Motion recovery from image sequences: Discrete viewpoint vs. differential viewpoint

  • Yi Ma
  • Jana Košecká
  • Shankar Sastry
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1407)


The aim of this paper is to explore intrinsic geometric methods of recovering the three dimensional motion of a moving camera from a sequence of images. Generic similarities between the discrete approach and the differential approach are revealed through a parallel development of their analogous motion estimation theories.

We begin with a brief review of the (discrete) essential matrix approach, showing how to recover the 3D displacement from image correspondences. The space of normalized essential matrices is characterized geometrically: the unit tangent bundle of the rotation group is a double covering of the space of normalized essential matrices. This characterization naturally explains the geometry of the possible number of 3D displacements which can be obtained from the essential matrix.

Second, a differential version of the essential matrix constraint previously explored by [19, 20] is presented. We then present the precise characterization of the space of differential essential matrices, which gives rise to a novel eigenvector-decomposition-based 3D velocity estimation algorithm from the optical flow measurements. This algorithm gives a unique solution to the motion estimation problem and serves as a differential counterpart of the SVD-based 3D displacement estimation algorithm from the discrete case.

Finally, simulation results are presented evaluating the performance of our algorithm in terms of bias and sensitivity of the estimates with respect to the noise in optical flow measurements.


optical flow epipolar constraint motion estimation 


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  1. 1.
    Michael J. Brooks, Wojciech Chojnacki, and Luis Baumela. Determining the egomotion of an uncalibrated camera from instantaneous optical flow. in press, 1997.Google Scholar
  2. 2.
    A. R. Brass and B. K. Horn. Passive navigation. Computer Graphics and Image Processing, 21:3–20, 1983.Google Scholar
  3. 3.
    D. J. Heeger and A. D. Jepson. Subspace methods for recovering rigid motion I: Algorithm and implementation. International Journal of Computer Vision, 7(2):95–117, 1992.CrossRefGoogle Scholar
  4. 4.
    A. D. Jepson and D. J. Heeger. Linear subspace methods for recovering translation direction. Spatial Vision in Humans and Robots, Cambridge Univ. Press, pages 39–62, 1993.Google Scholar
  5. 5.
    K. Kanatani. 3D interpretation of optical flow by renormalization. International Journal of Computer Vision, 11(3):267–282, 1993.CrossRefGoogle Scholar
  6. 6.
    Kenichi Kanatani. Geometric Computation for Machine Vision. Oxford Science Publications, 1993.Google Scholar
  7. 7.
    H. C. Longuet-Higgins. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133–135, 1981.CrossRefGoogle Scholar
  8. 8.
    Waxman A. M., Kamgar-Parsi B., and Subbarao M. Closed form solutions to image flow equations for 3d structure and motion. International Journal of Computer Vision 1, pages 239–258, 1987.CrossRefGoogle Scholar
  9. 9.
    Yi Ma, Jana Kosecká, and Shankar Sastry. Motion recovery from image sequences: Discrete viewpoint vs. differential viewpoint. Electronic Research Laboratory Memorandum, UC Berkeley, UCB/ERL, June 1997.Google Scholar
  10. 10.
    Yi Ma, Jana Kosecká, and Shankar Sastry. Vision guided navigation for a nonholonomic mobile robot. Electronic Research Laboratory Memorandum, UC Berkeley, UCB/ERL(M97/42), June 1997.Google Scholar
  11. 11.
    Stephen Maybank. Theory of Reconstruction from Image Motion. Springer Series in Information Sciences. Springer-Verlag, 1993.Google Scholar
  12. 12.
    Philip F. McLauchlan and David W. Murray. A unifying framework for structure and motion recovery from image sequences. In Proceeding of Fifth International Conference on Computer Vision, pages 314–320, Cambridge, MA, USA, 1995. IEEE Comput. Soc. Press.Google Scholar
  13. 13.
    Richard M. Murray, Zexiang Li, and Shankar S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC press Inc., 1994.Google Scholar
  14. 14.
    S. Soatto, R. Frezza, and P. Perona. Motion estimation via dynamic vision. IEEE Transactions on Automatic Control, 41(3):393–413, March 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    T. Y. Tian, C. Tomasi, and D. J. Heeger. Comparison of approaches to egomotion computation. In Proceedings of 1996 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 315–20, Los Alamitos, CA, USA, 1996. IEEE Comput. Soc. Press.Google Scholar
  16. 16.
    Carlo Tomasi and Takeo Kanade. Shape and motion from image streams under orthography. Intl. Journal of Computer Vision, 9(2):137–154, 1992.CrossRefGoogle Scholar
  17. 17.
    G. Toscani and O. D. Faugeras. Structure and motion from two noisy perspective images. Proceedings of IEEE Conference on Robotics and Automation, pages 221–227, 1986.Google Scholar
  18. 18.
    Roger Y. Tsai and Thomas S. Huang. Uniqueness and estimation of threedimensional motion parameters of rigid objects with curved surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(1):13–27, January 1984.CrossRefGoogle Scholar
  19. 19.
    T. Vieville and O. D. Faugeras. Motion analysis with a camera with unknown, and possibly varying intrinsic parameters. Proceedings of Fifth International Conference on Computer Vision, pages 750–756, June 1995.Google Scholar
  20. 20.
    Xinhua Zhuang and R. M. Haralick. Rigid body motion and optical flow image. Proceedings of the First International Conference on Artificial Intelligence Applications, pages 366–375, 1984.Google Scholar
  21. 21.
    Xinhua Zhuang, Thomas S. Huang, and Narendra Ahuja. A simplified linear optic flow-motion algorithm. Computer Vision, Graphics and Image Processing, 42:334–344, 1988.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Yi Ma
    • 1
  • Jana Košecká
    • 1
  • Shankar Sastry
    • 1
  1. 1.Electronics Research LaboratoryUniversity of California at BerkeleyBerkeleyUSA

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