Journal of Mathematical Imaging and Vision

, Volume 19, Issue 3, pp 237–253

The Classical Theory of Invariants and Object Recognition Using Algebraic Curve and Surfaces

  • Hakan Civi
  • Colin Christopher
  • Aytul Ercil
Article

Abstract

Combining implicit polynomials and algebraic invariants for representing and recognizing complicated objects proves to be a powerful technique. In this paper, we explore the findings of the classical theory of invariants for the calculation of algebraic invariants of implicit curves and surfaces, a theory largely disregarded in the computer vision community by a shadow of skepticism. Here, the symbolic method of the classical theory is described, and its results are extended and implemented as an algorithm for computing algebraic invariants of projective, affine, and Euclidean transformations. A list of some affine invariants of 4th degree implicit polynomials generated by the proposed algorithm is presented along with the corresponding symbolic representations, and their use in recognizing objects represented by implicit polynomials is illustrated through experiments. An affine invariant fitting algorithm is also proposed and the performance is studied.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M.M. Blane, Z. Lei, and D.B. Cooper, “The 3L algorithm for fitting implicit polynomial curves and surfaces to data,” Brown University LEMS Lab. Technical Report, No. 160, Feb. 1997.Google Scholar
  2. 2.
    J.A. Dieudonn'e and J.B. Carrell, Invariant Theory, Old and New, Academic Press, 1971.Google Scholar
  3. 3.
    L.J.V. Gool, T. Moons, E. Pauvels, and A. Oosterlinck, “Foundations of semi-differential invariants,” Int. Journal of Computer Vision, Jan. 1993.Google Scholar
  4. 4.
    J.H. Grace and A. Young, The Algebra of Invariants, Cambridge Univ. Press, 1903.Google Scholar
  5. 5.
    G.B. Gurevich, Foundations of the Theory of Algebraic Invariants, P. Noordhoff, 1964.Google Scholar
  6. 6.
    D. Hilbert, Theory of Algebraic Invariants, Cambridge University Press, 1993.Google Scholar
  7. 7.
    C.M. Hoffman, “Implicit curves and surfaces in CAGD,” IEEE Computer Graphics and Applications, January 1993.Google Scholar
  8. 8.
    D. Keren, “Using symbolic computation to find algebraic invariants,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 16, No. 11, pp. 1143-1149, 1994.CrossRefGoogle Scholar
  9. 9.
    J.P.S. Kung and G.-C. Rota, “Invariant theory of binary forms,” Bull. of the Amer. Math. Soc., Vol. 10, No. 1, pp. 26-85, 1984.Google Scholar
  10. 10.
    Z. Lei and H. Civi, “Closed-form object pose estimation using algebraic shape representation,” Brown University LEMS Lab. Technical Report, No. 161, March 1997.Google Scholar
  11. 11.
    Z. Lei, H. Civi, and D.B. Cooper, “Free-form object modeling and inspection,” in Proceedings, Automated Optical Inspection for Industry, SPIE's Photonics China' 96, Beijing, China, Nov. 1996.Google Scholar
  12. 12.
    J.L. Mundy and A. Zisserman, Geometric Invariance in Machine Vision, MIT Press, 1992.Google Scholar
  13. 13.
    P. Olver, G. Shapiro, and A. Tennenbaum, “Differential invariant signatures and flows in computer vision: A symmetry group approach,” in Geometry-Driven Diffusion in Computer Vision, B.M.H. Romeny (Ed.), Kluwer Academic Press, 1995.Google Scholar
  14. 14.
    J. Ponce, D.J. Kriegman, S. Petitjean, S. Sullivan, G. Taubin, and B. Vijayakumar, “Representations and algorithms for 3D curved object recognition,” in Three-Dimensional Object Recognition Systems, A.K. Jain and P.J. Flynn (eds.), Elsevier Science Publishers, 1993, pp. 327-352.Google Scholar
  15. 15.
    G. Rayna, REDUCE: Software for Algebraic Computation, Springer-Verlag, 1987.Google Scholar
  16. 16.
    L.S. Shapiro, A. Zisserman, and M. Brady, “3D motion recovery via affine epipolar geometry,” Int. Journal of Computer Vision, Vol. 16, pp. 147-182, 1995.Google Scholar
  17. 17.
    B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, 1993.Google Scholar
  18. 18.
    J. Subrahmonia, D.B. Cooper, and D. Keren, “Practical reliable bayesian recognition of 2D and 3D objects using implicit polynomials and algebraic invariants,” IEEE Transactions on Pattern Analysis and Machine Intelligence, May 1996, pp. 505-519.Google Scholar
  19. 19.
    G. Taubin and D.B. Cooper, “2D and 3D object recognition and positioning with algebraic invariants and covariants,” in Symbolic and Numerical Comput. for Artif. Intelligence, B.R. Donald, D. Kapur, and J.L. Mundy (Eds.), Academic Press, 1992.Google Scholar
  20. 20.
    I. Weiss, “Geometric invariants and object recognition,” Int. Journal of Computer Vision, Vol. 10, No. 3, pp. 207-231, 1993.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Hakan Civi
    • 1
  • Colin Christopher
    • 2
  • Aytul Ercil
    • 3
  1. 1.AT Kearny ConsultingIstanbulTurkey
  2. 2.Department of MathematicsBogazici University BebekIstanbulTurkey
  3. 3.Engineering and Natural SciencesSabanci UniversityIstanbulTurkey

Personalised recommendations