Camera calibration using the dual double-contact property of circles

Rong Zeng; Yue Zhao; Yuyang Chen

doi:10.1364/JOSAA.493181

Journal of the Optical Society of America A
Vol. 40,
Issue 11,
pp. 2084-2095
(2023)
•https://doi.org/10.1364/JOSAA.493181

Camera calibration using the dual double-contact property of circles

Rong Zeng, Yue Zhao, and Yuyang Chen

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Rong Zeng, Yue Zhao, and Yuyang Chen, "Camera calibration using the dual double-contact property of circles," J. Opt. Soc. Am. A 40, 2084-2095 (2023)

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Abstract

To mitigate the influence of the relative positional relationship between circles on camera calibration, this study explores the double contact of dual circles with dual circular points. We demonstrate that one of the generalized eigenvectors of two dual circles corresponds to the line passing through their respective centers. Moreover, we establish a methodology to identify this line from the three generalized eigenvectors and show that, regardless of the positional arrangement of the three co-planar circles, the image of the circular points can be constructed by calculating the image of this line. Consequently, the applicability of circles as calibration templates is expanded, enabling the development of a novel optimization technique for fitting circular images with enhanced calibration precision.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Equations (43)

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	Methods
	CNC Method	OP-CNC Method	CC Method	OP-CC Method	Ying and Zha [4]	Gurdjos et al. [8]	Liang and Zhao [12]
Running time	0.43	0.82	1.28	1.65	1.39	0.47	0.49

	Camera Intrinsic Parameters
Method	$f_{u}$	$f_{v}$	$u_{0}$	$v_{0}$	$s$
Ground truth	1214.7701	1153.3216	529.9063	420.3315	5.1209
CNC method	1216.1611	1154.1249	529.0122	421.7701	4.9847
OP-CNC method	1215.2344	1152.9926	529.4933	420.4244	5.0923
CC method	1221.6018	1159.0144	526.5729	416.1048	5.4997
OP-CC method	1216.8812	1157.5416	528.1527	417.3909	5.3641
Ying and Zha [4]	1222.0137	1159.0124	534.7318	425.6192	5.6917
Gurdjos et al. [8]	1216.4811	1154.0100	528.8387	421.4902	5.2492
Liang and Zhao [12]	1217.9204	1155.7311	532.9171	422.3917	5.3422

Input of Eq. (A12)				Output of Eq. (A13)
$a$	$C_{1}$ and $C_{2}$	$d_{i} (i = 1 \sim 4)$ and $a$	$d_{i} (i = 1 \sim 4)$	$\partial_{i} (i = 1, 2, 3)$
$p + 1 \leq a < + \infty$	Non-overlapping or exterior contact	$\frac{\partial d_{1}}{\partial a} > 0$ , $\frac{\partial d_{2}}{\partial a} < 0$ , $\frac{\partial d_{3}}{\partial a} < 0$ , $\frac{\partial d_{4}}{\partial a} > 0$ .	$1 \leq d_{1} < + \infty$ , $0 < d_{2} \leq 1$ , $0 < d_{3} \leq p$ , $p \leq d_{4} < + \infty$ .	$\partial_{1} = 4$ , $\partial_{2} = 2$ , $\partial_{3} = 2$ .
$1 - p < a < p + 1$	Intersection	No required condition	$ξ < 0$	$\partial_{1} = 4$ , $\partial_{2} = 0$ , $\partial_{3} = 0$ .
$a = 1 - p$	Interior contact	No required condition	$d_{1} = d_{2} = 1$ , $d_{3} = d_{4} = p$ .	$\partial_{1} = 4$ , $\partial_{2} = 2$ .
$\partial_{3} = 2$	Inclusion	$0 < a < 1 - p$ , $\frac{\partial d_{1}}{\partial a} > 0$ , $\frac{\partial d_{2}}{\partial a} < 0 \frac{\partial d_{3}}{\partial a} > 0$	$\frac{\partial d_{4}}{\partial a} < 0$ , $1 < d_{1} < + \infty$ , $0 < d_{2} < 1$ , $p < d_{3} < + \infty$ .	$0 < d_{4} < p$ , $\partial_{1} = 4$ , $\partial_{2} = 0$ .

