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The Absolute Line Quadric and Camera Autocalibration

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Abstract.

We introduce a geometrical object providing the same information as the absolute conic: the absolute line quadric (ALQ). After the introduction of the necessary exterior algebra and Grassmannian geometry tools, we analyze the Grassmannian of lines of P from both the projective and Euclidean points of view. The exterior algebra setting allows then to introduce the ALQ as a quadric arising very naturally from the dual absolute quadric. We fully characterize the ALQ and provide clean relationships to solve the inverse problem, i.e., recovering the Euclidean structure of space from the ALQ. Finally we show how the ALQ turns out to be particularly suitable to address the Euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters, providing new linear and non-linear algorithms for this problem. We also provide experimental results showing the good performance of the techniques.

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References

  • Bayro-Corrochano, E. and Rosenhahn, B. 2002. A geometric approach for the analysis and computation of the intrinsic parameters. Pattern Recognition, 35:169–186.

    Article  Google Scholar 

  • Barnabei, M., Brini, A., and Rota, G.C. 1985. On the exterior calculus of invariant theory. Journal of Algebra, 96:120–160.

    MathSciNet  Google Scholar 

  • Bourbaki, N. 1970. Éléments de mathématique. Algèbre. Chapitres 1 à 3, Paris, Hermann.

  • Carlsson, S. 1993. The double algebra: An effective tool for computing invariants in computer vision, Applications of Invariance in Computer Vision: Joint European – US Workshop, Azores, Portugal, 145–164.

  • Dummit, D.S. and Foote, R.M. 1999. Abstract Algebra. Wiley, NJ, USA.

    Google Scholar 

  • Faugeras, O. and Luong, Q.T. 2001. The Geometry of Multiple Images, The MIT Press: Cambridge, Massachusetts.

    Google Scholar 

  • Gill, P., Murray, W., and Wright, M. 1981. Practical Optimization. Academic Press.

  • Harris, J. 1992. Algebraic Geometry. A First Course. Graduate Texts in Mathematics 133, Springer-Verlag: New York.

  • Hartley, R.I. and Zisserman, A. 2000. Multiple View Geometry in Computer Vision. Cambridge University Press: Cambridge, UK.

    Google Scholar 

  • Heyden, A. 1998. A common framework for multiple-view tensors. In European Conference on Computer Vision, Freiburg, Germany.

  • Heyden, A. and Aström, K. 1998. Minimal conditions on intrinsic parameters for Euclidean reconstruction. Asian Conference on Computer Vision, Hong Kong.

  • Kaminski, J.Y., Fryers, M., Shashua, A., and Teicher, M. 2001. Multiple view geometry of non-planar algebraic curves. Computer Vision, 2001. ICCV 2001, no. 2, pp. 181–186.

  • Maybank, S.J. and Faugeras, O. 1992. A theory of self-calibration of a moving camera. International Journal of Computer Vision, 8:123–152.

    Google Scholar 

  • Pollefeys, M., Koch, R., and Van Gool, L. 1998. Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters, In Proc. ICCV'98.

  • Ponce, J. 2000. On computing metric upgrades of projective reconstructions under the rectangular pixel assumption. Proc. SMILE.

  • Pottmann, H. and Wallner, J. 2001. Computational Line Geometry, Springer-Verlag: Germany.

    Google Scholar 

  • Semple, J.G. and Kneebone, G.T. 1998. Algebraic Projective Geometry, Oxford University Press: Oxford.

    Google Scholar 

  • Seo, Y. and Heyden, A. Autocalibration from the orthogonality constraints. In ICCV'00.

  • Triggs, W. 1995. Matching constraints and the joint image. In IEEE Int. Conf. Computer Vision, Cambridge: MA.

  • Triggs, W. 1997. Auto-calibration and the absolute quadric. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, pp. 609–614.

  • Valdés, A. and Ronda, J. I. 2005. Camera autocalibration and the calibration pencil. Journal of Mathematical Imaging and Vision, 23:167–174.

    Article  MathSciNet  Google Scholar 

  • Wells, R.O. 1980. Differential Analysis on Complex Manifolds, Springer-Verlag: New York.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid, E-28040, Spain

    Antonio Valdés

  2. Grupo de Tratamiento de Imágenes, Universidad Politécnica de Madrid, Madrid, E-28040, Spain

    José Ignacio Ronda & Guillermo Gallego

Authors
  1. Antonio Valdés
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  2. José Ignacio Ronda
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  3. Guillermo Gallego
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Corresponding author

Correspondence to Antonio Valdés.

Additional information

This work has been partly supported by the Comisión Interministerial de Ciencia y Tecnología (CICYT) of the Spanish Government.

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Valdés, A., Ronda, J.I. & Gallego, G. The Absolute Line Quadric and Camera Autocalibration. Int J Comput Vision 66, 283–303 (2006). https://doi.org/10.1007/s11263-005-3677-y

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  • DOI: https://doi.org/10.1007/s11263-005-3677-y

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