Subspace segmentation with outliers: A grassmannian approach to the maximum consensus subspace

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.