Advertisement

Least Square for Grassmann-Cayley Agelbra in Homogeneous Coordinates

  • Vincent Lesueur
  • Vincent Nozick
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8334)

Abstract

This paper presents some tools for least square computation in Grassmann-Cayley algebra, more specifically for elements expressed in homogeneous coordinates. We show that building objects with the outer product from k-vectors of same grade presents some properties that can be expressed in term of linear algebra and can be treated as a least square problem. This paper mainly focuses on line and plane fitting and intersections computation, largely used in computer vision. We show that these least square problems written in Grassmann-Cayley algebra have a direct reformulation in linear algebra, corresponding to their standard expression in projective geometry and hence can be solved using standard least square tools.

Keywords

Grassmann-Cayley algebra least square line fitting plane fitting intersection 

References

  1. 1.
    Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory: Ix combinatorial methods in invariant theory. Studies in Applied Mathematics 53(3), 185–216 (1974)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Barnabei, M., Brini, A., Rota, G.-C.: On the exterior calculus of invariant theory. Journal of Algebra 96(1), 120–160 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Carlsson, S.: The Double Algebra: An Effective Tool for Computing Invariants in Computer Vision. In: Mundy, J.L., Zisserman, A., Forsyth, D. (eds.) AICV 1993. LNCS, vol. 825, pp. 145–164. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  4. 4.
    Förstner, W., Brunn, A., Heuel, S.: Statistically testing uncertain geometric relations. In: Mustererkennung 2000, pp. 17–26. Springer, Berlin (2000)CrossRefGoogle Scholar
  5. 5.
    Gebken, C., Perwass, C., Sommer, G.: Parameter Estimation from Uncertain Data in Geometric Algebra. In: Advances in Applied Clifford Algebras, vol. 18(3-4), pp. 647–664 (2008)Google Scholar
  6. 6.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press (2004)Google Scholar
  7. 7.
    Lu, F., Hartley, R.: A fast optimal algorithm for L 2 triangulation. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds.) ACCV 2007, Part II. LNCS, vol. 4844, pp. 279–288. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Kanatani, K., Sugaya, Y., Niitsuma, H.: Optimization without Minimization Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation. IEICE Transactions 93-D, 2836–2845 (2010)Google Scholar
  9. 9.
    Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science: an object-oriented approach to geometry, revised 1st edn. Morgan Kaufmann series in computer graphics. Elsevier, San Francisco (2009)Google Scholar
  10. 10.
    Dorst, L., Lasenby, J.: Guide to geometric algebra in practice. Springer (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Vincent Lesueur
    • 1
  • Vincent Nozick
    • 1
  1. 1.Gaspard Monge Institute, UMR 8049Université Paris-Est Marne-la-ValléeFrance

Personalised recommendations