Riemannian geometry of Grassmann manifolds with a view on algorithmic computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in ℝn. In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.

A Grassman-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh Quotient Iteration (RQI) computes a 1-dimensional invariant subspace of a symmetric matrix A with cubic convergence. We propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. The geometry of the algorithm on the Grassmann manifold Gr(p,n) is developed to show cubic convergence and to draw connections with recently proposed Newton algorithms on Riemannian manifolds.

Coarse-to-fine vision-based localization by indexing scale-invariant features

This paper presents a novel coarse-to-fine global localization approach inspired by object recognition and text retrieval techniques. Harris-Laplace interest points characterized by scale-invariant transformation feature descriptors are used as natural landmarks. They are indexed into two databases: a location vector space model (LVSM) and a location database. The localization process consists of two stages: coarse localization and fine localization. Coarse localization from the LVSM is fast, but not accurate enough, whereas localization from the location database using a voting algorithm is relatively slow, but more accurate. The integration of coarse and fine stages makes fast and reliable localization possible. If necessary, the localization result can be verified by epipolar geometry between the representative view in the database and the view to be localized. In addition, the localization system recovers the position of the camera by essential matrix decomposition. The localization system has been tested in indoor and outdoor environments. The results show that our approach is efficient and reliable. © 2006 IEEE.

Image-based localization and pose recovery using scale invariant features

In this paper, we propose a vision based mobile robot localization strategy. Local scale-invariant features are used as natural landmarks in unstructured and unmodified environment. The local characteristics of the features we use prove to be robust to occlusion and outliers. In addition, the invariance of the features to viewpoint change makes them suitable landmarks for mobile robot localization. Scale-invariant features detected in the first exploration are indexed into a location database. Indexing and voting allow efficient recognition of global localization. The localization result is verified by epipolar geometry between the representative view in database and the view to be localized, thus the probability of false localization will be decreased. The localization system can recover the pose of the camera mounted on the robot by essential matrix decomposition. Then the position of the robot can be computed easily. Both calibrated and un-calibrated cases are discussed and relative position estimation based on calibrated camera turns out to be the better choice. Experimental results show that our approach is effective and reliable in the case of illumination changes, similarity transformations and extraneous features. © 2004 IEEE.