Abstract
The application of Cayley transformation to enhance the numerical stability of camera calibration is investigated. First, a new calibration equation, called the standard calibration equation, is introduced using the Cayley transformation and its analytical solution is obtained. The standard calibration equation is equivalent to the classical calibration equation, but it exhibits remarkable better numerical stability. Second, a one-parameter calibration family, called the Cayley calibration family which is equivalent to the standard calibration equation, is obtained using also the Cayley transformation and it is found that this family is composed of those infinite homographies whose rotation has the same axis with the rotation between the two given views. The condition number of equations in the Cayley calibration family varies with the parameter value, and an algorithm to determine the best parameter value is provided. Third, the generalized Cayley calibration families equivalent to the standard calibration equation are also introduced via generalized Cayley transformations. An example of the generalized Cayley transformations is illustrated, called the S-Cayley calibration family. As in the Cayley calibration family, the numerical stability of equations in a generalized Cayley calibration family also depends on the parameter value. In addition, a more generic calibration family is also proposed and it is proved that the standard calibration equation, the Cayley calibration family and the S-Cayley calibration family are all some special cases of this generic calibration family.
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References
Brown, D. C. (1971). Close-range camera calibration. Photogrammetric Engineering, 37(8), 855–866.
Bell, S. (1992). The Cayley transform, potential theory and conformal mapping. Boca Raton: CRC Press.
Faig, W. (1975). Calibration of close-range photogrammetry systems: Mathematical formulaton. Photogrammetric Engineering and Remote Sensing, 41(21), 1479–1486.
Faugeras, O. (1993). Three-dimensional computer vision: A geometric viewpoint. Cambridge: MIT Press.
Faugeras, O., Luong, Q.-T., & Manybank, S. J. (1992). Camera self-calibration: Theroy and experiments. In Proc. of European conference on computer vision (pp. 321–334).
Hartley, R. (1997a). Kruppa’s equations derived from the fundamental matrix. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(2), 133–135.
Hartley, R. (1997b). Self-calibration of stationary cameras. International Journal of Computer Vision, 22(1), 5—23.
Hartley, R. (1997c). In defense of the eight-point algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(6), 580–593.
Hartley, R., & Zisserman, A. (2000). Multiple view geometry in computer vision. Cambridge: Cambridge University Press.
Harris, C., & Stephens, M. J. (1988). A combined corner and edge detector. In Proceeding 4th alvey vision conference.
Luong, Q. T., & Faugeras, O. (1997). Self-calibration of a moving camera from point correspondence and fundamental matrices. International Journal of Computer Vision, 22(3), 261–289.
Lowe, D. G. (2004). Distinctive image feature from scale invariant keypoint. International Journal of Computer Vision, 60(2), 91–110.
Maybank, S. J., & Faugeras, O. (1992). A theory of self-calibration of a moving camera. International Journal of Computer Vision, 8(2), 123–152.
Oliensis, J., & Hartley, R. (2007). Iterative extensions of the Sturm/Trigs algorithm: Convergence and nonconvergence. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(12), 2217–2233.
Pollefeys, M., & Gool, L. (1999). Stratified self-calibration with the modulus constraint. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(8), 707–724.
Pollefeys, M., Gool, L., & Osterlinck, A. (1996). The modulus constraint: A new constraint for self-calibration. In: Proc. of international conference of pattern recognition (pp. 31–42).
Ponce, J., McHenry, K., Papadopoulo, T., Teillaud, M., & Teiggs, B. (2000). On the absolute quadratic complex and its application to autocalibration. In Proc. of computer vision and pattern recognition (pp. 780–787).
Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. (1988). Numerical recipes in C: The art of scientific computing. Cambridge: Cambridge University Press.
Sturm, P., & Maybank, S. J. (1999). On plane-based camera calibration: A general algorithm, singularities, applications. In Proc. of computer vision and pattern recognition (pp. 432–437).
Sturm, P., & Triggs, B. (1996). A factorization based algorithm for mulit-image projective structure and motion. In Proc. of European conference on computer vision (pp. 709–720).
Tsai, R. (1986). An efficient and accurate camera calibration technique for 3D machine vision. In Proc. of computer vision and pattern recognition (pp. 364–374). Miami Beach, USA.
Triggs, B. (1997). Auto-calibration and the absolute quadric. In Proc. of computer vision and pattern recognition (pp. 609–614).
Wang, L., Wu, F. C., & Hu, Z. Y. (2007). Multi-camera calibration with one-dimensional object under general motions. In Proc. of international conference on computer vision (pp. 1–7).
Wu, F. C., Hu, Z. Y., & Zhu, H. J. (2005). Camera calibration with moving one-dimensional objects. Pattern Recognition, 38(5), 755–765.
Zhang, Z. (1999). Flexible camera calibration by viewing a plane from unknown orientations. In Proc. of international conference on computer vision (pp. 666–673).
Zhang, Z. (2000). A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11), 1330–1334.
Zhang, Z. (2004). Camera calibration with one-dimensional objects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(7), 892–899.
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Wu, F.C., Wang, Z.H. & Hu, Z.Y. Cayley Transformation and Numerical Stability of Calibration Equation. Int J Comput Vis 82, 156–184 (2009). https://doi.org/10.1007/s11263-008-0193-x
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DOI: https://doi.org/10.1007/s11263-008-0193-x