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Two-Way Ambiguity in 2D Projective Reconstruction from Three Uncalibrated 1D Images
February 2001 (vol. 23 no. 2)
pp. 212-216

Abstract—We show that there is, in general, a two-way ambiguity for 2D projective reconstruction from three uncalibrated 1D views, independent of the number of point correspondences. The two distinct projective reconstructions are exactly related by a quadratic transformation with the three camera centers as fundamental points. Unique 2D reconstruction is possible only when the three camera centers are aligned. By Carlsson duality, there is a dual two-way ambiguity for 2D projective reconstruction from six point correspondences, independent of the number of 1D views. The theoretical results are demonstrated on numerical examples.

[1] K. Åström, “Invariance Methods for Points, Curves, and Surfaces in Computational Vision,” PhD thesis, Lund Univ., 1996.
[2] K. Åström and M. Oskarsson, “Solutions and Ambiguities of the Structure and Motion Problem for 1D Retinal Vision,” J. Math. Imaging and Vision, vol. 12, no. 12, pp. 1-135, 2000.
[3] T. Buchanan, “The Twisted Cubic and Camera Calibration,” Computer Vision, Graphics, and Image Processing, vol. 42, no. 1, pp. 130–132, Apr. 1988.
[4] S. Carlsson, “Duality of Reconstruction and Positioning from Projective Views,” Proc. Workshop Representation of Visual Scenes, pp. 85-92, June 1995.
[5] S. Carlsson and D. Weinshall, Dual Computation of Projective Shape and Camera Positions from Multiple Images Int'l J. Computer Vision, vol. 27, no. 3, pp. 227-241, May 1998.
[6] 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.
[7] O. Faugeras and B. Mourrain, "On the Geometry and Algebra of the Point and Line Correspondences Between N Images," Proc. Int'l Conf. Computer Vision, pp. 951-956, 1995.
[8] 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.
[9] R.I. Hartley, "A Linear Method for Reconstruction From Lines and Points," Proc. Int'l Conf. Computer Vision, 1995, pp. 882-887.
[10] S. Maybank, Theory of Reconstruction from Image Motion. Springer-Verlag, 1993.
[11] S.J. Maybank and A. Shashua, “Ambiguity in Reconstruction from Images of Six Points,” Proc. Sixth Int'l Conf. Computer Vision, pp. 703-708, 1998.
[12] L. Quan, Invariants of Six Points and Projective Reconstruction from Three Uncalibrated Images IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 1, pp. 34-46, Jan. 1995.
[13] L. Quan, “Uncalibrated 1D Projective Camera and 3D Affine Reconstruction of Lines,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 60-65, June 1997.
[14] L. Quan, “Inherent Two-Way Ambiguity in 2D Projective Reconstruction from Three Uncalibrated 1D Images,” Proc. Seveth Int'l Conf. Computer Vision, pp. 344-349, 1999.
[15] 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.
[16] J.G. Semple and G.T. Kneebone, Algebraic Projective Geometry. Oxford Science Publication, 1952.
[17] A. Shashua, “Algebraic Functions for Recognition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-789, 1995.
[18] M. Spetsakis and J. Aloimonos, “A Unified Theory of Structure from Motion,” Proc. DARPA Image Understanding Workshop, pp. 271-283, 1990.
[19] B. Triggs, Matching Constraints and the Joint Image Proc. Int'l Conf. Computer Vision, pp. 338-343, 1995.

Index Terms:
1D camera, vision geometry, ambiguity, reconstruction.
Citation:
Long Quan, "Two-Way Ambiguity in 2D Projective Reconstruction from Three Uncalibrated 1D Images," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 2, pp. 212-216, Feb. 2001, doi:10.1109/34.908971
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