Advertisement

Journal of Mathematical Imaging and Vision

, Volume 22, Issue 1, pp 27–48 | Cite as

Pose Estimation in Conformal Geometric Algebra Part I: The Stratification of Mathematical Spaces

  • Bodo Rosenhahn
  • Gerald Sommer
Article

Abstract

2D-3D pose estimation means to estimate the relative position and orientation of a 3D object with respect to a reference camera system. This work has its main focus on the theoretical foundations of the 2D-3D pose estimation problem: We discuss the involved mathematical spaces and their interaction within higher order entities. To cope with the pose problem (how to compare 2D projective image features with 3D Euclidean object features), the principle we propose is to reconstruct image features (e.g. points or lines) to one dimensional higher entities (e.g. 3D projection rays or 3D reconstructed planes) and express constraints in the 3D space. It turns out that the stratification hierarchy [11] introduced by Faugeras is involved in the scenario. But since the stratification hierarchy is based on pure point concepts a new algebraic embedding is required when dealing with higher order entities. The conformal geometric algebra (CGA) [24] is well suited to solve this problem, since it subsumes the involved mathematical spaces. Operators are defined to switch entities between the algebras of the conformal space and its Euclidean and projective subspaces. This leads to another interpretation of the stratification hierarchy, which is not restricted to be based solely on point concepts. This work summarizes the theoretical foundations needed to deal with the pose problem. Therefore it contains mainly basics of Euclidean, projective and conformal geometry. Since especially conformal geometry is not well known in computer science, we recapitulate the mathematical concepts in some detail. We believe that this geometric model is useful also for many other computer vision tasks and has been ignored so far. Applications of these foundations are presented in Part II [36].

