A unified language for computer vision and robotics

  • Eduardo Bayro-Corrochano
  • Joan Lasenby
Processing of the 3D Visual Space
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1315)


Geometric algebra is an universal mathematical language which provides very comprehensive techniques for analyzing the complex geometric situations occurring in artificial Perception Action Cycle systems. In the geometric algebra framework such a system is both easier to analyze and to control in real time computations. This paper describes the application of rotors and motors for tasks involving the algebra of the 3D kinematics. Using purely geometric derivations and the constraints for point and line correspondences in n-views projective invariants are computed and the projective depth is discussed in terms of the generalized cross-ratio.


Clifford algebra geometric algebra robotics hand-eye calibration computer vision projective invariants projective depth 


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  1. 1.
    Bayro-Corrochano, E. and Lasenby, J. 1995. Object modelling and motion analysis using Clifford algebra. In Proceedings of Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision, Ed. Roger Mohr and Wu Chengke, Xi'an, China, April, pp. 143–149.Google Scholar
  2. 2.
    Bayro-Corrochano, E., Daniilidis, K. and Sommer, G. 1997. Hand-Eye calibration in terms of motion of lines using Geometric Algebra. In Proc. of the 10th Scandinavian Conference on Image Analysis SCIA'97, Lappeenranta, Finland, June 9–11, Vol. I, pp. 397–404.Google Scholar
  3. 3.
    Bayro-Corrochano E., Lasenby J., Sommer G. Geometric Algebra: A framework for computing point and line correspondences and projective structure using n uncalibrated cameras IEEE Proceedings of ICPR'96 Viena, Austria, Vol. I, pages 334–338, August, 1996.Google Scholar
  4. 4.
    Bayro-Corrochano E., Buchholz S., Sommer G. 1996. Selforganizing Clifford neural network IEEE ICNN'96 Washington, DC, June, pp. 120–125.Google Scholar
  5. 5.
    Blaschke, W. 1960. Kinematik und Quaternionen. VEB Deutscher Verlag der Wissenschaften, Berlin 1960.Google Scholar
  6. 6.
    Carlsson, S. 1994. The Double Algebra: and effective tool for computing invariants in computer vision. Applications of Invariance in Computer Vision, Lecture Notes in Computer Science 825; Proceedings of 2nd-joint Europe-US Workshop, Azores, October 1993. Eds. Mundy, Zisserman and Forsyth. Springer-Verlag.Google Scholar
  7. 7.
    Chen H. A screw motion approach to uniqueness analysis of head-eye geometry. In IEEE Conf. Computer Vision and Pattern Recognition, pages 145–151, Maui, Hawaii, June 3–6, 1991.Google Scholar
  8. 8.
    Chou J.C.K. and Kamel M. Finding the position and orientation of a sensor on a robot manipulator using quaternions. Intern. Journal of Robotics Research, 10(3):240–254, 1991.Google Scholar
  9. 9.
    Clifford, W.K. 1878. Applications of Grassmann's extensive algebra. Am. J. Math. 1: 350–358.Google Scholar
  10. 10.
    Clifford, W.K. 1873. Preliminary sketch of bi-quaternions. Proc. London Math. Soc., 4:381–395.Google Scholar
  11. 11.
    Csurka, G. and Faugeras, O. 1995. Computing three-dimensional projective invariants from a pair of images using the Grassmann-Cayley algebra. In Proceedings of Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision, Ed. Roger Mohr and I/Vu Chengke, Xi'an, China, April, pp. 150–157.Google Scholar
  12. 12.
    Danifidis K. and Bayro-Corrochano E. The dual quaternion approach to hand-eye calibration. IEEE Proceedings of ICPR'96 Viena, Austria, Vol. I, pages 318–322, August, 1996.Google Scholar
  13. 13.
    Grassmann, H. 1877. Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehre. Math. Ann., 12: 375.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Hestenes, D. 1986. New Foundations for Classical Mechanics D. Reidel, Dordrecht.Google Scholar
  15. 15.
    Hestenes, D. 1991. The design of linear algebra and geometry. Acta Applicandae Mathematicae, 23: 65–93.CrossRefGoogle Scholar
  16. 16.
    Hestenes, D. and Sobczyk, G. 1984. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. D. Reidel, Dordrecht.Google Scholar
  17. 17.
    Hestenes, D. and Ziegler, R. 1991. Projective geometry with Clifford algebra. Acta Applicandae Mathematicae, 23: 25–63.CrossRefGoogle Scholar
  18. 18.
    Horaud R. and Dornerika F. Hand-eye calibration. Intern. Journal of Robotics Research, 14:195–210, 1995.Google Scholar
  19. 19.
    Lasenby, J., Fitzgerald, W.J., Lasenby, A.N. and Doran, C.J.L. 1997. New geometric methods for computer vision. To appear in International Journal of Computer Vision.Google Scholar
  20. 20.
    Lasenby, J., Bayro-Corrochano, E., Lasenby, A. and Sommer, G. 1996. A New Methodology for the Computation of Invariants in Computer Vision. Cambridge University Engineering Department Technical Report, CUED /F-INENG/TR.244.Google Scholar
  21. 21.
    Lasenby, J., Bayro-Corrochano E.J., Lasenby, A. and Sommer G. 1996. A new methodology for computing invariants in computer vision. IEEE Proceedings of ICPR'96, Viena, Austria, Vol. I, pages 393–397, August, 1996.Google Scholar
  22. 22.
    Lasenby, J., Bayro-Corrochano E.J.. 1997. Computing 3D projective invariants from points and lines. To appear in Int. Conference on Analysis of Images and Patterns CAIP'97, Kiel, Germany, September, 1997.Google Scholar
  23. 23.
    Longuet-Higgins, H.C. 1981. A computer algorithm for reconstructing a scene from two projections. Nature, 293: 133–138.Google Scholar
  24. 24.
    Luong, Q-T. and Faugeras, O.D. 1996. The fundamental matrix: theory, algorithms and stability analysis. IJCV, 17: 43–75.CrossRefGoogle Scholar
  25. 25.
    McCarthy J.M. Dual orthogonal matrices in manipulator kinematics IJRR, Vol.5, Number 2, 1986.Google Scholar
  26. 26.
    Shashua, A. 1994. Projective structure from uncalibrated images: structure from motion and recognition PAMI, 16(8), 778:790.Google Scholar
  27. 27.
    Shiu Y.C. and Ahmad S. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Trans. Robotics and Automation, 5:16–27, 1989.CrossRefGoogle Scholar
  28. 28.
    Tsai R.Y. and Lenz R.K. A new technique for fully autonomous and efficient 3D robotics hand/eye calibration. IEEE Trans. Rob. and Autom., 5:345–358, 1989.CrossRefGoogle Scholar
  29. 29.
    Sommer G., Bayro-Corrochano E. and Bülow T. 1997. Geometric algebra as a framework for the perception-action cycle. Workshop on Theoretical Foundation of Computer Vision, Dagstuhl, March 13–19, 1996, Springer Wien.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Eduardo Bayro-Corrochano
    • 1
  • Joan Lasenby
    • 2
  1. 1.Computer Science InstituteChristian Albrechts UniversityKielGermany
  2. 2.Department of EngineeringCambridge

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