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Abstract

As one of the classical theories of projective geometry, Pascal’s theorem is closely related to the conic, which is the image of a circle. A circle is the geometric element typically used as a template for camera calibration methods. In this work, we propose a calibration method based on Pascal’s theorem and its inference, in which a circle is used as the calibration template. The proposed calibration method can be applied to solve the image of the circular points via the Newton iteration method. The intrinsic parameters of the camera can then be determined based on the constraints of the images of the circular points and the image of the absolute conic. The results of simulations and experiments conducted verify the validity and feasibility of the proposed calibration method.

Keywords

Computer vision projective geometry Pascal’s theorem camera intrinsic parameters the image of circular points Newton iteration method

Introduction

Computer vision is widely used for industrial detection, medical image analysis, and global positioning systems.¹ In addition, the camera pose can be obtained from its absolute orientation,² in which camera calibration plays a vital role. The main purpose of camera calibration is to determine the intrinsic parameters of the camera and solve the extrinsic parameters to position the photographic objective automatically. Cameras can be calibrated for motion scenes via the self-calibration method.³ Calibration accuracy is one of the most crucial elements of camera calibration methods. Wu et al.⁴ employed a T-shaped checkerboard and the iterative closest point algorithm to improve calibration accuracy. A circle is a common geometric element in a scene and is therefore typically used as the calibration template. The geometric properties or other projective invariance relations between two or more circles have also been used extensively in calibration methods, where the relative location and main position relationships of the camera may be separated, intersecting, concentric, or tangent. The image of the circular points can be obtained based on the common self-polar triangle of the separated circles to determine the vanishing point.⁵ The center of the circle in the imaging plane can be determined based on the coplanar arranging circles⁶ or the common self-polar triangle of concentric circles.⁷ Bin⁸ obtained the intrinsic parameters of the camera based on the image of the circular points. The common diameter was used to solve the vanishing point based on the invariance of the cross ratio algebraic projective geometry. Chen and Zhao⁹ proposed location relationships for three tangent circles to solve the image of their centers. Based on the invariance of the cross ratio, they obtained one vanishing point by combining the image of a tangent point with its corresponding image point on the diameter, while the other vanishing point was simultaneously obtained from two parallel tangent lines. The intrinsic parameters of the camera were determined from a linear relationship. Rudakova and Monasse¹⁰ first rectified the distorted image and used the equation coefficients of the circle image to calculate the homography matrix via the least squares method to determine the intrinsic parameters of the camera. Zhang¹¹ estimated the homography matrix between the images to determine the intrinsic and extrinsic parameters, which were continuously optimized and the radial distortions corrected simultaneously. Based on the projection of the two extremities of a chord for each pair of images, El Abderrahmani and Satori¹² estimated the homography matrix between each pair of images and determined the intrinsic parameters from a nonlinear relationship. Liu et al.¹³ used the images of longitude and latitude circles on a sphere to determine the initial intrinsic parameters. They selected the intersection points between the images of the latitude and used longitude circle nonlinear optimization to optimize the intrinsic and extrinsic parameters of the camera. However, as the accuracy of the solutions was unsatisfactory, application of the method is restricted to situations where the intrinsic parameters were obtained by solving the images of the circular points. Meng and Hu¹⁴ used a circle and several lines through the center of the circle as the calibration template. However, in their method, the image of the circular points is not obtained directly and there are superfluous lines. Wu et al.¹⁵ proposed a camera calibration method involving the affine scaling invariance of parallel circles. In their proposed method, the intersections of the parallel circles are computed to obtain the circular points and, thus, determine the intrinsic parameters of the camera. However, in the method, it is necessary to identify which intersections of the parallel circles are the images of the circular points.

In all the calibration methods outlined above, the intrinsic parameters are obtained in relation to an infinite element. Rosenhahn and Bayro-Corrochano¹⁶ implemented a different approach, in which the image features of the absolute conic are extracted based on Pascal’s theorem and algebraic expressions are used to solve the intrinsic parameters of the camera. However, in their approach, only four intrinsic parameters can be determined and the algebraic expressions are inherently abstract.

