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Issue No.07 - July (2010 vol.32)
pp: 1197-1210
Ming Zhao , The Chinese University of Hong Kong, Hong Kong
The problem we address is: Given line correspondences over three views, what is the condition of the line correspondences for the spatial relation of the three associated camera positions to be uniquely recoverable? The observed set of lines in space is called critical if there are multiple projectively nonequivalent configurations of the camera positions that can picture the same image triplet of the lines. We tackle the problem from the perspective of trifocal tensor, a quantity that captures the relative pose of the cameras in relation to the captured views. We show that the rank of a matrix that leads to the estimation of the tensor is reduced to 7, 11, 15 if the observed lines come from a line pencil, a line bundle, and a line field, respectively, which are line families belonging to linear line space; and 12, 19, 23 if the lines come from a general linear ruled surface, a general linear line congruence, and a general linear line complex, which are subclasses of linear line structures. We show that the above line structures, with the exception of linear line congruence and linear line complex, ought to be critical line structures. All of these structures are quite typical in reality, and thus, the findings are important to the validity and stability of practically all algorithms related to structure from motion and projective reconstruction using line correspondences.
Line structure, critical configurations, trifocal tensor.
Ming Zhao, "Rank Classification of Linear Line Structures from Images by Trifocal Tensor Determinability", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 7, pp. 1197-1210, July 2010, doi:10.1109/TPAMI.2009.103
[1] R.I. Hartley, "Lines and Points in Three Views and the Trifocal Tensor," Int'l J. Computer Vision, vol. 22, no. 2, pp. 125-140, 1997.
[2] A. Shashua and M. Werman, "On the Trilinear Tensor of Three Perspective Views and Its Underlying Geometry," Proc. Int'l Conf. Computer Vision, 1995.
[3] T. Papadopoulo and O.D. Faugeras, "A New Characterization of the Trifocal Tensor," Proc. Fifth European Conf. Computer Vision, vol. 1, pp. 109-123, 1998.
[4] M. Leung, Y. Liu, and T. Huang, "Estimating 3D Vehicle Motion in an Outdoor Scene from Monocular and Stereo Image Sequences," Proc. IEEE Workshop Visual Motion, pp. 62-68, Oct. 1991.
[5] Y. Liu, "Rigid Object Motion Estimation from Intensity Images Using Straight Line Correspondences," PhD dissertation, Univ. of Illinois at Urbana-Champaign, 1990.
[6] N. Navab and O. Faugeras, "Monocular Pose Determination from Lines: Critical Sets and Maximum Number of Solutions," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 254-260, 1993.
[7] O. Faugeras and B. Mourrain, "On the Geometry and Algebra of the Point and Line Correspondences between n Images," Proc. Fifth Int'l Conf. Computer Vision, p. 951, 1995.
[8] R.I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, second ed. Cambridge Univ. Press, 2004.
[9] R. Hartley and F. Kahl, "Critical Configurations for Projective Reconstruction from Multiple Views," Int'l J. Computer Vision, vol. 71, no. 1, pp. 5-47, 2007.
[10] A. Shashua and S. Maybank, "Degenerate n Point Configurations of Three Views: Do Critical Surfaces Exist," technical report, Hebrew Univ. of Jerusalem, 1996.
[11] R.I. Hartley, "Ambiguous Configurations for 3-View Projective Reconstruction," Proc. Sixth European Conf. Computer Vision, pp. 922-935, 2000.
[12] F. Kahl, R. Hartley, and K. Astrom, "Critical Configurations for N-View Projective Reconstruction," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 158-163, 2001.
[13] R. Hartley and F. Kahl, "A Critical Configuration for Reconstruction from Rectilinear Motion," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, p. 511, 2003.
[14] J. Weng, T. Huang, and N. Ahuja, "Motion and Structure from Line Correspondences: Closed-Form Solution, Uniqueness, and Optimization," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 3, pp. 318-336, Mar. 1992.
[15] A. Bartoli and P. Sturm, "The 3D Line Motion Matrix and Alignment of Line Reconstructions," Int'l J. Computer Vision, vol. 57, no. 3, pp. 159-178, May 2004.
[16] A. Bartoli, R.I. Hartley, and F. Kahl, "Motion from 3D Line Correspondences: Linear and Non-Linear Solutions," Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, p. 477, 2003.
[17] A. Bartoli and P. Sturm, "Structure-from-Motion Using Lines: Representation, Triangulation, and Bundle Adjustment," Computer Vision and Image Understanding, vol. 100, no. 3, pp. 416-441, 2005.
[18] T. Buchanan, "Critical Sets for 3D Reconstruction Using Lines," Proc. Second European Conf. Computer Vision, pp. 730-738, 1992.
[19] T. Buchanan, "On the Critical Set for Photogrammetric Reconstruction Using Line Tokens in p3(c)," Geometriae Dedicata, vol. 44, pp. 223-232, 1992.
[20] Y. Liu and T. Huang, "A Linear Algorithm for Motion Estimation Using Straight Line Correspondences," Proc. Ninth Int'l Conf. Pattern Recognition, vol. I, pp. 213-219, 1988.
[21] S.J. Maybank, "The Critical Line Congruence for Reconstruction from Three Images," Applicable Algebra in Eng., Comm., and Computing, vol. 6, pp. 89-113, 1993.
[22] N. Navab and O. Faugeras, "The Critical Sets of Lines for Camera Displacement Estimation: A Mixed Euclidean-Projective and Constructive Approach," Int'l J. Computer Vision, vol. 23, pp. 17-44, 1997.
[23] G.P. Stein and A. Shashua, "On Degeneracy of Linear Reconstruction from Three Views: Linear Line Complex and Applications," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 3, pp. 244-251, Mar. 1999.
[24] H. Pottmann and J. Wallner, Computational Line Geometry. Springer, 2001.
[25] J. Semple and G. Kneebone, Algebraic Projective Geometry. Oxford Clarendon Press/Oxford Univ. Press, 1952.
[26] G. Strang, Introduction to Linear Algebra, third ed. Wellesley Cambridge Press, Mar. 2003.
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