Calibration and rectification of vertically aligned binocular omnistereo vision systems
- Chengtao Cai^{1},
- Xiangyu Weng^{1}Email author,
- Bing Fan^{1} and
- Qidan Zhu^{1}
https://doi.org/10.1186/s13640-017-0195-0
© The Author(s). 2017
Received: 15 March 2017
Accepted: 26 June 2017
Published: 10 July 2017
Abstract
Omnidirectional stereo vision systems have been widely used as primary vision sensors in intelligent robot 3D measurement tasks, which require stereo calibration and rectification. Current stereo calibration and rectification methods suffer from complex calculations or a lack of accuracy. This paper establishes a simple and effective equivalency between an omnidirectional stereo vision system and a perspective vision system by studying stereo calibration and rectification methods. First, we improved the stereo calibration method. By applying the essential matrix, the complicated calibration process of the original method is simplified. By using a manual extraction method to extract corner points, noise error is eliminated and high precision is ensured. Second, we propose a new rectification method. By using the proposed simple rectification model and calibration data, the baseline length and an accurate column-aligned image pair are easily obtained, which reduces the computation time. The proposed stereo calibration and rectification method can simply and effectively obtain two key parameters of the triangulation formula for 3D measurement tasks: baseline length and parallax. Using real data captured by equipment, we performed experiments covering all the necessary stages to obtain a high-performance omnidirectional stereo vision system. Statistical analyses of the experimental results demonstrate the effectiveness of the proposed method.
Keywords
Robot Stereo calibration Rectification Stereo vision system1 Introduction
Omnidirectional stereo (omnistereo) vision systems composed of omnidirectional cameras offer the possibility of providing 3D measurement information for a 360° field of view. Several interesting configurations of omnistereo systems, such as binocular omnistereo [1], N-ocular omnistereo [2], circular projection omnistereo [3], and dynamic omnistereo [4], have been designed to achieve different mission requirements. Vertically aligned binocular (V-binocular) omnistereo vision systems, composed of two vertical coaxial catadioptric omnidirectional cameras, provide certain advantages over other types of omnistereo vision systems. (a) These systems possess a simple epipolar geometry correspondence. (b) The depth accuracy of the V-binocular omnistereo vision system is isotropic, and there are no occlusions of the image pair due to the coaxial installation. Due to the above advantages, V-binocular omnistereo vision systems have been widely used in many intelligent robot tasks [5–9]. In our research, to obtain a high-performance V-binocular omnistereo vision system, we focused on stereo calibration and rectification.
For a stereo system, calibration is the process of calibrating the camera intrinsic parameters and the camera-camera extrinsic relationship. There are two categories of current stereo calibration methods for omnistereo vision systems. One category is the calibration of the relative parameters between the camera and calibration boards. These parameters are then transformed into the camera-camera relationship [10]. Such methods provide high precision but typically require multiple calibration stereo pairs, and the Levenberg-Marquardt [11] iterative algorithm is required to reduce errors. Thus, significantly more work is required to configure the control points and measurement process. The other category consists of methods that calibrate the absolute parameters in the world coordinates [12] based on epipolar geometry [13]. The method uses only self-point correspondences in one image pair without requiring prior knowledge about the scene. However, accuracy suffers, making self-point correspondences in one image pair unsuitable for 3D information measurement tasks [14].
Stereo rectification aligns the corresponding points on the same column [15]. Current omnistereo rectification models also suffer from various defects. Some are limited to articular mirrors and produce heavily distorted images. Other models are not scan-line methods, and thereby lose the important advantage of simplified stereo matching. Y. Wang et al. proposed an omnistereo rectification method [14], which is a scan-line method and avoids heavy distortion; however, the rectification model is complicated and not suitable for real-time 3D-information measurement tasks.
To overcome these difficulties and obtain a high-performance omnistereo system, first, we improved the stereo calibration method [10] based on an epipolar geometry [12], which requires only a few matching points manually extracted from one image pair to reduce the complexity of the calibration and ensure accuracy. Second, we propose a simple rectification method. The calculation of baseline length and accurate column-aligned image pairs are easily achieved by using the proposed simple rectification model and calibration data. The proposed rectification method can reduce effort while ensuring a real-time calculation. After the proposed procedure, two key parameters of the triangulation formula for 3D measurement tasks, baseline length and parallax, are easily obtained.