In this work, we developed a calibration method in which the intrinsic parameters of a pinhole camera are determined based on the projective invariant of Pascal’s theorem, where Pascal’s theorem and its inference are extended to the affine plane in the complex field. According to Pascal’s theorem, for a pinhole camera model (in which a circle is used as the calibration template), the intersection points of the pairs at the opposite sides of the hexagon inscribed in the conic are collinear, where the six vertices of the hexagon can be any four points on the circle image and its circular points. In this manner, we can obtain the image of the circular points on the image plane. Further, the equations for the image of the circular points can be obtained directly to calibrate the camera based on the constraints of the images of the circular points and the image of the absolute conic.

The remainder of this article is organized as follows. The underlying theory of our calibration method is described in the “Preliminaries” section. The camera calibration method proposed in this work is based on the Pascal’s theorem and its inference is presented in the “Calibration of the pinhole camera” section. The simulations and experiments performed to verify the validity of our proposed method are presented and discussed in the “Experiments” section. In addition, the results obtained from our method are compared with those obtained from classical methods developed by Huang et al.⁵ and Zhang.¹¹ Finally, the conclusions drawn based on the findings of this work are presented in the “Conclusions” section of this article.

Preliminaries

In Figure 1, let $O_{c} - x_{c} y_{c} z_{c}$ and $O_{w} - x_{w} y_{w} z_{w}$ represent the camera and world coordinate systems, respectively. For the pinhole camera model, the conic c on the image plane π^′ is the image of the circle C on the plane π through the camera optical center $O_{c}$ . The plane π^′ is perpendicular to the optical axis (z _c axis) and therefore, the projection point $m_{A}$ of arbitrary point A on the circle C can be expressed as

m_{A} = K [R | t] A

Figure 1.

Conic c on the image plane π^′ is the image of the circle C on the template plane π for the pinhole camera model.

where the rotation matrix R and the translation vector t represent the transformation relationship between the camera coordinate system $O_{c} - x_{c} y_{c} z_{c}$ and the world coordinate system $O_{w} - x_{w} y_{w} z_{w}$ . K is a matrix consisting of the intrinsic parameters of the camera, where¹⁷

K = (\begin{matrix} f_{u} & s & u_{0} \\ 0 & f_{v} & v_{0} \\ 0 & 0 & 1 \end{matrix})

${[u_{0} v_{0} 1]}^{T}$ denotes the homogeneous coordinate matrix of the principal point of the camera, s represents the skew, and f_u and f_v represent the scale factors in the u and v directions of the image plane, respectively. We present the following theorems based on the projective geometry of conics.

Theorem 1

For a simple, arbitrary six-point graph inscribed in a conic, the points of the intersection of the opposite sides of the hexagon are collinear. This line is Pascal’s line.¹⁸

Theorem 2

The circular points I, J must be on the circle.¹⁹

Lemma 1

If there are five different intersections of the nonsingular conic with another nonsingular conic, the two conics are coincident.²⁰

Based on theorem 1, theorem 2, and lemma 1, we formulate the following proposition.

Proposition 1

In Figure 2(a), the simple six-point graph is composed of four arbitrary points, A, B, C, D, and circular points, I, J, on circle C . The intersection points of the pairs of opposite sides are collinear and satisfy Pascal’s theorem.

Figure 2.

(a) Pascal’s line in circle C , where the three points L, M, N are collinear. (b) Pascal’s line in conic c corresponding to the circle C , where the three points m _L, m _M, m _N are collinear.

Proof

In Figure 2(a), let the equation of circle C be f(x, y) = 0, where x, y are the homogeneous coordinates of the points in circle C . The inscribed hexagon ABCDIJ is formed from six different points on circle C , where A, B, C, D are four real points in the circle and I, J are the circular points. Let the equations of the lines AB, BC, CD, DI, IJ, JA be $L_{i} = 0 (i = 1, 2, \dots, 6)$ , where L _i are the first-order functional relationships between x and y. For L ₅ = 0, the line IJ is the line at infinity. Let the equation of the line AD be β = 0, where β is the first-order functional relationship between x and y.

