Journal of Mathematical Imaging and Vision

, Volume 50, Issue 3, pp 179–198 | Cite as

Autocalibration with the Minimum Number of Cameras with Known Pixel Shape

  • José I. RondaEmail author
  • Antonio Valdés
  • Guillermo Gallego


In 3D reconstruction, the recovery of the calibration parameters of the cameras is paramount since it provides metric information about the observed scene, e.g., measures of angles and ratios of distances. Autocalibration enables the estimation of the camera parameters without using a calibration device, but by enforcing simple constraints on the camera parameters. In the absence of information about the internal camera parameters such as the focal length and the principal point, the knowledge of the camera pixel shape is usually the only available constraint. Given a projective reconstruction of a rigid scene, we address the problem of the autocalibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters. We propose an algorithm that only requires 5 cameras (the theoretical minimum), thus halving the number of cameras required by previous algorithms based on the same constraint. To this purpose, we introduce as our basic geometric tool the six-line conic variety (SLCV), consisting in the set of planes intersecting six given lines of 3D space in points of a conic. We show that the set of solutions of the Euclidean upgrading problem for three cameras with known pixel shape can be parameterized in a computationally efficient way. This parameterization is then used to solve autocalibration from five or more cameras, reducing the three-dimensional search space to a two-dimensional one. We provide experiments with real images showing the good performance of the technique.


Camera autocalibration Varying parameters Square pixels Three-dimensional reconstruction Absolute conic Six line conic variety 



The authors thank the anonymous reviewers for their helpful comments, in particular for suggesting the comparisons of the SLCV algorithm with the 3D search algorithms, and for pointing out references [8] and [32].


