Conjugate gradient on Grassmann manifolds for robust subspace estimation

a r t i c l e i n f o Most geometric computer vision problems involve orthogonality constraints. An important subclass of these problems is subspace estimation, which can be equivalently formulated into an optimization problem on Grassmann manifolds. In this paper, we propose to use the conjugate gradient algorithm on Grassmann man-ifolds for robust subspace estimation in conjunction with the recently introduced generalized projection based M-Estimator (gpbM). The gpbM method is an elemental subset-based robust estimation algorithm that can process heteroscedastic data without any user intervention. We show that by optimizing the orthogonal parameter matrix on Grassmann manifolds, the performance of the gpbM algorithm improves significantly. Results on synthetic and real data are presented. Orthogonality constraints arise frequently in geometric computer vision problems. Linear subspace estimation naturally falls into this category. Orthogonal matrices representing linear subspaces of Euclidean space can be represented as points on Grassmann manifolds. Studying the geometric properties of Grassmann manifolds can therefore prove very useful in solving many vision problems. In the recent past, the problem of subspace estimation has been formulated in many different ways. Important methods include robust regression based approaches [34, 22], spectral clustering based approaches [6, 41] and clustering random hypothesis on Grassmann manifolds [35, 2]. Robust regression has been an active field of research in computer vision. It corresponds to estimating multiple, noisy inlier structures present in the data corrupted with gross outliers. Following RAndom SAmple Consensus (RANSAC) [7], many algorithms like MLESAC, LO-RANSAC, PROSAC, QDEGSAC, have been proposed. See [27] for a brief description of these methods. Recently, the projection based M-estimator (pbM) of [34] was extended to the generalized pbM (gpbM) [22]. The main advantage of pbM and gpbM over RAN-SAC and RANSAC-like regression algorithms is that both pbM and gpbM do not require from the user an estimate of the scale of the noise corrupting inlier points. While pbM uses a MAD-based scale estimate that is dependent on the choice of a particular hypothesis, gpbM estimates the true scale of the noise beforehand. In this paper, we extend the work of [22] to robustly estimate sub-spaces by using concepts of Riemannian geometry. While the main idea of work follows that of [34], it is different from [34] in a number of ways. Like [34], we also refine the estimate of the subspace obtained from the gpbM algorithm by using conjugate gradient on Grassmann manifolds. But, in …