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                          Segmentation of Dynamic Scenes 
                      from the Multibody Fundamental Matrix

                                      Rene Vidal
                                     UC Berkeley

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We present a geometric approach for the analysis of dynamic scenes containing
multiple rigidly moving objects seen 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 moving objects. We derive
a rank constraint on the image points from which one can estimate the number
of independent motions and linearly solve for the multibody fundamental
matrix. We prove that the epipoles of each independent motion lie exactly in
the intersection of the left null space of the multibody fundamental matrix
with the so-called Veronese surface. We then show that individual epipoles and
epipolar lines can be uniformly and efficiently computed by using a novel
polynomial factorization technique. Given the epipoles and epipolar lines,
the estimation of individual fundamental matrices becomes a linear
problem. Then, motion and feature point segmentation is automatically
obtained from either the epipoles and epipolar lines or the individual
fundamental matrices.