Self-calibration of a General Radially Symmetric Distortion Model

  • Jean-Philippe Tardif
  • Peter Sturm
  • Sébastien Roy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3954)


We present a new approach for self-calibrating the distortion function and the distortion center of cameras with general radially symmetric distortion. In contrast to most current models, we propose a model encompassing fisheye lenses as well as catadioptric cameras with a view angle larger than 180°.

Rather than representing distortion as an image displacement, we model it as a varying focal length, which is a function of the distance to the distortion center. This function can be discretized, acting as a general model, or represented with e.g. a polynomial expression.

We present two flexible approaches for calibrating the distortion function. The first one is a plumbline-type method; images of line patterns are used to formulate linear constraints on the distortion function parameters. This linear system can be solved up to an unknown scale factor (a global focal length), which is sufficient for image rectification. The second approach is based on the first one and performs self-calibration from images of a textured planar object of unknown structure. We also show that by restricting the camera motion, self-calibration is possible from images of a completely unknown, non-planar scene.

The analysis of rectified images, obtained using the computed distortion functions, shows very good results compared to other approaches and models, even those relying on non-linear optimization.


Interest Point Optical Center View Angle Camera Model Distortion Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Baker, S., Nayar, S.K.: A Theory of Single-Viewpoint Catadioptric Image Formation. IJCV 35(2), 1–22 (1999)CrossRefGoogle Scholar
  2. 2.
    Barreto, J.P., Daniilidis, K.: Unifying image plane liftings for central catadioptric and dioptric cameras. In: OMNIVIS (2004)Google Scholar
  3. 3.
    Born, M., Wolf, E.: Principles of Optics. Pergamon Press, Oxford (1965)Google Scholar
  4. 4.
    Brown, D.C.: Close-Range Camera Calibration. Photogrammetric Engineering 37(8), 855–866 (1971)Google Scholar
  5. 5.
    Claus, D., Fitzgibbon, A.W.: Rational Function Model for Fish-eye Lens Distortion. In: CVPR (2005)Google Scholar
  6. 6.
    Devernay, F., Faugeras, O.: Straight lines have to be straight: Automatic calibration and removal of distortion from scenes of structured environments. In: MVA (2001)Google Scholar
  7. 7.
    Fitzgibbon, A.W.: Simultaneous linear estimation of multiple view geometry and lens distortion. In: CVPR (2001)Google Scholar
  8. 8.
    Fleck, M.M.: Perspective Projection: The Wrong Imaging Model. TR 95–01, University of Iowa (1995)Google Scholar
  9. 9.
    Geyer, C., Daniilidis, K.: Catadioptric Camera Calibration. In: ICCV (1999)Google Scholar
  10. 10.
    Grossberg, M.D., Nayar, S.K.: A general imaging model and a method for finding its parameters. In: ICCV (2001)Google Scholar
  11. 11.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  12. 12.
    Hartley, R.I., Kang, S.B.: Parameter-free Radial Distortion Correction with Centre of Distortion Estimation. In: ICCV (2005)Google Scholar
  13. 13.
    Intel Open Source Computer Vision Library,
  14. 14.
    Micusik, B., Pajdla, T.: Autocalibration & 3D Reconstruction with Non-central Catadioptric Cameras. In: CVPR (2004)Google Scholar
  15. 15.
    Shah, S., Aggarwal, J.K.: Intrinsic Parameter Calibration Procedure for A (High-Distortion) Fish-Eye Lens Camera with Distortion Model and Accuracy Estimation. Pattern Recognition 29(11), 1775–1788 (1996)CrossRefGoogle Scholar
  16. 16.
    Stevenson, D.E., Fleck, M.M.: Nonparametric correction of distortion. TR 95–07, University of Iowa (1995)Google Scholar
  17. 17.
    Sturm, P., Ramalingam, S.: A generic concept for camera calibration. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3022, pp. 1–13. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Swaminathan, R., Grossberg, M., Nayar, S.: Caustics of catadioptric cameras. In: ICCV (2001)Google Scholar
  19. 19.
    Tardif, J.-P., Sturm, P.: Calibration of Cameras with Radially Symmetric Distortion. In: OMNIVIS 2005 (2005)Google Scholar
  20. 20.
    Thirthala, S., Pollefeys, M.: The Radial Trifocal Tensor. A tool for calibrating the radial distortion of wide-angle cameras. In: CVPR 2005 (2005)Google Scholar
  21. 21.
    Thirthala, S., Pollefeys, M.: Multi-View Geometry of 1D Radial Cameras and its Application to Omnidirectional Camera Calibration. In: ICCV (to appear, 2005)Google Scholar
  22. 22.
    Triggs, B.: Autocalibration from planar scenes. In: Burkhardt, H.-J., Neumann, B. (eds.) ECCV 1998. LNCS, vol. 1406, pp. 89–108. Springer, Heidelberg (1998)Google Scholar
  23. 23.
    Ying, X., Hu, Z.: Can we consider central catadioptric cameras and fisheye cameras within a unified imaging model. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3021, pp. 442–455. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  24. 24.
    Zhang, Z.: A Flexible New Technique for Camera Calibration. PAMI 22(11), 1330–1334 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jean-Philippe Tardif
    • 1
  • Peter Sturm
    • 2
  • Sébastien Roy
    • 1
  1. 1.DIROUniversité de MontréalCanada
  2. 2.INRIA Rhône-AlpesMontbonnot St MartinFrance

Personalised recommendations