Keywords

2D-3D pose estimation stratification hierarchy conformal geometric algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Araujo, R.L. Carceroni, and C.M. Brown, “A fully projective formulation to improve the accuracy of Lowe’s pose-estimation algorithm,” Computer Vision and Image Understanding CVIU, Vol. 70, pp. 227–238, 1998.Google Scholar
  2. 2.
    E. Bayro-Corrochano, “The geometry and algebra of kinematics,” Vol. 40, pp. 457–472, 2001.Google Scholar
  3. 3.
    E. Bayro-Corrochano, K. Daniilidis, and G. Sommer, “Motor algebra for 3D kinematics: The case of the hand-eye calibration,” Journal of Mathematical Imaging and Vision, Vol. 13, pp. 79–100, 2000.Google Scholar
  4. 4.
    J.R. Beveridge, “Local search algorithms for geometric object recognition: Optimal correspondence and pose,” Technical Report CS 93–5, University of Massachusetts, 1993.Google Scholar
  5. 5.
    W. Blaschke, “Kinematik und Quaternionen, Mathematische Monographien 4,” Deutscher Verlag der Wissenschaften, 1960.Google Scholar
  6. 6.
    C. Bregler and J. Malik, “Tracking people with twists and exponential maps,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Santa Barbara, California, 1998, pp. 8–15.Google Scholar
  7. 7.
    A. Chiuso and G. Picci, “Visual tracking of points as estimation on the unit sphere,” in The Confluence of Vision and Control, Springer-Verlag, 1998, pp. 90–105.Google Scholar
  8. 8.
    CLU Library, “A C++ Library for Clifford Algebra,” available at http://www.perwass.de/CLU Cognitive Systems Group, University Kiel, 2001.
  9. 9.
    L. Dorst, “‘The inner products of geometric algebra,” in Applied Geometric Algebras for Computer Science and Engineering (AGACSE), L. Dorst, C. Doran, and J. Lasenby (Eds.), Birkhäuser Verlag, 2001, pp. 35–46.Google Scholar
  10. 10.
    L. Dorst, “Honing geometric algebra for its use in the computer sciences,” Vol. 40, pp. 127–152, 2001.Google Scholar
  11. 11.
    O. Faugeras, “Stratification of three-dimensional vision: Projective, affine and metric representations,” Journal of Optical Society of America, Vol. 12, No. 3, pp. 465–484, 1995.Google Scholar
  12. 12.
    J. Gallier, Geometric Methods and Applications for Computer Science and Engineering, Springer Verlag: New York, 2001.Google Scholar
  13. 13.
    J.E. Gilbert and M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, 1991.Google Scholar
  14. 14.
    W.E.L. Grimson, Object Recognition by Computer, The MIT Press: Cambridge, MA, 1990.Google Scholar
  15. 15.
    D. Hestenes, “The design of linear algebra and geometry,” Acta Applicandae Mathematicae, Vol. 23, pp. 65–93, 1991.Google Scholar
  16. 16.
    D. Hestenes, “Invariant body kinematics: I. Saccadic and compensatory eye movements,” Neural Networks, Vol. 7, pp. 65–77, 1994.Google Scholar
  17. 17.
    D. Hestenes, H. Li, and A. Rockwood, “New algebraic tools for classical geometry,” Vol. 40, pp. 3–23, 2001.Google Scholar
  18. 18.
    D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, D. Reidel Publ. Comp., Dordrecht, 1984.Google Scholar
  19. 19.
    D. Hestenes and R. Ziegler, “Projective geometry with Clifford algebra,” Acta Applicandae Mathematicae, Vol. 23, pp. 25–63, 1991.Google Scholar
  20. 20.
    J.R. Holt and A.N. Netravali, “Uniqueness of solutions to structure and motion from combinations of point and line correspondences,” Journal of Visual Communication and Image Representation, Vol. 7, No. 2, pp. 126–136, 1996.Google Scholar
  21. 21.
    H.H. Homer, “Pose determination from line-to-plane correspondences: Existence condition and closed-form solutions,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 6, pp. 530–541, 1991.Google Scholar
  22. 22.
    R. Horaud, T.Q. Phong, and P.D. Tao, “Object pose from 2-d to 3-d point and line correspondences,” International Journal of Computer Vision, Vol. 15, pp. 225–243, 1995.Google Scholar
  23. 23.
    H. Klingspohr, T. Block, and R.-R. Grigat, “A passive real-time gaze estimation system for human-machine interfaces,” in Computer Analysis of Images and Patterns (CAIP), G. Sommer, K. Daniilidis, and J. Pauli (Eds.), LNCS 1296, Springer-Verlag Heidelberg, 1997, pp. 718–725.Google Scholar
  24. 24.
    H. Li, D. Hestenes, and A. Rockwood, “Generalized homogeneous coordinates for computational geometry,” Vol. 40, pp. 27–52, 2001.Google Scholar
  25. 25.
    R.M. Murray, Z. Li, and S.S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, 1994.Google Scholar
  26. 26.
    T. Needham, Visual Complex Analysis, Oxford University Press, 1997.Google Scholar
  27. 27.
    C. Perwass and J. Lasenby, “A novel axiomatic derivation of geometric algebra,” Technical Report CUED/F—INFENG/TR.347, Cambridge University Engineering Department, 1999.Google Scholar
  28. 28.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C, Cambridge University Press, 1993.Google Scholar
  29. 29.
    Project homepage, Geometric Algebras and the Perception-Action Cycle See links at http://www.ks.informatik.uni-kiel.de/modules.php?name=Projekte&file=index&func=hp&prid=14
  30. 30.
    J. Rooney, “A comparison of representations of general screw displacement,” Environment and Planning, Vol. B5, pp. 45–88, 1978.Google Scholar
  31. 31.
    B. Rosenhahn, O. Granert, and G. Sommer, “Monocular pose estimation of kinematic chains,” in Applied Geometric Algebras for Computer Science and Engineering, Birkhäuser Verlag, L. Dorst, C. Doran, and J. Lasenby (Eds.), 2001, pp. 373–383.Google Scholar
  32. 32.
    B. Rosenhahn and J. Lasenby, “Constraint equations for 2D-3D pose estimation in conformal G algebra,” Technical Report CUED/F - INFENG/TR.396, Cambridge University Engineering Department, 2000.Google Scholar
  33. 33.
    B. Rosenhahn, Ch. Perwass, and G. Sommer, “Pose estimation of 3D free-form contours,” Technical Report 0207,Christian-Albrechts-Universität zu Kiel, Institut für Informatik und Praktische Mathematik, 2002.Google Scholar
  34. 34.
    B. Rosenhahn, Ch. Perwass, and G. Sommer, “Pose estimation of 3D free-form contours in conformal geometry,” in Proceedings of Image and Vision Computing (IVCNZ), D. Kenwright (Ed.), New Zealand, 2002, pp. 29–34.Google Scholar
  35. 35.
    B. Rosenhahn and G. Sommer, “Adaptive pose estimation for different corresponding entities,” in Pattern Recognition, 24th DAGM Symposium, L. Van Gool (Ed.), Springer-Verlag: Berling Heidelberg, LNCS 2449, 2002, pp. 265–273.Google Scholar
  36. 36.
    B. Rosenhahn and G. Sommer, “Pose estimation in conformal geometric algebra. Part II: Real-time pose estimation using extended feature concepts,” Journal of Mathematical Imaging and Vision, Vol. 22, No. 1, pp. 49–70, 2005.Google Scholar
  37. 37.
    B. Rosenhahn, Y. Zhang, and G. Sommer, “Pose estimation in the language of kinematics,” in Second International Workshop, Algebraic Frames for the Perception-Action Cycle, G. Sommer and Y.Y. Zeevi (Eds.), LNCS 1888, Springer-Verlag, Heidelberg, 2000, pp. 284–293.Google Scholar
  38. 38.
    A. Ruf and R. Horaud, “Vision-based guidance and control of robots in projective space,”’ in Proceedings, 6th European Conference on Computer Vision (ECCV), Part II, Vernon D. (Ed.), LNCS 1843, Springer-Verlag Heidelberg, 2000, pp. 50– 66.Google Scholar
  39. 39.
    F. Shevlin, “Analysis of orientation problems using Plücker lines,” International Conference on Pattern Recognition, Brisbane, Vol. 1, pp. 685–689, 1998.Google Scholar
  40. 40.
    G. Sommer (Eds.), Geometric Computing with Clifford Algebra, Springer Verlag Heidelberg, 2001.Google Scholar
  41. 41.
    G. Sommer, B. Rosenhahn, and Y. Zhang, “Pose estimation using geometric constraints,” in Multi-Image Search and Analysis, R. Klette, Th. Huang, and G. Gimmel’farb (Eds.), LNCS 2032, Springer-Verlag, Heidelberg, 2001, pp. 153–170.Google Scholar
  42. 42.
    A. Ude, “Filtering in a unit quaternion space for model-based object tracking,” Robotics and Autonomous Systems, Vol. 28, Nos. 2/3, pp. 163–172, 1999.Google Scholar
  43. 43.
    M.W. Walker and L. Shao, “Estimating 3-d location parameters using dual number quaternions,” CVGIP: Image Understanding, Vol. 54, No. 3, pp. 358–367, 1991.Google Scholar
  44. 44.
    I.M. Yaglom, Klein Felix, and Lie Sophus, Birkhäuser, Boston, 1988.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Cognitive Systems Group, Institute of Computer Science and Applied MathematicsChristian-Albrechts-University of KielKielGermany

Personalised recommendations