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.
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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.
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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
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