Using real data captured by our system, we performed experiments with the proposed stereo calibration and rectification method and compared the data with those from some existing methods. We also performed other necessary experiments to verify the high performance of the proposed method, including stereo matching, 3D reconstruction and depth estimation. Statistical analyses of the experimental results demonstrate the effectiveness of the system.
2 Stereo calibration method
2.1 Single-viewpoint system calibration
- 1.
Extrinsic parameters: The relationship between the plane calibration plate coordinate system and the panoramic-camera coordinate system can be expressed by the formula x = PX, where P = [R _{w}, T _{w}] is a 3 × 4 matrix, as shown in Fig. 2. The rotation matrix R _{w} is a quaternion notation, wherein W represents the projection process, and V _{1} = [q _{ o }, q _{1}, q _{2}, q _{3}, t _{1}, t _{2}, t _{3}] represents the unknown variables in the projection matrix P.
- 2.
Nonlinear projection transformation: Assuming that the projection coordinates under the mirror coordinate system are known, the projection point coordinates on the metric plane can be calculated. H represents the nonlinear projection equation, and V _{2} = [ξ] represents the unknown variable of the mirror.
- 3.Distortion: The model introduces two primary distortions: radial distortion, which is caused by changes in radial curvature, and eccentric distortion, which is caused by the incomplete co-linearity of the axes of the optical lens. These distortions occur in optical systems as a result of assembly errors. Distortions can be expressed by five parameters, in which three are radial distortion δ _{ r } factors:$$ {\delta}_r=1+{k}_1{\rho}^2+{k}_2{\rho}^4+{k}_5{\rho}^6 $$(1)where \( \rho =\sqrt{x^2+{y}^2} \). The remaining two parameters are eccentric distortion δ _{ d } factors:$$ {\delta}_d=\left[\begin{array}{c}\hfill 2{k}_3 xy+{k}_4\left({\rho}^2+2{x}^2\right)\hfill \\ {}\hfill {k}_3\left({\rho}^2+2{y}^2\right)+2{k}_4 xy\hfill \end{array}\right] $$(2)
D represents the distortion equation, and V _{3} = [k _{1}, k _{2}, k _{3}, k _{4}, k _{5},] represents the distortion variable.
- 4.Perspective camera model: the projection process from the normalized plane to the image plane can be expressed using the generalized camera projection matrix K _{ c }:$$ {\boldsymbol{K}}_c=\left[\begin{array}{c}\hfill {\gamma}_1\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \end{array}\kern1em \begin{array}{c}\hfill \alpha \hfill \\ {}\hfill {\gamma}_2\hfill \\ {}\hfill 0\hfill \end{array}\kern1em \begin{array}{c}\hfill {u}_0\hfill \\ {}\hfill {v}_0\hfill \\ {}\hfill 1\hfill \end{array}\right] $$(3)
where γ _{ i } = f _{ i }⋅η, f is the camera focal length, η is the mirror parameter, and V _{4} = [α, γ _{1}, γ _{2}, u _{0}, v _{0}] represents the unknown variable.
- 5.Final projection equation: G represents the all-projection equation, and V includes the 18 unknowns:$$ G={K}_c\times D\times H\times W,= V=\left[{V}_1,{V}_2,{V}_3,{V}_4\right] $$(4)Assuming that there are n images in the calibration board, each image has m corner points, and all the unknown parameters’ maximum likelihood solutions can be obtained by calculating the minimum value of the formula$$ {\displaystyle \sum_{i=1}^n{{\displaystyle \sum_{j=1}^m\left\Vert G\left({V}_{1 i},{V}_2,{V}_3,{V}_4,{X}_{i j}\right)-{m}_{i j}\right\Vert}}^2} $$(5)
where G(V _{1i }, V _{2}, V _{3}, V _{4}, X _{ ij }) is the projection of the calibration board’s corner points, and X _{ ij } and m _{ ij } contain the corresponding image coordinates. Equation 5 is a nonlinear optimization equation that can be solved using the Levenberg-Marquardt optimization algorithm. The initial-value selection problem has been well analyzed in Mei and Rives [19] and will not be discussed in this study.