Let the points of the intersection of the opposite sides of the hexagon be L, N, M, which are given by

L_{1} \land L_{4} = L

L_{3} \land L_{6} = N

L_{2} \land L_{5} = M

By taking another point P on circle C , the curve Γ ₁ can be obtained from the nonzero constant

λ_{1} = \frac{L_{2} (P) β (P)}{L_{1} (P) L_{3} (P)}

That is

L_{2} β - λ_{1} L_{1} L_{3} = 0

Hence, curve Γ ₁ and circle C both pass through the points A, B, C, D, P. From lemma 1, it is obvious that curve Γ ₁ and circle C are the same circle. Curve Γ ₂ can be obtained from the other nonzero constant

λ_{2} = \frac{L_{5} (P) β (P)}{L_{4} (P) L_{6} (P)}

that is

L_{5} β - λ_{2} L_{4} L_{6} = 0

From the equations for lines L ₅ and L ₄, I is one of the circular points on the lines L ₅ and L ₄. Similarly, J is one of the circular points on the lines L ₅ and L ₆. Hence, both I and J are in curve Γ ₂. According to theorem 2, the circular points are located on the circle, and curve Γ ₂ and circle C both pass through the three points A, D, P. Hence, from lemma 1, this proves the equivalence between curve Γ ₂ and circle C , and we can formulate the following equations

L_{2} β - λ_{1} L_{1} L_{3} \equiv μ_{1} f (x, y) = 0

L_{5} β - λ_{2} L_{4} L_{6} \equiv μ_{2} f (x, y) = 0

β can be eliminated from equations (7) and (8) and thus, we can obtain

λ_{1} L_{1} L_{3} L_{5} - λ_{2} L_{2} L_{4} L_{6} \equiv (μ_{2} L_{2} - μ_{1} L_{5}) f (x, y) = 0

Because the coordinates of point A are not the only solution of the equation

{\begin{matrix} L_{1} = 0 \\ L_{6} = 0 \end{matrix}

but it also satisfies equation (9), point A is on the curve given by equation (9). Similarly, points B, C, D, I, J, L, M, N are also on the curve given by equation (9). Because points A, B, C, D, I, J are on the circle f(x, y) = 0 whereas the three points L, M, N are not on the circle, the points L, M, N must be on the line $(μ_{2} L_{2} - μ_{1} L_{5}) = 0$ , and the three points L, M, N are collinear.

Inference 1

From proposition 1, the simple six-point graph, composed of five arbitrary points on circle C and one of the circular points I, J, satisfies Pascal’s theorem.

It is evident that an inference is established according to proposition 1 and therefore, the proof process can be omitted.

Calibration of the pinhole camera

Solving the image of the circular points

For the pinhole camera model, the four points m _A, m _B, m _C, m _D on the circle image c correspond to the images of points A, B, C, D and the images of the circular points I, J are m _I, m _J. Hence, we formulated the following proposition for a simple six-point graph composed of six points m _A, m _B, m _C, m _D, m _I, m _J.

Proposition 2

In Figure 2(b), the intersection points of the pairs of opposite sides of m _A, m _B, m _C, m _D, m _I, m _J inscribed in the conic c are collinear and thus, the images m _I, m _J of the circular points can be solved.

