International Journal of Computer Vision

, Volume 68, Issue 1, pp 7–25

Two-View Multibody Structure from Motion

  • René Vidal
  • Yi Ma
  • Stefano Soatto
  • Shankar Sastry

DOI: 10.1007/s11263-005-4839-7

Cite this article as:
Vidal, R., Ma, Y., Soatto, S. et al. Int J Comput Vision (2006) 68: 7. doi:10.1007/s11263-005-4839-7


We present an algebraic geometric approach to 3-D motion estimation and segmentation of multiple rigid-body motions from noise-free point correspondences in two perspective views. Our approach exploits the algebraic and geometric properties of the so-called multibody epipolar constraint and its associated multibody fundamental matrix, which are natural generalizations of the epipolar constraint and of the fundamental matrix to multiple motions. We derive a rank constraint on a polynomial embedding of the correspondences, from which one can estimate the number of independent motions as well as linearly solve for the multibody fundamental matrix. We then show how to compute the epipolar lines from the first-order derivatives of the multibody epipolar constraint and the epipoles by solving a plane clustering problem using Generalized PCA (GPCA). Given the epipoles and epipolar lines, the estimation of individual fundamental matrices becomes a linear problem. The clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices. Although our approach is mostly designed for noise-free correspondences, we also test its performance on synthetic and real data with moderate levels of noise.


multibody structure from motion 3-D motion segmentation multibody epipolar constraint multibody fundamental matrix Generalized PCA (GPCA) 

Copyright information

© Springer Science + Business Media, LLC. 2006

Authors and Affiliations

  • René Vidal
    • 1
  • Yi Ma
    • 2
  • Stefano Soatto
    • 3
  • Shankar Sastry
    • 4
  1. 1.Center for Imaging Science, Department of Biomedical EngineeringJohns Hopkins UniversityBaltimore
  2. 2.Department of ECEUniversity of Illinois at Urbana-ChampaignUrbana
  3. 3.Computer Science DepartmentUniversity of California at Los AngelesLos Angeles
  4. 4.Department of EECSUniversity of California at BerkeleyBerkeley

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