Perceiving dynamic scenes of rigid bodies, through affine projections of moving 3D point clouds, boils down to clustering the rigid motion subspaces supported by the points' image trajectories. For a physically meaningful interpretation, clusters must be consistent with the geometry of the underlying subspaces. Most of the existing measures for subspace clustering are ambiguous, or geometrically inconsistent. A practical consequence is that methods based on such (dis)similarities are unstable when the number of rigid bodies increase. This paper introduces the Normalized Subspace Inclusion (NSI) criterion to resolve these issues. Relying on this similarity, we propose a robust methodology for rigid motion segmentation, and test it, extensively, on the Hopkins155 database. The geometric consistency of the NSI assures the method's accuracy when the number of rigid bodies increases, while robustness proves to be suitable for dealing with challenging imaging conditions.

Segmenting arbitrary unions of linear subspaces is an important tool for computer vision tasks such as motion and image segmentation, SfM or object recognition. We segment subspaces by searching for the orthogonal complement of the subspace supported by the majority of the observations, i.e., the maximum consensus subspace. It is formulated as a Grassmannian optimization problem: a smooth, constrained but nonconvex program is immersed into the Grassmann manifold, resulting in a low dimensional and unconstrained program solved with an efficient optimization algorithm. Nonconvexity implies that global optimality depends on the initialization. However, by finding the maximum consensus subspace, outlier rejection becomes an inherent property of the method. Besides robustness, it does not rely on prior global detection procedures (e.g., rank of data matrices), which is the case of most current works. We test our algorithm in both synthetic and real data, where no outlier was ever classified as inlier.