2.2 A modification theory
2.3 Cylindrical expansion model
where θ = 2πi/W.
2.4 Epipolar geometry and essential matrix
where E = R[T]_{×} is an essential 3 × 3 matrix with a rank of 2. Notably, the essential matrix has only 5° of freedom. Based on Eq. 10, each pair of points can provide two linear constraint equations of the essential matrix; thus, calculating the essential matrix requires a minimum of eight pairs of points. Because of the homogeneity of Eq. 10, the essential matrix E can be obtained only with the difference of a non-zero factor, indicating that the equation \( \boldsymbol{E}=\left[\boldsymbol{R},\overline{\boldsymbol{T}}\right] \) _{×} with the motion parameters [R, T] can only be obtained as \( \left[\boldsymbol{R},\overline{\boldsymbol{T}}\right] \), that is, with the difference of a non-zero factor, where \( \boldsymbol{T}=\lambda \overline{\boldsymbol{T}} \). Here, the physical meaning of the non-zero factor λ is the baseline length.
2.5 Relative positional calculation of the omnidirectional, stereo vision system
The symbol ≈ indicates that there is a difference in the proportional constant factor λ, which can be calculated by the real translation vector \( \boldsymbol{T}=\lambda \overline{\boldsymbol{T}} \), where \( \overline{\boldsymbol{T}} \) is the unitized vector of the translation vector solution in Eq. 11. In this case, the physical meaning of λ is the baseline length. The calculation method of λ and the real translation vector T will be described after the stereo rectification method is described.
3 Stereo rectification method
3.1 A new rectification model
An ideal, vertical-baseline, omnidirectional, stereo pair has a linear, epipolar, geometric relationship in the radial direction; however, this assumption is ideal in that inevitable misalignment errors exist between the two optical axes when applied in practice. Our rectification is a procedure used to obtain a column alignment image pair in our vertically arranged systems. A standard, stereo, cylindrical image pair can be generated using the rotation matrix R, and the unit transition vector \( \overline{\boldsymbol{T}} \) can be calculated using the essential matrix. This significantly improves the speed of the algorithm making it suitable for real-time applications. After applying our proposed procedure, two important parameters, namely, (a) baseline length (obtained via stereo calibration) and (b) pixel disparity of the vertical axis (easily obtained from a paralleled cylindrical image pair obtained by rectification), can be used in Eq. 12 to calculate metric scene measurements for robot tasks.
In Eq. 15, the calculation of the rotation matrix R _{u} does not rely on the real solution of the translation vector T; indeed, the rotation matrix R _{u} can be calculated using the unit translation vector \( \overline{\boldsymbol{T}} \), which is directly decomposed by the essential matrix E. Using Eqs. 8 and 13–15 to calculate the point’s 3D coordinates of the expanded cylindrical image in the mirror coordinate system, the unified sphere-imaging model’s projection formula can be applied to facilitate the cylindrical expansion of the upper and lower panoramic stereo image pair to obtain the standard, cylindrical, stereo image pair. Then, the standard, vertical-baseline, omnidirectional, stereo-vision-positioning model in Fig. 6 is used to calculate the 3D coordinates of the spatial points according to Eq. 12.
3.2 Baseline length calculation
Therefore, an accurate value of the proportional constant factor λ can be calculated from two corner points that are accurately located on the calibration board; the real value of the translation vector T can thus be obtained.
4 Results and discussion
4.1 V-binocular omnistereo vision system
Mirror parameters and camera parameters given by the manufacturer
Mirror parameters | Camera parameters | ||
---|---|---|---|
a (major axis) | 31.2888 mm | Maximum resolution | 2448 × 2048 pixels |
b (minor axis) | 51.1958 mm | Effective resolution | 1360 × 1360 pixels |
ξ (mirror parameter) | 0.82 | Frame rate | 10 frames/s |
Unilateral vertical viewing angle | 120° | Interface | 1394b |
The camera base was equipped with a 3°-of-freedom adjustment device. By using a single-view-point constraint determination method [20] and by adjusting the device, the single-view-point constraint was considered to be satisfied during camera-mirror assembly, and the installation accuracy of the mechanical structures was guaranteed. There were no changes in the extrinsic parameters of our hardware configuration. The baseline was defined as the distance from the focus of the two mirrors under the unified sphere-imaging model, which was accurately measured directly by a long Vernier caliper. The installation spacing of the vertical baseline was 332 mm, which was used as a ground truth to validate the accuracy of the stereo calibration method.