Proof

In Figure 2(b), the points m _A, m _B, m _C, m _D, m _I, m _J on conic c correspond to the points A, B, C, D, I, J, on circle C , respectively. Let the linear equations for m _A m _B, m _B m _C, m _C m _D, m _D m _I, m _I m _J, and m _J m _A be $l_{i} = 0 (i = 1, 2, \dots, 6)$ , and the homogeneous coordinates of m _I, m _J be ${[x_{1} + x_{2} i, x_{3} + x_{4} i, 1]}^{T}$ , ${[x_{1} - x_{2} i, x_{3} - x_{4} i, 1]}^{T}$ , respectively. According to proposition 1 and the geometric invariant under projective transformation, the intersection points of the opposite sides of the hexagon comprising six points m _A, m _B, m _C, m _D, m _I, m _J are the images m _L, m _M, m _N of the three points L, M, N, which can be expressed as

l_{1} \land l_{4} = m_{L}

l_{2} \land l_{5} = m_{M}

l_{6} \land l_{3} = m_{N}

The three points m _L, m _M, m _N are collinear and therefore

det (m_{L}, m_{M}, m_{N}) = 0

where $det (\cdot)$ denotes the determinant. Furthermore, the images m _I, m _J of the circular points must be in the conic c and therefore, we obtain the following constraint

m_{I}^{T} c m_{I} = 0

Because m _I is a complex point, we obtain the following equations from equations (13) and (14)

{\begin{array}{l} Re (det (m_{L}, m_{M}, m_{N})) = 0 \\ Im (det (m_{L}, m_{M}, m_{N})) = 0 \\ Re (m_{I}^{T} c m_{I}) = 0 \\ Im (m_{I}^{T} c m_{I}) = 0 \end{array}

From equation (15), the images m _I, m _J of the circular points can be solved using the Newton iteration method.²¹

Inference 2

From proposition 2, the simple six-point graph is composed of five arbitrary points and one of the images of the circular points on the circle image satisfies Pascal’s theorem and therefore, one of the images of the circular points can be solved.

It is apparent that an inference is established according to proposition 2 and therefore, the proof process can be omitted.

Computation of the camera intrinsic parameters

The image of the absolute conic is $ω = K^{- T} K^{- 1}$ ,²² and m _I, m _J represent a pair of conjugate complex points. Therefore, the two linear constraints of the camera intrinsic parameters can be expressed as

{\begin{matrix} Re (m_{I}^{T} ω m_{I}) = 0 \\ Im (m_{I}^{T} ω m_{I}) = 0 \end{matrix}

where Re and Im denote the real and imaginary parts, respectively. The image of the absolute conic ω has five degrees of freedom and hence, at least three images are needed to obtain three pairs of images of the circular points. The image of the absolute conic ω can be obtained by computing the singular value decomposition (SVD) of the coefficient matrix. Following this, K can be determined by solving the inverse after performing Cholesky decomposition of ω.

Based on proposition 2 and inference 2, we propose the following camera calibration procedure:

Step 1: Capture three images from different orientations.

Step 2: Extract the feature points using Canny edge detector after segmenting the image by a threshold²³ and fit the feature points to get the conic using the least squares method.²⁴

Step 3: Solve the appropriate points on the fitted conic according to proposition 2 or inference 2 and compute the images of the circular points according to equation (15).

Step 4: Obtain ω using equation (16) and the SVD method. Determine K by solving the inverse after performing Cholesky decomposition of the solution ω.

Experiments

To verify the effectiveness of our proposed calibration method and test its sensitivity to noise, we performed a large number of experiments using simulated data and real images.

Computer simulations

For the computer simulations, the intrinsic parameters used with the camera were f_u = 500, f_v = 600, u ₀ = 400, v ₀ = 300, and s = 0.

Effect of positions of points on the intrinsic parameters

By choosing four feature points on the circle image and adding circular points to form a hexagon, the diverse connective sequence of the intersection points of the pairs of opposite sides of the hexagon may have 60 different Pascal lines. With the help of Pascal’s theorem and its inference, the general constraint equations on the conic are transformed into the form of the constraint about Pascal’s line. No point matching is needed between the images, but the relative position of the four points will influence the accuracy of the calibration results. Hence, we investigated the effect of Gaussian noise on the standard deviations of f_u , f_v , u ₀, v ₀, and s for different test cases of the relative positions of the four feature points. Because the conic is an ellipse on the image plane, the conic is divided into four parts according to the major axis and minor axis of the ellipse. Five cases were considered for the distributions of the four feature points: average (Figure 3(a)), three-one (Figure 3(b)), half (Figure 3(c)), two-two (Figure 3(d)), and all (Figure 3(e)). Gaussian noise with a mean of zero and variance of σ was added to every pixel in the conic for the five test cases. The variance σ was varied from zero to two pixels. We performed 50 independent simulations for each value of σ and determined the deviations of the camera intrinsic parameters for the five test cases, as shown in Figure 4(a) to (e). The simulations were similar to the situation given in inference 2, and therefore, the tests can be omitted.