	Methods
	CNC Method	OP-CNC Method	CC Method	OP-CC Method	Ying and Zha [4]	Gurdjos et al. [8]	Liang and Zhao [12]
Running time	0.43	0.82	1.28	1.65	1.39	0.47	0.49

	Camera Intrinsic Parameters
Method	$f_{u}$	$f_{v}$	$u_{0}$	$v_{0}$	$s$
Ground truth	1214.7701	1153.3216	529.9063	420.3315	5.1209
CNC method	1216.1611	1154.1249	529.0122	421.7701	4.9847
OP-CNC method	1215.2344	1152.9926	529.4933	420.4244	5.0923
CC method	1221.6018	1159.0144	526.5729	416.1048	5.4997
OP-CC method	1216.8812	1157.5416	528.1527	417.3909	5.3641
Ying and Zha [4]	1222.0137	1159.0124	534.7318	425.6192	5.6917
Gurdjos et al. [8]	1216.4811	1154.0100	528.8387	421.4902	5.2492
Liang and Zhao [12]	1217.9204	1155.7311	532.9171	422.3917	5.3422

Algorithm: Camera calibration algorithm using the dual double-contact property of circles
Require: Take at least three pictures including three coplanar circles from different angles.
Ensure: Output intrinsic parameter matrix $K$ ; Rotation matrix $R$ ; Translation vector $T$ .
1: Extract 200 random points on three conics $c_{i} (i = 1, 2, 3)$ to fit their
contours using the elliptic least-squares method [20].
2: Determine $l_{ij} (i, j = 1, 2, 3; i < j)$ according to Proposition 3 and
Proposition 4.
3: If the centers of the three coplanar circles, $o_{i} (i = 1, 2, 3)$ , are
non-collinear then
4: Obtain $o_{i} (i = 1, 2, 3)$ using Eq. (12) (CNC method).
5: Else
6: Obtain $o_{i} (i = 1, 2, 3)$ using Eqs. (13), (17), and (18)
(CC method).
7: End if
8: Determine $l_{\infty}$ using Eq. (20).
9: Obtain $m_{I}, m_{J}$ by solving Eq. (23) to compute $ω$ using Eq. (24) and
$c_{\infty}^{*}$ [11].
10: Compute matrix $K$ by performing Cholesky decomposition and
matrix inversion for $ω$
11: Compute the homography matrix $H$ by performing the SVD-like
decomposition of $c_{\infty}^{*}$ [18],
12: Compute $R$ and $T$ by solving Eqs. (26) and (27).
13: Optimize $K$ , $R$ , and $T$ using Eq. (28) (the optimized CNC and CC
methods are denoted as OP-CNC and OP-CC methods,
respectively.)

Algorithm: Camera calibration algorithm using the dual double-contact property of circles
Require: Take at least three pictures including three coplanar circles from different angles.
Ensure: Output intrinsic parameter matrix $K$ ; Rotation matrix $R$ ; Translation vector $T$ .
1: Extract 200 random points on three conics $c_{i} (i = 1, 2, 3)$ to fit their
contours using the elliptic least-squares method [20].
2: Determine $l_{ij} (i, j = 1, 2, 3; i < j)$ according to Proposition 3 and
Proposition 4.
3: If the centers of the three coplanar circles, $o_{i} (i = 1, 2, 3)$ , are
non-collinear then
4: Obtain $o_{i} (i = 1, 2, 3)$ using Eq. (12) (CNC method).
5: Else
6: Obtain $o_{i} (i = 1, 2, 3)$ using Eqs. (13), (17), and (18)
(CC method).
7: End if
8: Determine $l_{\infty}$ using Eq. (20).
9: Obtain $m_{I}, m_{J}$ by solving Eq. (23) to compute $ω$ using Eq. (24) and
$c_{\infty}^{*}$ [11].
10: Compute matrix $K$ by performing Cholesky decomposition and
matrix inversion for $ω$
11: Compute the homography matrix $H$ by performing the SVD-like
decomposition of $c_{\infty}^{*}$ [18],
12: Compute $R$ and $T$ by solving Eqs. (26) and (27).
13: Optimize $K$ , $R$ , and $T$ using Eq. (28) (the optimized CNC and CC
methods are denoted as OP-CNC and OP-CC methods,
respectively.)

Abstract

Data availability

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Figures (12)

Tables (4)

Equations (43)

Journal of the Optical Society of America A