4.2 Single camera calibration experiment
Intrinsic calibration parameters of the omnidirectional cameras
The upper camera | The lower camera | |
---|---|---|
Main point position [u _{0}, v _{0}]/(pixels) | [673.59, 683.82] | [669.98, 676.12] |
Equivalent focal length [γ _{1}, γ _{2}] | [410.44, 411.28] | [412.37, 414.78] |
Mirror parameter ξ | 0.83176 | 0.84279 |
Radial distortion [k _{1}, k _{2}] | [-0.08661, 0.00732] | [-0.09310, 0.00961] |
Tangential distortion [k _{3}, k _{4}] | [-0.00131, 0.00279] | [0.00445, 0.00387] |
Re-projection error | [0.63084, 0.59738] | [0.99979, 0.95924] |
4.3 Stereo calibration
Rotation matrix and unit translation vector results of the improved method
Essential matrix | Rotation matrix R | Unit translation vector \( \overline{\boldsymbol{T}} \) |
\( \boldsymbol{E}=\left[\begin{array}{c}\hfill 0.0257\hfill \\ {}\hfill 0.9996\hfill \\ {}\hfill 0.0084\hfill \end{array}\kern1em \begin{array}{c}\hfill 0.9993\hfill \\ {}\hfill -0.0255\hfill \\ {}\hfill 0.0283\hfill \end{array}\kern1em \begin{array}{c}\hfill 0.0040\hfill \\ {}\hfill -0.0011\hfill \\ {}\hfill 0.0001\hfill \end{array}\right] \) | \( \boldsymbol{R}=\left[\begin{array}{ccc}\hfill 0.9993\hfill & \hfill 0.0259\hfill & \hfill -0.0269\hfill \\ {}\hfill -0.0255\hfill & \hfill 0.9996\hfill & \hfill 0.0131\hfill \\ {}\hfill 0.0273\hfill & \hfill -0.0124\hfill & \hfill 0.9996\hfill \end{array}\right] \) | \( \overline{\boldsymbol{T}=\left[\begin{array}{c}\hfill 0.0010\hfill \\ {}\hfill 0.0040\hfill \\ {}\hfill -1.000\hfill \end{array}\right]} \) |
Comparison results of our stereo calibration method with the original method
Rotation matrix R | Translation vector T | |
---|---|---|
Stereo calibration results of the improved method | \( \boldsymbol{R}=\left[\begin{array}{ccc}\hfill 0.9993\hfill & \hfill 0.0259\hfill & \hfill -0.0269\hfill \\ {}\hfill -0.0255\hfill & \hfill 0.9996\hfill & \hfill 0.0131\hfill \\ {}\hfill 0.0273\hfill & \hfill -0.0124\hfill & \hfill 0.9996\hfill \end{array}\right] \) | \( \boldsymbol{T} =\left[\begin{array}{c}\hfill 0.3301\hfill \\ {}\hfill 1.3205\hfill \\ {}\hfill \hbox{-} 330.1302\hfill \end{array}\right] \) |
Stereo calibration results of the original method | \( \boldsymbol{R}=\left[\begin{array}{c}\hfill 0.9997\hfill \\ {}\hfill -0.0215\hfill \\ {}\hfill -0.0120\hfill \end{array}\kern1em \begin{array}{c}\hfill 0.0217\hfill \\ {}\hfill 0.9995\hfill \\ {}\hfill 0.0221\hfill \end{array}\kern1em \begin{array}{c}\hfill 0.0115\hfill \\ {}\hfill -0.0223\hfill \\ {}\hfill 0.9997\hfill \end{array}\right] \) | \( \boldsymbol{T}=\left[\begin{array}{c}\hfill -7.9351\hfill \\ {}\hfill 4.4249\hfill \\ {}\hfill -322.8688\hfill \end{array}\right] \) |
The proportional constant factor λ was calculated using Eq. 18. The result of the improved method was λ = 330.1302 and \( \boldsymbol{T}=\lambda \overline{\boldsymbol{T}} \); therefore, the final translation vector obtained using our calibration method was Τ = [0.3301, 1.3205, ‐ 330.1302]. The modulus of vector T is the calibrated baseline length. The baseline length of our system was 332 mm. The deviation between our calibration and true value was 1.8698 mm with an error of 0.56%. The translation vector calculated by the original method in Table 4 was Τ = [‐ 7.9351, 4.4249, ‐ 322.8688], whose model is 322.9966. The deviation from the true value of 332 mm was −9.0034 mm, giving an error of 2.71%. Thus, the improved method has better performance.
