Nonrigid Shape and Motion from Multiple Perspective Views

  • René Vidal
  • Daniel Abretske
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3952)


We consider the problem of nonrigid shape and motion recovery from point correspondences in multiple perspective views. It is well known that the constraints among multiple views of a rigid shape are multilinear on the image points and can be reduced to bilinear (epipolar) and trilinear constraints among two and three views, respectively. In this paper, we generalize this classic result by showing that the constraints among multiple views of a nonrigid shape consisting of K shape bases can be reduced to multilinear constraints among K + ⌈ (K + 1)/2⌉, ⋯, 2K + 1 views. We then present a closed form solution to the reconstruction of a nonrigid shape consisting of two shape bases. We show that point correspondences in five views are related by a nonrigid quintifocal tensor, from which one can linearly compute nonrigid shape and motion. We also demonstrate the existence of intrinsic ambiguities in the reconstruction of camera translation, shape coefficients and shape bases. Examples show the effectiveness of our method on nonrigid scenes with significant perspective effects.


Null Space Camera Motion Multiple View Shape Base Point Correspondence 
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.


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  1. 1.
    Aanaes, H., Kahl, F.: Estimation of deformable structure and motion. In: ECCV Workshop on Vision and Modelling of Dynamic Scenes (2002)Google Scholar
  2. 2.
    Brand, M.: Morphable 3D models from video. In: IEEE Conf. on Computer Vision & Pattern Recognition, pp. 456–463 (2001)Google Scholar
  3. 3.
    Brand, M., Bhotika, R.: Flexible flow for 3D nonrigid tracking and shape recovery. In: IEEE Conf. on Computer Vision & Pattern Recognition, pp. 315–322 (2001)Google Scholar
  4. 4.
    Bregler, C., Hertzmann, A., Biermann, H.: Recovering non-rigid 3D shape from image streams. In: IEEE Conf. on Computer Vision & Pattern Recognition, pp. 2690–2696 (2000)Google Scholar
  5. 5.
    Doretto, G., Chiuso, A., Wu, Y., Soatto, S.: Dynamic textures. International Journal of Computer Vision 51(2), 91–109 (2003)CrossRefMATHGoogle Scholar
  6. 6.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, Cambridge (2000)Google Scholar
  7. 7.
    Ma, Y., Huang, K., Vidal, R., Košecká, J., Sastry, S.: Rank conditions on the multiple view matrix. International Journal of Computer Vision 59(2), 115–137 (2004)CrossRefGoogle Scholar
  8. 8.
    Tomasi, C., Kanade, T.: Shape and motion from image streams under orthography. International Journal of Computer Vision 9(2), 137–154 (1992)CrossRefGoogle Scholar
  9. 9.
    Torresani, L., Bregler, C.: Space-time tracking. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 801–812. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Torresani, L., Yang, D., Alexander, E., Bregler, C.: Tracking and modeling non-rigid objects with rank constraints. In: IEEE Conf. on Computer Vision and Pattern Recognition, pp. 493–500 (2001)Google Scholar
  11. 11.
    Vidal, R., Ma, Y., Soatto, S., Sastry, S.: Two-view multibody structure from motion. International Journal of Computer Vision (2006)Google Scholar
  12. 12.
    Vidal, R., Ravichandran, A.: Optical flow estimation and segmentation of multiple moving dynamic textures. In: IEEE Conf. on Computer Vision & Pattern Recognition, pp. 516–521 (2005)Google Scholar
  13. 13.
    Xiao, J., Chai, J.-x., Kanade, T.: A closed-form solution to non-rigid shape and motion recovery. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 573–587. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • René Vidal
    • 1
    • 2
  • Daniel Abretske
    • 2
  1. 1.Center for Imaging Science, Department of BMEJohns Hopkins UniversityUSA
  2. 2.Department of Computer ScienceJohns Hopkins UniversityUSA

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