(Professor S. Shankar Sastry)

(ONR) N00014-00-1-0621

In [1,2] 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 are automatically obtained from either the epipoles and epipolar lines or the individual fundamental matrices.

In [3] we present an algebraic geometric approach for segmenting both static and dynamic scenes from image intensities. We introduce the multibody affine constraint as a geometric relationship between the motion of multiple objects and the image intensities generated by them. This constraint is satisfied by all the pixels, regardless of the body to which they belong and regardless of depth discontinuities or perspective effects. We propose a polynomial factorization technique that estimates the number of affine motion models as well as their motion parameters in polynomial time. The factorization technique is used to initialize a nonlinear algorithm that minimizes the algebraic error defined by the multibody affine constraint. Our approach is based solely on image intensities, hence it does not require feature tracking or correspondences. It is therefore a natural generalization of the so-called direct methods in a single-body structure from motion to multiple moving objects.

Figure 1: Segmentation of three independently moving objects

Figure 2: Segmentation of two moving robots

- [1]
- R. Vidal, Y. Ma, S. Soatto, and S. Sastry, "Two-View Multibody Structure from Motion,"
*Int. J. Computer Vision*(submitted). - [2]
- R. Vidal, S. Soatto, Y. Ma, and S. Sastry, "Segmentation of Dynamic Scenes from the Multibody Fundamental Matrix,"
*ECCV Workshop on Vision and Modeling of Dynamic Scenes,*Copenhagen, Denmark, May 2002. - [3]
- R. Vidal and S. Sastry, "Segmentation of Dynamic Scenes from Image Intensities,"
*IEEE Workshop on Vision and Motion Computing,*Orlando, FL, December 2002.

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