When stereo calibration is performed using the contrast method, the calibrated results for each stereo pair will be slightly different. This is because the method [10] does not allow for the manual extraction of grid points. Mei and Rives only considered images wherein the grid points were successfully extracted, which increases noise and rounding errors. The results obtained using Eq. 20 can only be used as an initial approximation of the real results. The Levenberg-Marquardt iterative algorithm is then used to perform the calculation, which minimizes the projection error. Such methods typically require multiple calibration stereo pairs, and thus, significantly more work is required to configure the control points and measurement process, which requires rigorous and complex calculations.
In contrast, in our method, we only need to calculate one stereo pair to obtain the rotation matrix and translation vector via our manual extraction method, therein allowing the methods to eliminate noise error to obtain the maximum amount of available data. Our calibration method is easier to implement.
4.4 Stereo rectification
A rectification experiment of the upper and lower images in Fig. 10 was performed using the calibrated intrinsic parameters, the calculated rotation matrix, the unit translation vector, and our proposed stereo rectification method. We saved the rectification transforms as lookup tables. The results are shown in Fig. 11b left. Figure 11b right presents the details from the rectified image pair. We can observe that the stereo correspondences fall on the same line and that the pixels are aligned. Comparing the details before rectification with those after rectification, we can conclude that our rectification method is effective.
Stereo rectification accuracy
Upper image (pixel) | Lower image (pixel) | Row coordinate parallax (pixel) | Column coordinate parallax (pixel) | ||
---|---|---|---|---|---|
x coordinates | y coordinates | x coordinates | y coordinates | ||
52.13 | 113.16 | 52.38 | 64.42 | −0.26 | 48.74 |
63.35 | 113.08 | 63.26 | 64.47 | 0.09 | 48.61 |
73.72 | 113.10 | 74.20 | 64.55 | −0.48 | 48.55 |
363.22 | 177.98 | 362.57 | 73.40 | 0.65 | 104.58 |
389.30 | 177.51 | 388.68 | 72.71 | 0.61 | 104.79 |
414.88 | 176.49 | 414.38 | 73.03 | 0.49 | 103.46 |
887.12 | 159.46 | 887.14 | 55.00 | −0.02 | 104.47 |
907.97 | 157.41 | 907.42 | 51.14 | 0.54 | 106.27 |
928.93 | 155.11 | 928.52 | 47.51 | 0.41 | 107.60 |
1144.03 | 136.51 | 1144.16 | 40.67 | −0.13 | 95.84 |
1163.51 | 136.89 | 1163.31 | 41.79 | 0.20 | 95.10 |
1181.96 | 137.40 | 1182.18 | 43.42 | −0.22 | 93.98 |
With the same picture taken during our experiment and the rectification method proposed by Wang et al. [14], a contrast experiment can be performed under the same conditions. We cannot visually see the difference between the algorithm proposed in this paper and the contrast algorithm visually from the image pair, so we compared the quantized data. The mean value of the abscissa parallax of 63 corresponding corner points was 0.9225 pixels. This finding indicates that our rectification method provides a higher pixel-alignment accuracy. We used 500 pictures to record the computation time of our proposed algorithm and contrast algorithm. The average time of our algorithm was 97 ms per frame, while the average calculation time of the contrast algorithm was 151 ms per frame. Thus, our algorithm requires less computational time than the comparison algorithm.
