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Reconstruction from Projections Using Grassmann Tensors

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Abstract

In this paper a general procedure is given for reconstruction of a set of feature points in an arbitrary dimensional projective space from their projections into lower dimensional spaces. This extends the methods applied in the well-studied problem of reconstruction of scene points in ℘3 given their projections in a set of images. In this case, the bifocal, trifocal and quadrifocal tensors are used to carry out this computation. It is shown that similar methods will apply in a much more general context, and hence may be applied to projections from ℘n to ℘m, which have been used in the analysis of dynamic scenes, and in radial distortion correction. For sufficiently many generic projections, reconstruction of the scene is shown to be unique up to projectivity, except in the case of projections onto one-dimensional image spaces (lines), in which case there are two solutions.

Projections from ℘n to ℘2 have been considered by Wolf and Shashua (in International Journal of Computer Vision 48(1): 53–67, 2002), where they were applied to several different problems in dynamic scene analysis. They analyzed these projections using tensors, but no general way of defining such tensors, and computing the projections was given. This paper settles the general problem, showing that tensor definition and retrieval of the projections is always possible.

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References

  • Faugeras, O. D., Quan, L., & Sturm, P. (2000). Self-calibration of a 1D projective camera and its application to the self-calibration of a 2D projective camera. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(10), 1179–1185.

    Article  Google Scholar 

  • Hartley, R. I. (1997). In defense of the eight-point algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(6), 580–593.

    Article  Google Scholar 

  • Hartley, R. I. (1998a). Computation of the quadrifocal tensor. In Lecture notes in computer science : Vol. 1406. Proceedings of the 5th European conference on computer vision (pp. 20–35). Freiburg, Germany. Berlin: Springer.

    Google Scholar 

  • Hartley, R. I. (1998b). Minimizing algebraic error. Philosophical Transactions of the Royal Society of London, Series A, 356(1740), 1175–1192.

    Article  MATH  MathSciNet  Google Scholar 

  • Hartley, R., & Vidal, R. (2008). Perspective nonrigid shape and motion recovery. In Proceedings of the European conference on computer Vision (pp. 276–289).

  • Hartley, R. I., & Zisserman, A. (2004). Multiple view geometry in computer vision (2nd ed.). Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Heyden, A. (1998a). A common framework for multiple-view tensors. In Proceedings of the 5th European conference on computer vision (pp. 3–19). Freiburg, Germany.

  • Heyden, A. (1998b). Reduced multilinear constraints: Theory and experiments. International Journal of Computer Vision, 30(1), 5–26.

    Article  Google Scholar 

  • Quan, L. (2001). Two-way ambiguity in 2d projective reconstruction from three uncalibrated 1d images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(2), 212–216.

    Article  Google Scholar 

  • Quan, L., & Kanade, T. (1997). Affine structure from line correspondences with uncalibrated affine cameras. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(8), 834–845.

    Article  Google Scholar 

  • Semple, J. G., & Kneebone, G. T. (1979). Algebraic projective geometry. Oxford: Oxford University Press.

    Google Scholar 

  • Thirthala, S., & Pollefeys, M. (2005a). Multi-view geometry of 1d radial cameras and its application to omnidirectional camera calibration. In Proceedings of the 10th international conference on computer vision (pp. 1539–1546). Beijing, China.

  • Thirthala, S., & Pollefeys, M. (2005b). The radial trifocal tensor: A tool for calibrating the radial distortion of wide-angle cameras. In Proceedings of the IEEE conference on computer vision and pattern recognition (Vol. 1, pp. 321–328) San Diego.

  • Wolf, L., & Shashua, A. (2002). On projection matrices ℘k→℘2,k=3,…,6, and their applications in computer vision. International Journal of Computer Vision, 48(1), 53–67.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. National ICT Australia and the Australian National University, Canberra, Australia

    Richard I. Hartley

  2. Australian National University, Canberra, Australia

    Frederik Schaffalitzky

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  1. Richard I. Hartley
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  2. Frederik Schaffalitzky
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Correspondence to Richard I. Hartley.

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Hartley, R.I., Schaffalitzky, F. Reconstruction from Projections Using Grassmann Tensors. Int J Comput Vis 83, 274–293 (2009). https://doi.org/10.1007/s11263-009-0225-1

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  • DOI: https://doi.org/10.1007/s11263-009-0225-1

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