Abstract
Circular markers are planar markers which offer great performances for detection and pose estimation. For an uncalibrated camera with rectangular pixels, the images of at least two coplanar circles in one view are generally required to recover the circle poses. Unfortunately, detecting more than one ellipse in the image is tricky and time-consuming, especially regarding concentric circles. On the other hand, when the camera is calibrated, the pose of one circle can be computed with its image alone but the solution is twofold and cannot be a priori disambiguated. Our contribution is to put beyond this limit (i) by dealing with the case of a calibrated camera with “default parameters” (e.g., using \(2\times 80\%\) of the image diagonal as focal length) that sees only one circle and (ii) by defining a theoretical framework where the pose ambiguity can be investigated. Regarding (i), we empirically show the surprising observation that default calibration leads to a circle pose estimation with accurate reprojection results which is quite satisfactory for augmented reality. As for (ii), we propose a new geometric formulation that enables to show how to detect geometric configurations in which the ambiguity can be removed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
SO(3) refers to the 3D rotation group.
- 2.
\(\text {Sym}_3\) refers to the space of order-3 real symmetric matrices.
- 3.
Virtual conics have positive definite matrices, so, no real points on them.
- 4.
The signature of a conic is \(\sigma (\mathsf {C})=(\max (p,n),\min (p,n))\), where p and n count the positive and negative eigenvalues of its (real) matrix \(\mathsf {C}\). It is left unchanged by projective transformations.
- 5.
The 3D plane through the camera centre and parallel to the image plane.
- 6.
- 7.
References
Bergamasco, F., Albarelli, A., Rodolà, E., Torsello, A.: RUNE-Tag: a high accuracy fiducial marker with strong occlusion resilience. In: CVPR (2011)
Calvet, L., Gurdjos, P., Griwodz, C., Gasparini, S.: Detection and accurate localization of circular fiducials under highly challenging conditions. In: CVPR (2016)
Fiala, M.: Artag, a fiducial marker system using digital techniques. In: CVPR (2005)
Fiala, M.: Designing highly reliable fiducial markers. PAMI 32(7), 1317–1324 (2010)
Gurdjos, P., Sturm, P., Wu, Y.: Euclidean structure from N\( \ge \) 2 parallel circles: theory and algorithms. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3951, pp. 238–252. Springer, Heidelberg (2006). https://doi.org/10.1007/11744023_19
Hartley, R., Zisserman, A.: Multiple View Geometry. Cambridge University Press, Cambridge (2003)
Huang, H., Zhang, H., Cheung, Y.M.: The common self-polar triangle of concentric circles and its application to camera calibration. In: CVPR (2015)
Kim, J.S., Gurdjos, P., Kweon, I.S.: Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration. PAMI 27(4), 637–642 (2005)
Kopp, J.: Efficient numerical diagonalization of hermitian 3x3 matrices. Int. J. Modern Phys. C 19(03), 523–548 (2008)
Köser, K., Koch, R.: Differential spatial resection - pose estimation using a single local image feature. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008. LNCS, vol. 5305, pp. 312–325. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-88693-8_23
Pagani, A., Koehle, J., Stricker, D.: Circular Markers for camera pose estimation. In: Image Analysis for Multimedia Interactive Services (2011)
Pollefeys, M., Koch, R., Van Gool, L.: Self-calibration and metric reconstruction inspite of varying and unknown intrinsic camera parameters. IJCV 32(1), 7–25 (1999)
Rekimoto, J.: Matrix: a realtime object identification and registration method for augmented reality. In: Asian Pacific Computer Human Interaction (1998)
Sturm, P.: Algorithms for plane-based pose estimation. In: CVPR (2000)
Szeliski, R.: Computer Vision: Algorithms and Applications. Springer, London (2010). https://doi.org/10.1007/978-1-84882-935-0
Szpak, Z.L., Chojnacki, W., van den Hengel, A.: Guaranteed ellipse fitting with a confidence region and an uncertainty measure for centre, axes, and orientation. JMIV 52(2), 173–199 (2015)
Wagner, D., Schmalstieg, D.: Artoolkitplus for pose tracking on mobile devices. In: ISMAR (2007). https://doi.org/10.1.1.157.1879
Chen, Q., Wu, H., Wada, T.: Camera calibration with two arbitrary coplanar circles. In: Pajdla, T., Matas, J. (eds.) ECCV 2004. LNCS, vol. 3023, pp. 521–532. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24672-5_41
Wu, H., Chen, Q., Wada, T.: Conic-based algorithm for visual line estimation from one image. In: Automatic Face and Gesture Recognition (2004)
Zheng, Y., Ma, W., Liu, Y.: Another way of looking at monocular circle pose estimation. In: ICIP (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Mariyanayagam, D., Gurdjos, P., Chambon, S., Brunet, F., Charvillat, V. (2019). Pose Estimation of a Single Circle Using Default Intrinsic Calibration. In: Jawahar, C., Li, H., Mori, G., Schindler, K. (eds) Computer Vision – ACCV 2018. ACCV 2018. Lecture Notes in Computer Science(), vol 11363. Springer, Cham. https://doi.org/10.1007/978-3-030-20893-6_36
Download citation
DOI: https://doi.org/10.1007/978-3-030-20893-6_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20892-9
Online ISBN: 978-3-030-20893-6
eBook Packages: Computer ScienceComputer Science (R0)