This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Epipolar Geometry from Profiles under Circular Motion
June 2001 (vol. 23 no. 6)
pp. 604-616

Abstract—This paper addresses the problem of motion estimation from profiles (also known as apparent contours) of an object rotating on a turntable in front of a single camera. Its main contribution is the development of a practical and accurate technique for solving this problem from profiles alone, which is precise enough to allow for the reconstruction of the shape of the object. No correspondences between points or lines are necessary, although the method proposed can be used equally when these features are available without any further adaptation. Symmetry properties of the surface of revolution swept out by the rotating object are exploited to obtain the image of the rotation axis and the homography relating epipolar lines in two views in a robust and elegant way. These, together with geometric constraints for images of rotating objects, are then used to obtain first the image of the horizon, which is the projection of the plane that contains the camera centers, and then the epipoles, thus fully determining the epipolar geometry of the image sequence. The estimation of the epipolar geometry by this sequential approach (image of rotation axis—homography—image of the horizon—epipoles) avoids many of the problems usually found in other algorithms for motion recovery from profiles. In particular, the search for the epipoles, by far the most critical step, is carried out as a simple one-dimensional optimization problem. The initialization of the parameters is trivial and completely automatic for all stages of the algorithm. After the estimation of the epipolar geometry, the Euclidean motion is recovered using the fixed intrinsic parameters of the camera obtained either from a calibration grid or from self-calibration techniques. Finally, the spinning object is reconstructed from its profiles using the motion estimated in the previous stage. Results from real data are presented, demonstrating the efficiency and usefulness of the proposed methods.

