Skip to main content
SpringerLink
Log in

3D Motion recovery via affine Epipolar geometry

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

Algorithms to perform point-based motion estimation under orthographic and scaled orthographic projection abound in the literature. A key limitation of many existing algorithms is that they operate on the minimum amount of data required, often requiring the selection of a suitable minimal set from the available data to serve as a “local coordinate frame”. Such approaches are extremely sensitive to errors and noise in the minimal set, and forfeit the advantages of using the full data set. Furthermore, attention is seldom paid to the statistical performance of the algorithms.

We present a new framework that allowsall available features to be used in the motion computations, without the need to select a frame explicitly. This theory is derived in the context of theaffine camera, which preserves parallelism and generalises the orthographic, scaled orthographic and para-perspective models. We define the affine epipolar geometry for two such cameras, giving the fundamental matrix in this case. The noise resistant computation of the epipolar geometry is discussed, and a statistical noise model constructed so that confidence in the results can be assessed.

The rigid motion parameters are then determineddirectly from the epipolar geometry, using the novel rotation representation of Koenderink and van Doorn (1991). The two-view partial motion solution comprises the scale factor between views, the projection of the 3D axis of rotation and the cyclotorsion angle, while the addition of a third view allows the true 3D rotation axis to be computed (up to a Necker reversal). The computed uncertainties in these parameters permit optimal estimates to be obtained over time by means of a linear Kalman filter. Our theory extends work by Huang and Lee (1989), Harris (1990), and Koenderink and van Doorn (1991), and results are given on both simulated and real data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aloimonos, J. 1986. Detection of surface orientation from texture I: The case of planes. InIEEE Conference on Computer Vision and Pattern Recognition (CVPR'86), Florida, pp. 584–593.

  • Aloimonos, J. and Bandyopadhyay, A. 1985. Perception of structure from motion: lower bound results. Tech. Report 158, Dept. Computer Science, University of Rochester.

  • Aloimonos, J.Y. 1992. Perspective approximations.Image and Vision Computing, 8(3):179–192.

    Google Scholar 

  • Arnold, R.D. and Binford, T.O. 1980. Geometrical constraints in stereo vision. InProceedings S.P.I.E., 238:281–292.

    Google Scholar 

  • Bar-Shalom, Y. and Fortmann, T.E. 1988. Tracking and Data Association, Academic Press Inc., USA.

    Google Scholar 

  • Barnard, S.T. and Thompson, W.B. 1980. Disparity analysis of images.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2(4):333–340.

    Google Scholar 

  • Beardsley, P.A., Zisserman, A., and Murray, D.W. 1993. Projective structure from image sequences. Tech. Report OUEL 1985/93, Dept. Engineering Science, University of Oxford.

  • Beardsley, P.A., Zisserman, A., and Murray, D.W. 1994. Navigation using affine structure from motion. In J.O. Eklundh (ed.),Proceedings European Conference on Computer Vision (ECCV-94), II:85–96.

  • Bennett, B.M., Hoffman, D.D., Nicola, J.E., and Prakash, C. 1989. Structure from two orthographic views of rigid motion.Journal of Optical Society of America, 6(7): 1052–1069.

    Google Scholar 

  • Berger, M. 1980.Geometry I, Springer Verlag.

  • Bookstein, F.L. 1979. Fitting conic sections to scattered data.Computer Graphics and Image Processing, 9:56–71.

    Google Scholar 

  • Charnley, D., Harris, C., Pike, M., Sparks, E., and Stephens, M. 1988. The DROID 3D vision system: algorithms for geometric integration. Plessey Research, Roke Manor, Technical Note 72/88/N488U.

  • Chen, H.H. and Huang, T.S. 1991. Using motion from orthographic views to verify 3D point matches.IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(9):872–878.

    Google Scholar 

  • Demey, S., Zisserman, A., and Beardsley, P. 1992. Affine and projective structure from motion.Proceedings British Machine Vision Conference (BMVC'92), pp. 49–58.