Figure 3.

Test cases of the relative positions of the feature points: (a) average, (b) three-one, (c) half, (d) two-two, and (e) all.

Figure 4.

Standard deviation curves of (a) f_u , (b) f_v , (c) u ₀, (d) v ₀, and (e) sas a function of the Gaussian noise variance σ for the five test cases shown in Figure 3.

The standard deviation curves of the five intrinsic parameters (f_u , f_v , u ₀, v ₀, and s) as a function of the Gaussian noise variance σ are shown in Figure 4. It can be observed that the standard deviations of the intrinsic parameters increase in an almost linear fashion with increasing Gaussian noise variance. However, at the same Gaussian noise variance, the standard deviations are the lowest for the “average” case compared to all the test cases considered in this study. For this reason, the “average” case is the most suitable test case for the calibration method.

Comparison with other calibration methods

To assess the effects of Gaussian noise, we compared our calibration method with those developed by Huang et al.⁵ and Zhang¹¹ in recent years, where a circle is used as the calibration template. The camera intrinsic parameters were computed based on proposition 2 (PT1), inference 2 (PT2), and the methods proposed by Huang et al.⁵ and Zhang.¹¹ In our calibration methods (PT1 and PT2), the feature points were chosen based on the following criterion: The distributions of the feature points in the conic must be relatively uniform. Next, Gaussian noise with variance σ was added to each point in the conic. The variance σ was varied from zero to two pixels.

We conducted 50 experiments for each value of σ and then determined the average values of the camera intrinsic parameters. Further, we compared the standard deviation curves of f_u , f_v , u ₀, v ₀, and s obtained from our proposed calibration methods with those obtained by the calibration methods of Huang et al.⁵ and Zhang,¹¹ as shown in Figure 5. It can be seen that the standard deviation curves of f_u , f_v , u ₀, v ₀, and s increase in an almost linear fashion with increasing Gaussian noise variance σ. However, at the same Gaussian noise variance, the standard deviations of the intrinsic parameters are lower than those obtained using the methods of Huang et al.⁵ and Zhang.¹¹ It is evident that our calibration methods (PT1 and PT2) are more effective within the Gaussian noise variances investigated in this work.

Figure 5.

Standard deviation curves of (a) f_u , (b) f_v , (c) u ₀, (d) v ₀, and (e) s as a function of the Gaussian noise variance σ for different calibration methods.

Real data experiments

We also performed experiments to validate our proposed calibration method, in which the resolution of the camera was 2456 × 2058 pixels. The calibration template consisted of a disc and checkerboard. A robot manipulator was used to capture images of the calibration template from different orientations (Figure 6(a) to (c)), and three images were selected, as shown in Figure 7(a) to (c). Canny edge detector was used to extract the contour of the circular image in Figure 7(a) to (c); the results are shown in Figure 8(a) to (c). The least squares method was used to fit the equation of the contour. The four points were calculated from the equation fitted of the circular images that satisfy the “average” case for PT1 according to the major axis and minor axis of the conic equation. Thus, the intrinsic parameters of the camera were solved using our proposed calibration methods (PT1 and PT2) as well as those developed by Huang et al.⁵ and Zhang.¹¹ The intrinsic parameters of the camera obtained from the experiments are summarized in Table 1.

Figure 6.

(a) to (c) Photographs of the robot manipulator used to capture images of the calibration template (disc and checkerboard) at different orientations.

Figure 7.