[1] K. Åström, R. Cipolla, and P. Giblin, “Generalised Epipolar Constraints,” Int'l J. Computer Vision, vol. 33, no. 1, pp. 51-72, Sept. 1999.
[2] K. Åström, R. Cipolla, and P.J. Giblin, “Generalised Epipolar Constraints,” Proc. Fouth European Conf. Computer Vision, B.F. Buxton and R. Cipolla, eds., vol. II, pp. 97-108, Apr. 1996.
[3] E. Boyer and M.O. Berger, “3D Surface Reconstruction Using Occluding Contours,” Int'l J. Computer Vision, vol. 22, no. 3, pp. 219-233, Mar./Apr. 1997.
[4] J. Canny, “A Computational Approach to Edge Detection,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 6, pp. 679-698, June 1986.
[5] T.J. Cham and R. Cipolla, “Geometric Saliency of Curve Correspondences and Grouping of Symmetric Contours,” Proc. Fourth European Conf. Computer Vision, B. Buxton and R. Cipolla, eds., pp. 385-398, Apr. 1996.
[6] R. Cipolla, K. Åström, and P.J. Giblin, “Motion from the Frontier of Curved Surfaces,” Proc. Fifth Int'l Conf. Computer Vision, pp. 269-275, June 1995.
[7] R. Cipolla and A. Blake, “Surface Shape from the Deformation of Apparent Contours,” Int'l J. Computer Vision, vol. 9, no. 2, pp. 83-112, Nov. 1992.
[8] R. Cipolla and P.J. Giblin, Visual Motion of Curves and Surfaces. Cambridge, UK: Cambridge Univ. Press, 1999.
[9] R. Chipolla, Y. Okamoto, and Y. Kuno, "Robust Structure from Motion Using Motion Parallax," Int'l Conf. Computer Vision, pp. 374-382,Berlin, May 1993.
[10] H.S.M. Coxeter, Introduction to Geometry, second ed. New York: John Wiley and Sons, 1969.
[11] G. Cross, A. Fitzgibbon, and A. Zisserman, “Parallax Geometry of Smooth Surfaces in Multiple Views,” Proc. Seventh Int'l Conf. Computer Vision, vol. I, pp. 323-329, Sept. 1999.
[12] R.W. Curwen, C.V. Stewart, and J.L. Mundy, “Recognition of Plane Projective Symmetry,” Proc. Sixth Int'l Conf. Computer Vision, pp. 1115-1122, Jan. 1998.
[13] O.D. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint.Cambridge, Mass.: MIT Press, 1993.
[14] A.W. Fitzgibbon, G. Cross, and A. Zisserman, “Automatic 3D Model Construction for Turn-Table Sequences,” Proc. 3D Structure from Multiple Images of Large-Scale Environments, European Workshop, SMILE '98, R. Koch and L. Van Gool, eds., vol. 1506, pp. 155-170, June 1998.
[15] P.J. Giblin, F.E. Pollick, and J.E. Rycroft, “Recovery of an Unknown Axis of Rotation from the Profiles of a Rotating Surface,” J. Optical Soc. Am. A, vol. 11, no. 7, pp. 1976-1984, July 1994.
[16] P.J. Giblin and R.S. Weiss, “Reconstruction of Surfaces from Profiles,” Proc. First Int'l Conf. Computer Vision, pp. 136-144, June 1987.
[17] P.J. Giblin and R.S. Weiss, “Epipolar Fields on Surfaces,” Proc. Third European Conf. Computer Vision, J.-O. Eklundh, ed., pp. 14-23, May 1994.
[18] 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.
[19] T. Joshi, N. Ahuja, and J. Ponce, “Structure and Motion Estimation from Dynamic Silhouettes under Perspective Projection,” Int'l J. Computer Vision, vol. 31, no. 1, pp. 31-50, Feb. 1999.
[20] T. Kanade and J.R. Kender, “Mapping Image Properties into Shape Constraints: Skewed Symmetry, Affine-Transformable Patterns, and the Shape-from-Texture Paradigm,” Human and Machine Vision, J. Beck, B. Hope, and A. Rosenfeld, eds., pp. 237-257, 1983.
[21] R. Koch, M. Pollefeys, L. Van Gool, “Multi Viewpoint Stereo from Uncalibrated Video Sequences,” Proc. Fifth European Conf. Computer Vision, H. Burkhardt and B. Neumann, eds., pp. 55-71, June 1998.
[22] J.J. Koenderink, “What Does the Occluding Contour Tell Us about Solid Shape?” Perception, vol. 13, pp. 321-330, 1984.
[23] K.N. Kutulakos and S.M. Seitz, “A Theory of Shape by Space Carving,” Proc. Seventh Int'l Conf. Computer Vision, vol. I, pp. 307-314, Sept. 1999.
[24] J. Liu, J.L. Mundy, D.A. Forsyth, A. Zisserman, and C.A. Rothwell, “Efficient Recognition of Rotationally Symmetric Surface and Straight Homogeneous Generalized Cylinders,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 123-129, June 1993.
[25] Q.-T. Luong and O.D. Faugeras, “The Fundamental Matrix: Theory, Algorithms, and Stability Analysis,” Int'l J. Computer Vision, vol. 17, no. 1, pp. 43-75, Jan. 1996.
[26] P.R.S. Mendonça, K.-Y.K. Wong, and R. Cipolla, “Camera Pose Estimation and Reconstruction from Image Profiles under Circular Motion,” Proc. Sixth European Conf. Computer Vision, D. Vernon, ed., vol. II, pp. 864-877, June/July 2000.
[27] D.P. Mukherjee, A. Zisserman, and J.M. Brady, “Shape from Symmetry—Detecting and Exploiting Symmetry in Affine Images,” Phil. Trans. Royal Soc. London A, vol. 351, pp. 77-106, 1995.
[28] V.S. Nalwa, “Line-Drawing Interpretation: Bilateral Symmetry,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 10, pp. 1117-1120, Oct. 1989.
[29] J. Porrill and S.B. Pollard, “Curve Matching and Stereo Calibration,” Image and Vision Computing, vol. 9, no. 1, pp. 45-50, Feb. 1991.
[30] J.H. Rieger, “Three-Dimensional Motion from Fixed Points of a Deforming Profile Curve,” Optics Letters, vol. 11, no. 3, pp. 123-125, Mar. 1986.
[31] J. Sato and R. Cipolla, “Affine Reconstruction of Curved Surfaces from Uncalibrated Views of Apparent Contours,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 11, pp. 1188-1198, Nov. 1999.
[32] J.G. Semple and G.T. Kneebone, Algebraic Projective Geometry. Oxford, UK: Clarendon Press, 1998, originally published in 1952.
[33] R. Szeliski, Rapid Octree Construction from Image Sequences Proc. CVGIP: Image Understanding, vol. 58, no. 1, pp. 23-32, 1993.
[34] R. Szeliski and R. Weiss, “Robust Shape Recovery from Occluding Contours Using a Linear Smoother,” Int'l J. Computer Vision, vol. 28, no. 1, pp. 27-44, June 1998.
[35] C. Tomasi and T. Kanade, "Shape and Motion From Image Streams Under Orthography: A Factorization Method," Int'l J. Computer Vision, vol. 9, no. 2, pp. 137-154, 1992.
[36] B. Triggs, “Plane + Parallax, Tensors and Factorization,” Proc. Sixth European Conf. Computer Vision, D. Vernon, ed., pp. 522-538, June/July 2000.
[37] R. Vaillant and O.D. Faugeras, "Using Extremal Boundaries for 3-D Object Modelling," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 2, pp. 157-173, Feb. 1992.
[38] T. Vieville and D. Lingrand, “Using Specific Displacements to Analyze Motion without Calibration,” Int'l J. Computer Vision, vol. 31, no. 1, pp. 5-29, Feb. 1999.
[39] K.-Y.K. Wong, P.R.S. Mendonça, and R. Cipolla, “Reconstruction and Motion Estimation from Apparent Contours under Circular Motion,” Proc. British Machine Vision Conf., T. Pridmore and D. Elliman, eds., vol. 1, pp. 83-92, Sept. 1999.
[40] Z. Zhang, “Determining the Epipolar Geometry and Its Uncertainty—A Review,” Int'l J. Computer Vision, vol. 27, no. 2, pp. 161-195, 1998.
[41] A. Zisserman, D. Forsyth, J.L. Mundy, and C.A. Rothwell, “Recognizing General Curved Objects Efficiently,” Geometric Invariance in Computer Vision, J.L. Mundy and A. Zisserman, eds., Artificial Intelligence Series, chapter 11, Cambridge, Mass.: MIT Press, pp. 228-251, 1992.

Index Terms:
Structure and motion, epipolar geometry, profiles, apparent contours, circular motion.
Citation:
Paulo R.S. Mendonça, Kwan-Yee K. Wong, Roberto Cipolla, "Epipolar Geometry from Profiles under Circular Motion," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 6, pp. 604-616, June 2001, doi:10.1109/34.927461
Usage of this product signifies your acceptance of the Terms of Use.