principal point with pixel dimensions. As the principal point should theoretically coincide with the center of the image, u_{0} and v_{0} are chosen to evaluate the validity of the proposed optimal method. Figure 7 presents the optimal results of the initial values of the two elements u_{0} and v_{0} in the intrinsic parameter matrix. The dotted lines in Fig. 7 present the coordinates of the image center in the image coordinate system. Comparative experiments are performed on the perpendicularity method and Zhang’s method at the different distances.
According to Fig. 7, the initial values of u_{0} and v_{0} using the perpendicularity method approach to the coordinates of the center points of the images with the rising number of the images. The first few points are far away from the dotted lines. However, the optimal values of the perpendicularity optimal method are all near the dotted lines. The initial values of u_{0} and v_{0} based on Zhang’s and Tsai’s methods vary a lot with the increasing numbers of the images. Moreover, the optimal values are basically near the dotted lines as the number of the images is on the rise.
The means and the variances of the initial and optimal values of u_{0}, v_{0} are listed in Table 3. The calibrated intrinsic parameters of the camera at the different distances are shown in Table 4. Considering the experiment data above, the mean values of the initial u_{0} and v_{0} adopting the perpendicularity method are more close to the coordinates of the center point compared with the Zhang’s and Tsai’s methods. The optimal values of u_{0} and v_{0} of the perpendicularity method show an obviously decreasing trend. The mean optimal values of u_{0} and v_{0} are close to the coordinates of the center point of the image. The optimal values of u_{0} and v_{0} based on the Zhang’s and Tsai’s optimal methods are also close to the coordinates of the center point. However, the descending velocity of the perpendicularity method is higher than the velocity of Zhang’s and Tsai’s optimal methods. Moreover, the variances of the optimal values of u_{0} and v_{0} based on Zhang’s and Tsai’s methods are larger than the variances of the perpendicularity method, except the variances of the initial and optimal values of u_{0} at the distance of 400 mm, the optimal value of v_{0} at the distance of 500 mm, the initial value of u_{0}, the initial and the optimal values of v_{0} at the distance of 600 mm, and the initial value of u_{0} at the distance of 800 mm.
Discussion
In the experiment results, the linebased calibration method contributes an initial solution with higher accuracy. The perpendicularity method describes a better optimization approach. In the camera calibration process, the coordinates of the geometrical features significantly affect the accuracy of the camera calibration. In the perpendicularity method, the Hough transform is employed to extract the coordinates of the lines. As the Hough transform takes the advantage of the higher noise immunity than the point extraction method, the coordinates of lines are more accurate than the coordinates of points for the geometrical features. Furthermore, the 2D lines are stable features with respect to the variable distance. However, the 2D points that are identified in a close observation are smaller in a far observation. As the noise effects are the same on the images, the point feature tends to be recognized in the different locations for different distances. Finally, as the lines pass though the feature points, the objective function adopts the perpendicularity of the lines as the optimal object, which includes more geometrical information than the feature point superposition.
Methods
The previous method adopts points as the calibration features. Consequently, Harris corner detector^{38} is often chosen to extract point coordinates in the image to calibrate the camera. In this paper, the coordinates of the lines in the image coordinate system are generated from the Hough transform^{40,41}. The coordinates of a random 2D point in the Cartesian coordinate system correspond to a sinusoidal curve with two parameters, the radial coordinate ρ and the angular coordinate θ, in the polar coordinate system. Thus, a line in the Cartesian coordinate system is transferred to a series of sinusoidal curves in the polar coordinate system by the Hough transform. The polar coordinates of the crossing point of the sinusoidal curves relate to the correct coordinates of the line in the Cartesian coordinate system. Thus, Hough transform extracts the line in the image by solving the optimal values of the radial coordinate ρ and the angular coordinate θ in the parameter space and transferring them to the Cartesian coordinate system.
The calibration method includes two procedures, initial solution and optimal solution. The initial solutions of the homography matrix H_{a} and camera parameters are solved at first by the similar way to the pointbased calibration method^{37}. The line transform from the 2D world coordinate system to the image coordinate system is represented as^{43}
where a_{i} = [a_{i}, b_{i}, c_{i}]^{T}, = [, , ]^{T}, H_{a} = [h^{1} h^{2} h^{3}]^{T}, h^{jT} is the j^{th}row of H_{a}, H_{a} is a 3 × 3 transfer matrix of the camera.
As the cross product of two same vectors, and H_{a}a_{i}, is a zero vector 0, a 2D projective line in the image coordinate system and 2D line a_{i} in the world coordinate system satisfy
Equation (8) is rewritten as
where , , .
The singular value decomposition of the matrix G is expressed by^{44}
where B_{1} and C_{1} are orthogonal matrices, Λ_{1} is a diagonal matrix composed of the descending singular values.
From the right orthogonal matrix C_{1}, we have^{44}
where is the vector related to the smallest singular value in Λ_{1}. H_{a} is obtained by arranging vector h.
