Robust estimation for computer vision using grassmann manifolds

Real-world visual data are often corrupted and require the use of estimation techniques that are robust to noise and outliers. Robust methods are well studied for Euclidean spaces and their use has also been extended to Riemannian spaces. In this chapter, we present the necessary mathematical constructs for Grassmann manifolds, followed by two different algorithms that can perform robust estimation on them. In the first one, we describe a nonlinear mean shift algorithm for finding modes of the underlying kernel density estimate (KDE). In the second one, a user-independent robust regression algorithm, the generalized projection-based M-estimator (gpbM), is detailed. We show that the gpbM estimates are significantly improved if KDE optimization over the Grassmann manifold is also included. The results for a few real-world computer vision problems are shown to demonstrate the importance of performing robust estimation using Grassmann manifolds.