(a) to (c) Photographs of the calibration template (disc and checkerboard) captured from different orientations.

Figure 8.

(a) to (c) Extraction of the contour of the circular image and checkerboard using Canny edge detector.

Table 1.

Camera intrinsic parameters f_u , f_v , u ₀, v ₀, and s obtained from the proposed calibration methods (PT1 and PT2) and those developed by Huang et al.⁵ and Zhang.^11
,a

	f_u	f_v	s	u ₀	v ₀
PT1	2372.9392	2370.12695	0.2147	1222.2571	1058.6006
PT2	2373.6448	2371.2607	0.2335	1219.0149	1056.7128
Huang et al.⁵	2373.0192	2368.7361	0.2473	1218.6953	1052.9364
Zhang¹¹	2368.1386	2365.9431	0.2782	1215.3275	1050.2675

^aThe unit for all camera intrinsic parameters is pixels.

To further verify the validity of the camera intrinsic parameters presented in Table 1, the corners of the checkerboard in Figure 7(a) and (b) were extracted using the Harris algorithm,²⁵ as shown in Figure 9(a) and (b). The results in Table 1 were used to reconstruct the three-dimensional (3-D) information of the checkerboard in Figure 9 based on the polar geometric constraints²⁶ of point pairs in the two images. The 3-D reconstruction results are shown in Figure 10(a) to (d). We calculated the angle between any two straight lines in the parallel and perpendicular orientations by fitting the lines of each row and column in Figure 10 using the least squares method and determined the average values to verify the parallelism and orthogonality of the lines, as shown in Table 2. It should be noted that all the angles presented in Table 2 are approximations of the actual values. It is apparent that our proposed calibration methods (PT1 and PT2) are effective within a certain range of error.

Figure 9.

(a) and (b) Extraction of the corners of the checkerboard using Harris algorithm.

Figure 10.

3-D reconstruction results obtained based on the camera intrinsic parameters in Table 1 using four calibration methods: (a) PT1, (b) PT2, (c) Huang et al.,⁵ and (d) Zhang.¹¹

Table 2.

Average angle between any two straight lines in the parallel and perpendicular orientations in Figure 10.^a

Orientation	PT1	PT2	Huang et al.⁵	Zhang¹¹
Parallel	0.9461	0.8761	1.3398	1.5398
Perpendicular	88.6723	88.6723	88.3912	88.0912

^aThe unit for the average angles is degrees (°).

Conclusions

In this article, we proposed a calibration method in which the intrinsic parameters of a pinhole camera are determined based on Pascal’s theorem and its inference in order to expand the range of applications for the calibration method. In addition, we extended Pascal’s theorem and its inference to the affine plane in the complex field to obtain the constraints of the images of the circular points. Further, the general constraint equations on the conic are transformed into the form of the constraint about the Pascal’s line. Our calibration method directly uses the properties of a circle and infinite line without the need to consider the geometric position relationships between the circles. Based on the images of the circular points on the conic, we can obtain the nonlinear equations of the images of the circular points. We computed the images of the circular points using the Newton iteration method to determine the intrinsic parameters of the camera. The results of the simulations and experiments performed prove the effectiveness of our proposed camera calibration method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the National Natural Science Foundation of China (NSFC) [Grant Nos. 61663048, 11861075], the Program for Innovative Research Team (in Science and Technology) in Universities of Yunnan Province, and the Key Joint Project of the Science and Technology Department of Yunnan Province and Yunnan University [Grant No. 2018FY001 (-014)].

ORCID iD

Yue Zhao

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Camera calibration method based on Pascal’s theorem

Abstract

Keywords

Introduction

Preliminaries

Theorem 1

Theorem 2

Lemma 1

Proposition 1

Proof

Inference 1

Calibration of the pinhole camera

Solving the image of the circular points

Proposition 2

Proof

Inference 2

Computation of the camera intrinsic parameters

Experiments

Computer simulations

Effect of positions of points on the intrinsic parameters

Comparison with other calibration methods

Real data experiments

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References