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Self-Calibration of a 1D Projective Camera and Its Application to the Self-Calibration of a 2D Projective Camera
October 2000 (vol. 22 no. 10)
pp. 1179-1185

Abstract—We introduce the concept of self-calibration of a 1D projective camera from point correspondences, and describe a method for uniquely determining the two internal parameters of a 1D camera, based on the trifocal tensor of three 1D images. The method requires the estimation of the trifocal tensor which can be achieved linearly with no approximation unlike the trifocal tensor of 2D images and solving for the roots of a cubic polynomial in one variable. Interestingly enough, we prove that a 2D camera undergoing planar motion reduces to a 1D camera. From this observation, we deduce a new method for self-calibrating a 2D camera using planar motions. Both the self-calibration method for a 1D camera and its applications for 2D camera calibration are demonstrated on real image sequences.

[1] M. Armstrong, A. Zisserman, and R. Hartley, “Self-Calibration from Image Triplets,” Proc. Fourth European Conf. Computer Vision, B. Buxton and R. Cipolla, eds., vol. 1064, pp. 3-16, Apr. 1996.
[2] M. Armstrong, “Self-Calibration from Image Sequences,” PhD thesis, Univ. of Oxford, 1996.
[3] K. Åström, “Invariancy Methods for Points, Curves, and Surfaces in Computational Vision,” PhD thesis, Lund Univ., 1996.
[4] P.A. Beardsley and A. Zisserman, “Affine Calibration of Mobile Vehicles,” Proc. Europe-China Workshop Geometrical Modeling and Invariants for Computer Vision, R. Mohr and C. Wu, eds., pp. 214–221, Apr. 1995.
[5] T. Buchanan, “The Twisted Cubic and Camera Calibration,” Computer Vision, Graphics, and Image Processing, vol. 42, no. 1, pp. 130–132, Apr. 1988.
[6] O. Faugeras and G. Toscani, “Camera Calibration for 3D Computer Vision,” Proc. Int'l Workshop Machine Vision and Machine Intelligence, 1987.
[7] O. Faugeras, “Stratification of Three-Dimensional Vision: Projective, Affine, and Metric Representations,” J. Optical Soc. Am., vol. 12, pp. 465–484, 1995.
[8] O. Faugeras and S. Maybank,“Motion from point matches: Multiplicity of solutions,” International Journal of Computer Vision, vol. 3, no. 4, pp. 225-246, 1990.
[9] O. Faugeras and B. Mourrain, “About the Correspondences of Points Between$n$Images,” Proc. Workshop Representation of Visual Scenes, pp. 37–44, June 1995.
[10] O. Faugeras, L. Quan, and P. Sturm, “Self-Calibration of a 1D Projective Camera and Its Application to the Self-Calibration of a 2D Projective Camera,” Proc. European Conf. Computer Vision, June 1998.
[11] R.I. Hartley, “In Defense of the 8-Point Algorithm,” Proc. Fifth Int'l Conf. Computer Vision, pp. 1,064-1,070, June 1995.
[12] R.I. Hartley, “Euclidean Reconstruction from Uncalibrated Views,” Proc. Workshop Applications of Invariants in Computer Vision, pp. 187-202, Oct. 1993.
[13] R.I. Hartley, "A Linear Method for Reconstruction From Lines and Points," Proc. Int'l Conf. Computer Vision, 1995, pp. 882-887.
[14] A. Heyden, “Geometry and Algebra of Multiple Projective Transformations,” PhD thesis, Lund Univ., 1995.
[15] A. Heyden and K. Astrom, “Algebraic Properties of Multilinear Constraints,” Proc. Fourth European Conf. Computer Vision, B. Buxton and R. Cipolla, eds., pp. 671–682, Apr. 1996.
[16] A. Heyden, “A Common Framework for Multiple-View Tensors,” Proc. Fifth European Conf. Computer Vision, pp. 3–19, June 1998.
[17] Q.-T. Luong and O. Faugeras, “Camera Calibration, Scene Motion and Structure Recovery from Point Correspondences and Fundamental Matrices,” Int'l J. Computer Vision, vol. 22, no. 3, pp. 261-289, 1997.
[18] S.J. Maybank and O.D. Faugeras, “A Theory of Self-Calibration of a Moving Camera,” Int'l J. Computer Vision, vol. 8. no. 2, pp. 123-152, Aug. 1992.
[19] M. Pollefeys, R. Koch, and L. Van Gool, “Self-Calibration and Metric Reconstruction in Spite of Varying and Unknown Internal Camera Parameters,” Proc. Int'l Conf. Computer Vision, pp. 90-95, Jan. 1998.
[20] L. Quan and T. Kanade, Affine Structure From Line Correspondences with Uncalibrated Affine Cameras Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 8, pp. 834-845, Aug. 1997.
[21] J.G. Semple and G.T. Kneebone, Algebraic Projective Geometry. Oxford Science Publication, 1952.
[22] A. Shashua, “Algebraic Functions for Recognition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-789, 1995.
[23] M. Spetsakis and J. Aloimonos, “A Unified Theory of Structure from Motion,” Proc. DARPA Image Understanding Workshop, pp. 271–283, 1990.
[24] P. Sturm, “Vision 3D Non Calibrée: Contributionsàla Reconstruction Projective etétude des Mouvements Critiques Pour L'Auto-Calibrage,” PhD Thesis, INPG, Dec. 1997.
[25] P. Torr and A. Zisserman, “Performance Characterizaton of Fundamental Matrix Estimation under Image Degradation,” Machine Vision and Applications, vol. 9, pp. 321-333, 1997.
[26] B. Triggs, Matching Constraints and the Joint Image Proc. Int'l Conf. Computer Vision, pp. 338-343, 1995.
[27] B. Triggs, “Autocalibration and the Absolute Quadric,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 609-614, June 1997.
[28] Cyril Zeller and Olivier Faugeras, “Camera Self-Calibration from Video Sequences: The Kruppa Equations revisited,” Research report 2793, INRIA, Feb. 1996.
[29] Z. Zhang, R. Deriche, O. Faugeras, and Q.T. Luong, “A Rubust Technique for Matching Two Uncalibrated Images through the Recovery of the Unknown Epipolar Geometry,” Artificial Intelligence J., vol. 78, pp. 87-119, 1995.

Index Terms:
Vision geometry, camera model, self-calibration, planar motion, 1D camera.
Olivier Faugeras, Long Quan, Peter Strum, "Self-Calibration of a 1D Projective Camera and Its Application to the Self-Calibration of a 2D Projective Camera," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 10, pp. 1179-1185, Oct. 2000, doi:10.1109/34.879801
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