Image Sensor Model Using Geometric Algebra: From Calibration to Motion Estimation

  • Thibaud DebaeckerEmail author
  • Ryad Benosman
  • Sio H. Ieng


In computer vision image sensors have universally been defined as the nonparametric association of projection rays in the 3D world to pixels in the images. If the pixels’ physical topology can be often neglected in the case of perspective cameras, this approximation is no longer valid in the case of variant scale sensors, which are now widely used in robotics. Neglecting the nonnull pixel area and then the pixel volumic field of view implies that geometric reconstruction problems are solved by minimizing a cost function that combines the reprojection errors in the 2D images. This paper provides a complete and realistic cone-pixel camera model that equally fits constant or variant scale resolution together with a protocol to calibrate such a sensor. The proposed model involves a new characterization of pixel correspondences with 3D-cone intersections computed using convex hull and twists in Conformal Geometric Algebra. Simulated experiments show that standard methods and especially Bundle Adjustment are sometimes unable to reach the correct motion, because of their ray-pixel approach and the choice of reprojection error as a cost function which does not particularly fit the physical reality. This problem can be solved using a nonprojective cone intersection cost function as introduced below.


Motion Estimation Cone Intersection Geometric Algebra Minkowski Plane Bundle Adjustment 
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  1. 1.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benosman, R., Kang, S.: Panoramic Vision: Sensors, Theory, Applications. Springer, Berlin (2001) Google Scholar
  3. 3.
    Debaecker, T., Benosman, R.: Bio-inspired model of visual information codification for localization: from retina to the lateral geniculate nucleus. J. Integr. Neurosci. 6(3), 1–33 (2007) Google Scholar
  4. 4.
    Grossberg, M.D., Nayar, S.K.: A general imaging model and a method for finding its parameters. In: ICCV, pp. 108–115 (2001) Google Scholar
  5. 5.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2003) Google Scholar
  6. 6.
    Hestenes, D.: The design of linear algebra and geometry. Acta Appl. Math.: Int. Surv. J. Appl. Math. Math. Appl. 23, 65–93 (1991) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984) zbMATHGoogle Scholar
  8. 8.
    Kahl, F., Hartley, R.: Multiple view geometry under the l-infinity norm. In: PAMI (2008) Google Scholar
  9. 9.
    Ke, Q., Kanade, T.: Quasiconvex optimization for robust geometric reconstruction. In: ICCV (2005) Google Scholar
  10. 10.
    Kim, J.-H., Hartley, R.I., Frahm, J.-M., Pollefeys, M.: Visual odometry for non-overlapping views using second-order cone programming. In: ACCV (2), pp. 353–362 (2007) Google Scholar
  11. 11.
    Kim, J.-H., Li, H., Hartley, R.: Motion estimation for multi-camera systems using global optimization (2008) Google Scholar
  12. 12.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Perwass, C.: Applications of geometric algebra in computer vision. Ph.D. thesis, University of Cambridge (2000) Google Scholar
  14. 14.
    Perwass, C., Gebken, C., Sommer, G.: Geometry and kinematics with uncertain data. In: ECCV (1), pp. 225–237 (2006) Google Scholar
  15. 15.
    Rosenhahn, B.: Pose estimation revisited. Ph.D. thesis, Christian-Albrechts-Universitat zu Kiel, Institut für Informatik und Praktische Mathematik (2003) Google Scholar
  16. 16.
    Rosenhahn, B., Perwass, C., Sommer, G.: Free-form pose estimation by using twist representations. Algorithmica 38(1), 91–113 (2003) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Sommer, G., Rosenhahn, B., Perwass, C.: Twists—an operational representation of shape. In: IWMM GIAE, pp. 278–297 (2004) Google Scholar
  18. 18.
    Triggs, B., McLauchlan, P., Hartley, R., Fitzgibbon, A.: Bundle adjustment—a modern synthesis, pp. 298–375 (2000) Google Scholar
  19. 19.
    Zhang, Z.: Flexible camera calibration by viewing a plane from unknown orientations, vol. 1, pp. 666–673 (1999) Google Scholar
  20. 20.
    Zhang, Z.: A flexible new technique for camera calibration. IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  • Thibaud Debaecker
    • 1
    Email author
  • Ryad Benosman
    • 1
  • Sio H. Ieng
    • 1
  1. 1.Institut des Systèmes Intelligents et de Robotique (ISIR)ParisFrance

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