International Journal of Computer Vision

, Volume 68, Issue 1, pp 7-25

First online:

Two-View Multibody Structure from Motion

  • René VidalAffiliated withCenter for Imaging Science, Department of Biomedical Engineering, Johns Hopkins University Email author 
  • , Yi MaAffiliated withDepartment of ECE, University of Illinois at Urbana-Champaign
  • , Stefano SoattoAffiliated withComputer Science Department, University of California at Los Angeles
  • , Shankar SastryAffiliated withDepartment of EECS, University of California at Berkeley

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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)