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Optimization Criteria, Sensitivity and Robustness of Motion and Structure Estimation

  • Jana Košecká
  • Yi Ma
  • Shankar Sastry
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1883)

Abstract

The prevailing efforts to study the standard formulation of motion and structure recovery have been recently focused on issues of sensitivity and robustness of existing techniques. While many cogent observations have been made and verified experimentally, many statements do not hold in general settings and make a comparison of existing techniques difficult. With an ultimate goal of clarifying these issues we study the main aspects of the problem: the choice of objective functions, optimization techniques and the sensitivity and robustness issues in the presence of noise.

We clearly reveal the relationship among different objective functions, such as “(normalized) epipolar constraints”, “reprojection error” or “triangulation”, which can all be be unified in a new “ optimal triangulation” procedure formulated as a constrained optimization problem. Regardless of various choices of the objective function, the optimization problems all inherit the same unknown parameter space, the so called “essential manifold”, making the new optimization techniques on Riemanian manifolds directly applicable.

Using these analytical results we provide a clear account of sensitivity and robustness of the proposed linear and nonlinear optimization techniques and study the analytical and practical equivalence of different objective functions. The geometric characterization of critical points of a function defined on essential manifold and the simulation results clarify the difference between the effect of bas relief ambiguity and other types of local minima leading to a consistent interpretations of simulation results over large range of signal-to-noise ratio and variety of configurations.

Keywords

Objective Function Noise Level Motion Estimation High Noise Level Linear Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    K. Danilidis. Visual Navigation, chapter ”Understanding Noise Sensitivity in Structure from Motion”. Lawrence Erlbaum Associates, 1997.Google Scholar
  2. 2.
    K. Danilidis and H.-H. Nagel. Analytical results on error sensitivity of motion estimation from two views. Image and Vision Computing, 8:297–303, 1990.CrossRefGoogle Scholar
  3. 3.
    A. Edelman, T. Arias, and S. T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Analysis Applications, to appear.Google Scholar
  4. 4.
    R. Hartley and P. Sturm. Triangulation. Computer Vision and Image Understanding, 68(2):146–57, 1997.CrossRefGoogle Scholar
  5. 5.
    B. Horn. Relative orientation. International Journal of Computer Vision, 4:59–78, 1990.CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    K. Kanatani. Geometric Computation for Machine Vision. Oxford Science Publications, 1993.Google Scholar
  8. 8.
    J. Košecká, Y. Ma, and S. Sastry. Optimization criteria, sensitivity and robustness of motion and structure estimation. In Vision Algorithms Workshop, ICCV, pages 9–16, 1999.Google Scholar
  9. 9.
    H. C. Longuet-Higgins. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133–135, 1981.CrossRefGoogle Scholar
  10. 10.
    Y. Ma, J. Košecká, and S. Sastry. A mathematical theory of camera self-calibration. Electronic Research Laboratory Memorandum, UC Berkeley, UCB/ERL M98/64, October 1998.Google Scholar
  11. 11.
    Y. Ma, J. Košecká, and S. Sastry. Motion recovery from image sequences: Discrete viewpoint vs. differential viewpoint. In Proceeding of European Conference on Computer Vision, Volume II, (also Electronic Research Laboratory Memorandum M98/11, UC Berkeley), pages 337–53, 1998.Google Scholar
  12. 12.
    Y. Ma, J. Košecká, and S. Sastry. Linear differential algorithm for motion recovery: A geometric approach. Submitted to IJCV, 1999.Google Scholar
  13. 13.
    Y. Ma, J. Košecká, and S. Sastry. Optimization criteria and geometric algorithms for motion and structure estimation. submitted to IJCV, 1999.Google Scholar
  14. 14.
    S. Maybank. Theory of Reconstruction from Image Motion. Springer-Verlag, 1993.Google Scholar
  15. 15.
    J. Milnor. Morse Theory. Annals of Mathematics Studies no. 51. Princeton University Press, 1969.Google Scholar
  16. 16.
    R. M. Murray, Z. Li, and S. S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC press Inc., 1994.Google Scholar
  17. 17.
    S. Soatto and R. Brockett. Optimal and suboptimal structure from motion. Proceedings of International Conference on Computer Vision, to appear.Google Scholar
  18. 18.
    M. Spetsakis. Models of statistical visual motion estimation. CVIPG: Image Understanding, 60(3):300–312, November 1994.Google Scholar
  19. 19.
    T. Y. Tian, C. Tomasi, and D. Heeger. Comparison of approaches to egomotion computation. In CVPR, 1996.Google Scholar
  20. 20.
    J. Weng, T.S. Huang, and N. Ahuja. Motion and structure from two perspective views: Algorithms, error analysis, and error estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(5):451–475, 1989.CrossRefGoogle Scholar
  21. 21.
    J. Weng, T.S. Huang, and N. Ahuja. Motion and Structure from Image Sequences. Springer Verlag, 1993.Google Scholar
  22. 22.
    J. Weng, T.S. Huang, and N. Ahuja. Optimal motion and structure estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(9):864–884, 1993.CrossRefGoogle Scholar
  23. 23.
    Z. Zhang. Understanding the relationship between the optimization criteria in two-view motion analysis. In Proceeding of International Conference on Computer Vision, pages 772–777, Bombay, India, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Jana Košecká
    • 1
  • Yi Ma
    • 2
  • Shankar Sastry
    • 2
  1. 1.Computer Science DepartmentGeorge Mason UniversityFairfax
  2. 2.EECS DepartmentUniversity of California at BerkeleyBerkeley

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