There is no ground truth data for the rotation matrix. However, because the rectification method uses R and \( \overline{\boldsymbol{T}} \), the rectification accuracy can also laterally show the accuracy of the calculated rotation matrix R.
4.5 Off-line experiments in practice: stereo matching, 3D reconstruction, and depth estimation
We also conducted stereo matching, 3D reconstruction and off-line depth estimation experiments to show the accuracy of our proposed stereo calibration and rectification method.
4.5.1 Stereo matching
4.5.2 3D reconstruction
The distance between the calibration boards and the panorama system was less than 1–2 m. The origin was the center of the sensor; coordinates of all matching points were extracted from the generated picture. The corner points of the calibration boards are co-planar following reconstruction. By comparing the real distance of these corner points that we manually measured when taking these photos with the coordinates extracted from the generated picture, the average distance error of the 63 matching points was 1.16%. The 3D calculation results demonstrate the precision of our stereo calibration and rectification method.
4.5.3 Depth estimation
Depth estimation results compared with ground truth depth data
Selected points | P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | P_{7} | P_{8} | P_{9} |
Ground truth, m | 1.69 | 1.67 | 4.01 | 3.95 | 2.18 | 2.24 | 1.83 | 1.84 | 8.02 |
Estimated depth, m | 1.64 | 1.63 | 4.16 | 4.07 | 2.13 | 2.18 | 1.79 | 1.81 | 8.57 |
Depth error ratio, % | 2.96 | 2.40 | 3.74 | 3.04 | 2.29 | 2.68 | 2.19 | 1.63 | 6.86 |
Selected points | P_{10} | P_{11} | P_{12} | P_{13} | P_{14} | P_{15} | P_{16} | P_{17} | P_{18} |
Ground truth, m | 7.87 | 5.39 | 5.67 | 1.89 | 1.88 | 4.26 | 4.62 | 5.16 | 4.79 |
Estimated depth, m | 8.36 | 5.16 | 5.39 | 1.92 | 1.93 | 4.37 | 4.76 | 5.35 | 4.98 |
Depth error ratio, % | 6.22 | 4.27 | 4.94 | 1.59 | 2.66 | 2.58 | 3.03 | 3.68 | 3.97 |
According to Table 6, we calculated the average depth estimation error ratio with respect to the ground truth data to be 3.37%. This indicates that depth information can be effectively obtained following stereo calibration and rectification using our method.
5 Conclusions
In this paper, we have proposed a general, comprehensive stereo calibration and rectification method suitable for any V-binocular stereo vision system. We have provided the key techniques required to establish a simple and effective equivalency between an omnidirectional stereo vision system and a perspective vision system, including stereo calibration and rectification. The stereo calibration method was improved. The stereo calibration procedure was simplified based on epipolar geometry. The rounding error was reduced, and the accuracy was ensured by using a manual extraction method. The experimental results verified that the improved stereo calibration method is more accurate than the original method and reduces the complexity of the algorithm. We proposed a simple rectification model. The experimental results verified that the computation time of the proposed rectification method is shorter, and the accuracy is higher than the existing method, which makes it more suitable for real-time vision tasks. Other experiments, such as stereo matching, 3D reconstruction, and depth estimation were also conducted. Experimental results and analyses also support the effectiveness of our methods. In conclusion, our methods can effectively meet the requirements of high-precision vision sensors for robot tasks.
Declarations
Funding
This study was supported in part by the National Natural Science Foundation of China via grant number 61673129, the Natural Science Foundation of Heilongjiang Province of China via grant number F201414, and the Fundamental Research Funds for the Central Universities via grant number HEUCF160418 and HEUCF041703.
Authors’ contributions
The work presented in this paper was carried out in collaboration between all authors. CC, XW, and QZ conceived the research theme, designed and implemented the feature optimization procedure, and prepared the manuscript. XW and BF wrote the code. XW performed the experiments and analyzed the data. XW reviewed and edited the manuscript. All authors discussed the results and implications, commented on the manuscript at all stages, and approved the final version.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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