The transform matrix H_{a} stands for the projection between 2D world lines and 2D image lines, however, the camera parameters have been calculated by the pointbased transform matrix H_{m}. Therefore, the transform from the linebased matrix H_{a} to the pointbased matrix H_{m} should be performed on H_{a}. Although the relationship between H_{a} and H_{m} are given by ref. 43, we investigate the relationship by the following another way.
Two 2D points x_{1} and x_{2} are projected to the image points and by the pointbased transform matrix H_{m} as follows^{43}
The 2D line l determined by the 2D points x_{1} and x_{2} is carried out as
The projective line l′ determined by the image points and is
From equations (12), (13) and (15), the projective line l′ is
The right of equation (14) is transferred to
where is the adjoint matrix of H_{m}.
From equations (14), (16) and (17), we have
For a nonsingular projective matrix H_{m}, it is well known that
Stacking equations (7), (18) and (19), we obtain
where .
A projective matrix H_{m} is decomposed to^{37}
where H_{m} = [h_{1} h_{2} h_{3}], h_{i} is the i^{th} column of H_{m}, is the intrinsic parameter matrix of the camera, (r_{1} r_{2} t) is the extrinsic parameters that relates the position and posture of the camera in the world coordinate system.
The intrinsic parameters of the camera can be solved by^{37}
where , q_{ij} = [h_{i1}h_{j1}, h_{i1}h_{j2} + h_{i2}h_{j1}, h_{i2}h_{j2}, h_{i3}h_{j1} + h_{i1}h_{j3}, h_{i3}h_{j2} + h_{i2}h_{j3}, h_{i3}h_{j3}]^{T}, x = [x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}].
The singular value decomposition of the matrix Q that is derived from several homography matrices H_{m} is expressed by^{44}
where B_{2} and C_{2} are orthogonal matrices, Λ_{2} is a diagonal matrix with the descending singular values. From the orthogonal matrix C_{2}, we have^{44}
According to equation (24), the intrinsic parameters can be determined by^{37}
where α, β are the scale factors of the image, γ is the skew parameter of the image axes, λ is a scalar, (u_{0}, v_{0}) is the principal point with pixel dimensions.
The extrinsic parameters r_{1}, r_{2} and t are obtained from equations (21) and (25) and given by^{37}
The intrinsic parameters in the matrix A are considered as the initial solutions of the camera. The image information is affected by noises, illuminations, capture distance and other factors. For this reason, an objective function is constructed to solve the optimal solutions of the camera parameters. The perpendicularity of lines is not preserved under the perspective imaging. However, the property is invariable to the reconstructed lines in the world coordinate system. The parameterized lines in the world coordinate system are reconstructed by the lines in the image coordinate system and camera parameters. Then, the objective function is established by the sum of the dot products among the perpendicular reconstructed lines. Finally, the camera parameters are achieved by minimizing the objective function.
According to equations (7) and (21), the relationship between the coordinates of the lines in the world coordinate system and the coordinates of the lines in the image coordinate system can be represented as^{43}
where , are the coordinates of the image lines in the i^{th} image, is the transpose of the pointbased homography matrix of the i^{th} image, a_{i}, b_{i} are the reconstructed lines in the world coordinate system.
The sum of the dot products among the perpendicular reconstructed lines should be theoretically zero, then
where a_{i} and b_{i} indicate a vertical line and a horizontal line in the world coordinate system, respectively.
From equation (21), H_{mi}is written by the product of the intrinsic matrix and the extrinsic matrix as
Stacking equations (27)(30), , , , the objective function considering the perpendicularity of the reconstructed lines is given by
The optimal elements of the intrinsic parameters of the camera are obtained by minimizing the objective function and given by
where arg means the arguments correspond to the minimized function f(u_{0}, v_{0}, α, β, γ,).
Additional Information
How to cite this article: Xu, G. et al. A method to calibrate a camera using perpendicularity of 2D lines in the target observations. Sci. Rep. 6, 34951; doi: 10.1038/srep34951 (2016).
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Acknowledgements
This work was funded by National Natural Science Foundation of China under Grant No. 51478204, No. 51205164, Natural Science Foundation of Jilin Province under Grant No. 20150101027JC, and China Postdoctoral Science Special Foundation under Grant No. 2014T70284.
Author information
Affiliations
Traffic and Transportation College, Nanling Campus, Jilin University, Renmin Str. 5988#, Changchun, China
 Guan Xu
 , Anqi Zheng
 & Jian Su
Mechanical Science and Engineering College, Nanling Campus, Jilin University, Renmin Str. 5988#, Changchun, China
 Xiaotao Li
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Contributions
G.X., A.Z. and X.L. wrote the main manuscript text, G.X. and A.Z. did the experiments, the data analysis and the calibration algorithm, X.L. and J.S. prepared all the figures. All authors contributed to the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Xiaotao Li.
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