This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Exact Two-Image Structure from Motion
December 2002 (vol. 24 no. 12)
pp. 1618-1633

Abstract—For two-image structure from motion, we present a simple, exact expression for a least-squares image-reprojection error for finite motion that depends only on the motion. Optimal estimates of the structure and motion can be computed by minimizing this expression just over the motion parameters. Also, we present a solution to the triangulation problem: an exact, explicit expression for the optimal structure estimate given the motion. We identify a new ambiguity in recovering the structure for known motion. We study the exact error's properties experimentally and demonstrate that it often has several local minima for forward or backward motion estimates. Our experiments also show that the "reflected" local minimum of Oliensis and Soatto et al. occurs for large translational motions. Our exact results assume that the camera is calibrated and use a least-squares image-reprojection error that applies most naturally to a spherical imaging surface. We approximately extend our approach to the standard least-squares error in the image plane and uncalibrated cameras. We present an improved version of the Sampson error which gives better results experimentally.

[1] A. Chiuso, R. Brockett, and S. Soatto, “Optimal Structure from Motion: Local Ambiguities and Global Estimates,” Int'l J. Computer Vision, vol. 39, no. 3, pp. 195-228, 2000.
[2] R. Dutta, R. Manmatha, L.R. Williams, and E.M. Riseman, “A Data Set for Quantitative Motion Analysis,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 159–164, 1989.
[3] M.A. Fischler and R.C. Bolles, “Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography,” Graphics and Image Processing, vol. 24, no. 6, pp. 381–395, June 1981.
[4] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision. Cambridge Univ. Press, 2000.
[5] R.I. Hartley and P. Sturm, “Triangulation,” Computer Vision and Image Understanding, vol. 68, no. 2, pp. 146-157, 1997.
[6] R.I. Hartley, “Euclidean Reconstruction from Uncalibrated Views,” Proc. Workshop Applications of Invariants in Computer Vision, pp. 187-202, Oct. 1993.
[7] R. Hartley, “Projective Reconstruction and Invariants from Multiple Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 10, pp. 1036-1041, Oct. 1994.
[8] B.K.P. Horn, “Relative Orientation,” Int'l J. Computer Vision, vol. 4, pp. 59-78, 1990.
[9] A.D. Jepson and D.J. Heeger, “Linear Subspace Methods for Recovering Translational Direction,” Spatial Vision in Humans and Robots, pp. 39–62, Cambridge Univ. Press, 1993.
[10] K. Kanatani, Geometric Computation for Machine Vision. New York, NY: Oxford Univ. Press, 1993.
[11] R. Kumar and A.R. Hanson, “Sensitivity of the Pose Refinement Problem to Accurate Estimation of Camera Parameters,” Proc. Int'l Conf. Computer Vision, pp. 365 369, 1990.
[12] H.C. Longuet-Higgins, “A Computer Algorithm for Reconstructing a Scene from Two Projections,” Nature, vol. 293, pp. 133-135, 1981.
[13] D. Nister, “Automatic Dense Reconstruction from Uncalibrated Video Sequences,” KTH PhD thesis, 2001.
[14] J. Oliensis, “Exact Two-Image Structure from Motion,” technical report, NECI, 2001.
[15] J. Oliensis, “The Error Surface for Structure from Motion,” technical report, NEC, 2001.
[16] J. Oliensis and Y. Genc, “Fast and Accurate Algorithms for Projective Multi-Image Structure from Motion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 6, pp. 546-559, 2001.
[17] J. Oliensis, “A New Structure from Motion Ambiguity,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 185–191, 1999.
[18] J. Oliensis, “A Multi-Frame Structure from Motion Algorithm under Perspective Projection,” Int'l J. Computer Vision, vol. 34 nos. 2/3, pp. 163–192, 1999.
[19] J. Oliensis and Y. Genc, “New Algorithms for Two-Frame Structure from Motion,” NECI technical report, 2000. Expanded version of Proc. Int'l Conf. Computer Vision, pp. 737–744, 1999.
[20] S. Soatto and R. Brocket, “Optimal Structure from Motion: Local Ambiguities and Global Estimates,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 282–288, 1998.
[21] S. Soatto and R. Brocket, “Optimal and Suboptimal Structure from Motion,” technical report, Harvard Univ., 1997.
[22] S. Srinivasan, “Extracting Structure from Optical Flow Using the Fast Error Search Technique,” Int'l J. Computer Vision, vol. 37, no. 3, pp. 203-230, 2000.
[23] S. Srinivasan, “Fast Partial Search Solution to the 3D SFM Problem,” Proc. Int'l Conf. Computer Vision, pp. 528-535, 1999.
[24] P. Torr and D. Murray, “The Development and Comparison of Robust Methods for Estimating the Fundamental Matrix,” Int'l J. Computer Vision, vol. 3, no. 24, pp. 271-300, 1997.
[25] B Triggs, P.F. McLauchlan, R.I. Hartley, and A.W. Fitzgibbon, “Special Sessions—Bundle Adjustment: A Modern Synthesis,” Lecture Notes in Computer Science, vol. 1883, pp. 298-372, 2000.
[26] R. Vidal, Y. Ma, S. Shu, and S. Sastry, “Optimal Motion Estimation from Multiview Normalized Epipolar Constraint,” Proc. Int'l Conf. Computer Vision, vol. 1, pp. 34-41, 2001.
[27] J. Weng,T. Huang,, and N. Ahuja,“Motion and structure from two perspective views:Algorithms, error analysis and error estimation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 5, pp. 451-476, May 1989.
[28] Z. Zhang, “On the Optimization Criteria Used in 2-View Motion Analysis,” Trans. Pattern Analysis and Machine Intelligence, vol. 20, pp. 717–729, 1998, and Proc. Int'l Conf. Computer Vision, pp. 772–777, 1998.
[29] Z. Zhang, “Understanding the Relationship Between the Optimization Criteria in Two-View Motion Analysis,” Proc. Int'l Conf. Computer Vision, pp. 772-777, 1998.

Index Terms:
Structure from motion, two-image structure from motion, least-squares error, triangulation, ambiguity, spherical retina, coplanarity, Sampson error, local minima.
Citation:
John Oliensis, "Exact Two-Image Structure from Motion," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 12, pp. 1618-1633, Dec. 2002, doi:10.1109/TPAMI.2002.1114853
Usage of this product signifies your acceptance of the Terms of Use.