Pose estimation in conformal geometric algebra part I: The stratification of mathematical spaces

Journal of Mathematical Imaging and Vision

Volumn 22, Issue 1, 2005, Pages 27-48

  Rosenhahn, Bodo ^a   Sommer, Gerald ^a  

^a UNIVERSITY OF KIEL   (Germany)

Author keywords

2D 3D pose estimation; Conformal Geometric algebra; Stratification hierarchy

Indexed keywords

ALGORITHMS; CAMERAS; COMPUTER VISION; IMAGE RECONSTRUCTION; MATHEMATICAL MODELS; PROBLEM SOLVING;

2D-3D POSE ESTIMATION; CONFORMAL GEOMETRIC ALGEBRA (CGA); MATHEMATICAL SPACES; STRATIFICATION HIERARCHY;

ALGEBRA;

EID: 17044434183     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-005-4781-x     Document Type: Article

References (44)

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.
2. E. Bayro-Corrochano, "The geometry and algebra of kinematics," Vol. 40, pp. 457-472, 2001.
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.
4. J.R. Beveridge, "Local search algorithms for geometric object recognition: Optimal correspondence and pose," Technical Report CS 93-5, University of Massachusetts, 1993.
5. W. Blaschke, "Kinematik und Quaternionen, Mathematische Monographien 4," Deutscher Verlag der Wissenschaften, 1960.
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.
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.
8. CLU Library, "A C++ Library for Clifford Algebra," available at http://www.perwass.de/CLU Cognitive Systems Group, University Kiel, 2001.
9. L. Dorst, '"The inner products of geometric algebra," in Applied Geometric Algebras for Computer Science and Engineering (AGACSE), L. Dorst, C. Doran, and 5. Lasenby (Eds.), Birkhäuser Verlag, 2001, pp. 35-46.
10. L. Dorst, "Honing geometric algebra for its use in the computer sciences," Vol. 40, pp. 127-152, 2001.
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.
12. J. Gallier, Geometric Methods and Applications for Computer Science and Engineering, Springer Verlag: New York, 2001.
13. J.E. Gilbert and M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, 1991.
14. W.E.L. Grimson, Object Recognition by Computer, The MIT Press: Cambridge, MA, 1990.
15. D. Hestenes, "The design of linear algebra and geometry," Acta Applicandae Mathematicae, Vol. 23, pp. 65-93, 1991.
16. D. Hestenes, "Invariant body kinematics: I. Saccadic and compensatory eye movements," Neural Networks, Vol. 7, pp. 65-77, 1994.
17. D. Hestenes, H. Li, and A. Rockwood, "New algebraic tools for classical geometry," Vol. 40, pp. 3-23, 2001.
18. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, D. Reidel Publ. Comp., Dordrecht, 1984.
19. D. Hestenes and R. Ziegler, "Projective geometry with Clifford algebra," Acta Applicandae Mathematicae, Vol. 23, pp. 25-63, 1991.
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.
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.
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.
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.
24. H. Li, D. Hestenes, and A. Rockwood, "Generalized homogeneous coordinates for computational geometry," Vol. 40, pp. 27-52, 2001.
25. R.M. Murray, Z. Li, and S.S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, 1994.
26. T. Needham, Visual Complex Analysis, Oxford University Press, 1997.
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.
28. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C, Cambridge University Press, 1993.
29. Project homepage, Geometric Algebras and the Perception-Action Cycle See links at http ://www.ks.Informatik.unikiel.de/modules.php?name=Projekteifile= index &func=hp&prid=14
30. J. Rooney, "A comparison of representations of general screw displacement," Environment and Planning, Vol. B5, pp. 45-88, 1978.
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.
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.
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.
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.
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.
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.
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.
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.
39. F. Shevlin, "Analysis of orientation problems using Plücker lines," International Conference on Pattern Recognition, Brisbane, Vol. 1, pp. 685-689, 1998.
40. G. Sommer (Eds.), Geometric Computing with Clifford Algebra, Springer Verlag Heidelberg, 2001.
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.
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.
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.
44. I.M. Yaglom, Klein Felix, and Lie Sophus, Birkhäuser, Boston, 1988.