Robust Estimation for Computer Vision using Grassmann Manifolds Saket Anand 1 , Sushil Mittal 2 and Peter Meer 3 Abstract Real-world visual data is often corrupted and requires 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 mathemati- cal 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 ker- nel 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. 1 Introduction Estimation problems in geometric computer vision often require dealing with orthogonality constraints in the form of linear subspaces. Since orthogonal matrices representing linear subspaces of Euclidean space can be represented as points on Grassmann manifolds, understanding the geometric properties of these manifolds can prove very useful for solving many vision problems. Usually, the estimation process involves optimizing an objective function to find the regression coefficients that best describe the underlying constraints. Alternatively, given a distribution of sampled hypotheses of linear solutions, it could also be formulated as finding the cluster centers of those distributions as the dominant solutions to the underlying observations. A typical regression problem in computer vision involves discovering mul- tiple, noisy inlier structures present in the data corrupted with gross out- liers. Usually, very little or no information is available about the number of inlier structures, the nature of the noise corrupting each one of them and the amount of gross outliers. The original RAndom SAmple Cons- esus (RANSAC) [5] and its several variants like MLESAC, LO-RANSAC, 1. IIIT-Delhi, New Delhi, India. e-mail: anands@iiitd.ac.in 2. Scibler Corporation, Santa Clara, CA, USA. e-mail: mittal@scibler.com 3. Dept. of ECE, Rutgers University, NJ, USA. e-mail: meer@cronos.rutgers.edu 1