  1. 1.
    Bôcher, M.: Introduction to Higher Algebra. Dover Phoenix Editions. Dover, New York (2004) Google Scholar
  2. 2.
    Bradski, G.: The OpenCV Library. Dr. Dobb’s J. Softw. Tools (2000) Google Scholar
  3. 3.
    Carballeira, P., Ronda, J.I., Valdés, A.: 3D reconstruction with uncalibrated cameras using the six-line conic variety. In: IEEE International Conference on Image Processing, pp. 205–208 (2008) Google Scholar
  4. 4.
    Faugeras, O.: Three Dimensional Computer Vision. MIT Press, Cambridge (1993) Google Scholar
  5. 5.
    Faugeras, O.: Stratification of 3-D vision: projective, affine, and metric representations. J. Opt. Soc. Am. A 12(46), 548 (1995) Google Scholar
  6. 6.
    Faugeras, O., Luong, Q., Maybank, S.: Camera self-calibration: theory and experiments. In: European Conference on Computer Vision. Lecture Notes in Computer Science, vol. 588, pp. 321–334. Springer, Berlin (1992) Google Scholar
  7. 7.
    Faugeras, O., Luong, Q.T., Papadopoulou, T.: The Geometry of Multiple Images: the Laws that Govern the Formation of Images of a Scene and Some of Their Applications. MIT Press, Cambridge (2001) Google Scholar
  8. 8.
    Finsterwalder, S.: Die geometrischen grundlagen der photogrammetrie. Jahresber. Dtsch. Math.-Ver. 6, 1–42 (1897). Google Scholar
  9. 9.
    Furukawa, Y., Ponce, J.: Accurate, dense, and robust multi-view stereopsis. In: IEEE Conference on Computer Vision and Pattern Recognition (2007) Google Scholar
  10. 10.
    Hartley, R.: Projective reconstruction and invariants from multiple images. IEEE Trans. Pattern Anal. Mach. Intell. 16(10), 1036–1041 (1994) CrossRefGoogle Scholar
  11. 11.
    Hartley, R., Gupta, R., Chang, T.: Stereo from uncalibrated cameras. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 761–764 (1992) Google Scholar
  12. 12.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003) Google Scholar
  13. 13.
    Hartley, R.I.: Chirality. Int. J. Comput. Vis. 26(1), 41–61 (1998) CrossRefGoogle Scholar
  14. 14.
    Hartley, R.I., Hayman, E., de Agapito, L., Reid, I.: Camera calibration and the search for infinity. In: IEEE International Conference on Computer Vision, vol. 1, p. 510 (1999) Google Scholar
  15. 15.
    Hemayed, E.: A survey of camera self-calibration. In: Proc. IEEE Conference on Advanced Video and Signal Based Surveillance, pp. 351–357 (2003) CrossRefGoogle Scholar
  16. 16.
    Heyden, A., Åström, K.: Euclidean reconstruction from image sequences with varying and unknown focal length and principal point. In: IEEE Conference on Computer Vision and Pattern Recognition (1997) Google Scholar
  17. 17.
    Longuet-Higgins, H.C.: A computer algorithm for reconstructing a scene from two projections. Nature 293, 133–135 (1981) CrossRefGoogle Scholar
  18. 18.
    Lowe, D.: Object recognition from local scale-invariant features. In: International Conference on Computer Vision, pp. 1150–1157 (1999) Google Scholar
  19. 19.
    Luong, Q.T., Viéville, T.: Canonical representations for the geometries of multiple projective views. Comput. Vis. Image Underst. 64, 193–229 (1996) CrossRefGoogle Scholar
  20. 20.
    Ma, Y., Soatto, S., Kosecka, J., Sastry, S.: An Invitation to 3-D Vision. Springer, Berlin (2003) Google Scholar
  21. 21.
    Maybank, S.J., Faugeras, O.D.: A theory of self-calibration of a moving camera. Int. J. Comput. Vis. 8(2), 123–151 (1992) CrossRefGoogle Scholar
  22. 22.
    Mohr, R., Veillon, F., Quan, L.: Relative 3D reconstruction using multiple uncalibrated images. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 543–548 (1993) CrossRefGoogle Scholar
  23. 23.
    Pollefeys, M., Gool, L.V.: A stratified approach to metric self-calibration. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 407–412 (1997) Google Scholar
  24. 24.
    Pollefeys, M., Koch, R., van Gool, L.: Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters. Int. J. Comput. Vis. 1(32), 7–25 (1999) CrossRefGoogle Scholar
  25. 25.
    Ponce, J., McHenry, K., Papadopoulo, T., Teillaud, M., Triggs, B.: On the absolute quadratic complex and its application to autocalibration. In: IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 780–787 (2005) Google Scholar
  26. 26.
    Quan, L.: Invariants of 6 points from 3 uncalibrated images. In: Computer Vision—ECCV ’94. Lecture Notes in Computer Science, vol. 801, pp. 459–470. Springer, Berlin (1994) Google Scholar
  27. 27.
    Ronda, J.I., Valdés, A., Gallego, G.: Line geometry and camera autocalibration. J. Math. Imaging Vis. 32(2), 193–214 (2008) CrossRefGoogle Scholar
  28. 28.
    Schröcker, H.P.: Intersection conics of six straight lines. Beitr. Algebra Geom. 46(2), 435–446 (2005) zbMATHGoogle Scholar
  29. 29.
    Seo, Y., Heyden, A.: Auto-calibration from the orthogonality constraints. In: Proc. International Conference on Pattern Recognition, vol. 1, pp. 1067–1071 (2000) Google Scholar
  30. 30.
    Snavely, N., Seitz, S.M., Szeliski, R.: Modeling the world from Internet photo collections. Int. J. Comput. Vis. 80(2), 189–210 (2008) CrossRefGoogle Scholar
  31. 31.
    Sturm, P.: A case against Kruppa’s equations for camera self-calibration. IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1199–1204 (2000) CrossRefGoogle Scholar
  32. 32.
    Sturm, P.: A historical survey of geometric computer vision. In: Computer Analysis of Images and Patterns. LNCS, vol. 6854, pp. 1–8. Springer, Berlin (2011) CrossRefGoogle Scholar
  33. 33.
    Tresadern, P.A., Reid, I.D.: Camera calibration from human motion. Image Vis. Comput. 26(6), 851–862 (2008) CrossRefGoogle Scholar
  34. 34.
    Triggs, B.: The geometry of projective reconstruction I: Matching constraints and the joint image. Int. J. Comput. Vision, 338–343 (1995) Google Scholar
  35. 35.
    Triggs, B.: Autocalibration and the absolute quadric. In: Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 609–614 (1997) CrossRefGoogle Scholar
  36. 36.
    Triggs, B., McLauchlan, P., Hartley, R., Fitzgibbon, A.: Bundle adjustment—a modern synthesis. In: Vision Algorithms: Theory and Practice. Lecture Notes in Computer Science, vol. 1883, pp. 153–177. Springer, Berlin (2000) CrossRefGoogle Scholar
  37. 37.
    Valdés, A., Ronda, J.I., Gallego, G.: Linear camera autocalibration with varying parameters. In: IEEE International Conference on Image Processing, Singapore, vol. 5, pp. 3395–3398 (2004) Google Scholar
  38. 38.
    Valdés, A., Ronda, J.I., Gallego, G.: The absolute line quadric and camera autocalibration. Int. J. Comput. Vis. 66(3), 283–303 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • José I. Ronda
    • 1
    Email author
  • Antonio Valdés
    • 2
  • Guillermo Gallego
    • 1
  1. 1.Grupo de Tratamiento de ImágenesUniversidad Politécnica de MadridMadridSpain
  2. 2.Dep. de Geometría y TopologíaUniversidad Complutense de MadridMadridSpain

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