  • Durrant-Whyte, H.F. 1993.Methods and Systems in Data Fusion, in press.

  • Faugeras, O.D., Luong, Q.-T., and Maybank, S.J. 1992. Camera self-calibration: theory and experiments. In G. Sandini (ed.),Proceedings European Conference on Computer Vision (ECCV-92), pp. 321–334.

  • Faugeras, O.D. 1992. What can be seen in three dimensions with an uncalibrated stereo rig? In G. Sandini (ed.),Proceedings European Conference on Computer Vision (ECCV-92), pp. 563–578.

  • Gelb, M. 1974.Applied Optimal Estimation, MIT Press.

  • Harris, C. 1990. Structure-from-motion under orthographic projection. InProceedings European Conference on Computer Vision (ECCV-90), pp. 118–123.

  • Hartley, R.I. 1992. Estimation of relative camera positions for uncalibrated cameras. InProceedings European Conference on Computer Vision (ECCV-92), pp. 579–587.

  • Hollinghurst, N. and Cipolla, R. 1993. Uncalibrated stereo hand-eye coordination. InProceedings British Machine Vision Conference (BMVC'93), Surrey, pp. 389–398.

  • Hu, X. and Ahuja, N. 1991. Motion estimation under orthographic projection. InIEEE Transactions on Robotics and Automation, 7(6):848–853.

    Google Scholar 

  • Huang, T.S. and Lee, C.H. 1989. Motion and structure from orthographic projections.IEEE Trans. Pattern Anal. Machine Intell., PAMI-11(5):536–540.

    Google Scholar 

  • Kanatani, K. 1993.Geometric Computation for Computer Vision, Oxford University Press, UK.

    Google Scholar 

  • Koenderink, J.J. and van Doorn A.J. 1991. Affine structure from motion.Journal of Optical Society of America, 8(2):377–385.

    Google Scholar 

  • Lee, C. and Huang, T. 1990. Finding point correspondences and determining motion of a rigid object from two weak perspective views.Computer Vision, Graphics and Image Processing, 52: 309–327.

    Google Scholar 

  • Longuet-Higgins, H.C. 1981. A computer algorithm for reconstructing a scene from two projections.Nature, 293:133–135.

    Google Scholar 

  • Longuet-Higgins, H.C. 1991. A method of determining the relation positions of 4 points from 3 perspective projections. In P. Mowforth (ed.),Proceedings British Machine Vision Conference (BMVC'91), pp. 86–94.

  • Luong, Q.-T., Deriche, R., Faugeras, O., and Papadopoulo, T. 1993. On determining the fundamental matrix: analysis of different methods and experimental results. Tech. Report 1894, INRIA (Sophia Antipolis).

  • Mundy, J.L. and Zisserman A. (eds). 1992.Geometric Invariance in Computer Vision, MIT Press, USA.

    Google Scholar 

  • Olsen, S.I. 1992. Epipolar line estimation. In G. Sandini, (ed.), Proceedings European Conference on Computer Vision (ECCV-92), pp. 307–311.

  • Pollard, S.B., Mayhew, J.E.W., and Frisby, J.P. 1985. PMF: A Stereo Correspondence Algorithm using a Disparity Gradient Limit.Perception, 14:449–470.

    Google Scholar 

  • Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. 1988.Numerical Recipes in C, Cambridge University Press, USA.

    Google Scholar 

  • Quan, L. and Mohr, R. 1991. Towards structure from motion for linear features through reference points.IEEE Workshop on Visual Motion, New Jersey.

  • Reid, I.D. and Murray, D.W. 1993. Tracking foveated corner clusters using affine structure. InProceedings International Conference on Computer Vision (ICCV-4), Berlin, pp. 76–83.

  • Reid, I.D. and Murray, D.W. 1994. Active tracking of foveated feature clusters using affine structure. To appear IJCV.

  • Shapiro, L.S. 1993.Affine Analysis of Image Sequences, Ph.D. thesis, Dept. Engineering Science, Oxford University. Also 1995, Cambridge University Press, UK.

