Robust statistics over riemannian manifolds for computer vision

Abstract

The nonlinear nature of many compute vision tasks involves analysis over curved nonlinear spaces embedded in higher dimensional Euclidean spaces. Such spaces are known as manifolds and can be studied using the theory of differential geometry. In this thesis we develop two algorithms which can be applied over manifolds. The nonlinear mean shift algorithm is a generalization of the original mean shift, a popular feature space analysis method for vector spaces. Nonlinear mean shift can be applied to any Riemannian manifold and is provably convergent to the local maxima of an appropriate kernel density. This algorithm is used for motion segmentation with different motion models and for the filtering of complex image data. The projection based M-estimator is a robust regression algorithm which does not require a user supplied estimate of the scale, the level of noise corrupting the inliers. We build on the connections between kernel density estimation and robust M-estimators and develop data driven rules for scale estimation. The method can be generalized to handle heteroscedastic data and subspace estimation. The results of using pbM for affine motion estimation, fundamental matrix estimation and multibody factorization are presented. A new sensor fusion method which can handle heteroscedastic data and incomplete estimates of parameters is also discussed. The method is used to combine image based pose estimates with inertial sensors.