  • Shapiro, L.S. and Brady, J.M. 1993. Rejecting outliers and estimating errors in an orthogonal regression framework. Tech. Report OUEL 1974/93, Dept. Engineering Science, University of Oxford. Also inPhil. Tran. R. Soc. Lond., A (1995) 350: 407–439.

  • Shapiro, L.S., Wang H., and Brady, J.M. 1992. A matching and tracking strategy for independently moving objects. In D. Hogg and R. Boyle (eds.),Proceedings British Machine Vision Conference, Leeds, Springer-Verlag, U.K., pp. 306–315.

    Google Scholar 

  • Shapiro, L.S., Zisserman, A., and Brady, J.M. 1994. Motion from point matches using affine epipolar geometry. In J.O. Eklundh (ed.),Proceedings European Conference on Computer Vision (ECCV-94), II:73–84.

  • Strang, G. 1988.Linear Algebra and its Applications, 3rd ed., Harcourt Brace Jovanovich Inc., U.S.A.

    Google Scholar 

  • Thompson, D.W. and Mundy, J.L. 1987. Three dimensional model matching from an unconstrained viewpoint. In IEEE Conference on Robotics and Automation, Raleigh, NC, pp. 208–220.

  • Tomasi, C. and Kanade, T. 1992. Shape and motion from image streams under orthography: a factorization method.International Journal of Computer Vision, 9(2):137–154.

    Google Scholar 

  • Torr, P.H.S. 1993. Notes on epipole at infinity. Personal communication.

  • Torr, P.H.S. and Murray, D.W. 1993. Outlier detection and motion segmentation. In Schenker (ed.),Sensor Fusion VI, SPIE Vol. 2059, Boston, pp. 432–443.

  • Ullman, S. 1979.The Interpretation of Visual Motion, MIT Press, U.S.A.

    Google Scholar 

  • Ullman, S. and Basri, R. 1991. Recognition by linear combinations of models.IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(10):992–1006.

    Google Scholar 

  • Wang, H. and Brady, J.M. 1992. Corner detection: some new results.IEE Colloquium Digest of System Aspects of Machine Perception and Vision, pp. 1.1–1.4. London: IEE.

    Google Scholar 

  • Weinshall, D. and Tomasi, C. 1993. Linear and incremental acquisition of invariant shape models from image sequences. InProceedings International Conference on Computer Vision (ICCV-4), pp. 675–682.

  • Weinshall, D. 1993. Model-based invariants for 3-D vision.International Journal of Computer Vision, 10(1):27–42.

    Google Scholar 

  • Weng, J., Huang, T.S., and Ahuja, N. 1989. Motion and structure from two perspective views: algorithms, error analysis and error estimation.IEEE Trans. Pattern Anal. Machine Intell., PAMI-11(5):451–476.

    Google Scholar 

  • Weng, J., Ahuja, N., and Huang, T.S. 1993. Optimal motion and structure estimation.IEEE Trans. Pattern Anal. Machine Intell., PAMI-15(9):864–884.

    Google Scholar 

  • Xu, G., Nishimura, E., and Tsuji, S. 1993. Image correspondence and segmentation by epipolar lines: theory, algorithm and applications. Technical Report, Dept. Systems Engineering, Osaka University.

  • Zisserman, A. 1992.Notes on geometric invariance in vision: BMVC'92 tutorial, Leeds University.

Download references

Author information

Authors and Affiliations

  1. Robotics Research Group, Department of Engineering Science, Oxford University, Parks Road, OX1 3PJ, Oxford, UK

    Larry S. Shapiro, Andrew Zisserman & Michael Brady

Authors
  1. Larry S. Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew Zisserman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Brady
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shapiro, L.S., Zisserman, A. & Brady, M. 3D Motion recovery via affine Epipolar geometry. Int J Comput Vision 16, 147–182 (1995). https://doi.org/10.1007/BF01539553

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01539553

Keywords

Search

Navigation