Analytic manifolds were recently used for motion averaging, segmentation and robust estimation. Here we consider the epipolar constraint for calibrated cameras, which is the most general motion model for calibrated cameras and is encoded by the essential matrix. The set of all essential matrices forms the essential manifold. We provide a theoretical characterization of the geometry of the essential manifold and develop a parametrization which associates each essential matrix with a unique point on the manifold. Our work provides a more complete theoretical analysis of the essential manifold than previous work in this direction. We show the results of using this parametrization with real data sets, while previous work concentrated on theoretical analysis with synthetic data.

Discontinuity preserving filtering of images is an important low-level vision task. With the development of new imaging techniques like diffusion tensor imaging (DTI), where the data does not lie in a vector space, previous methods like the original mean shift are not applicable. In this paper, we use the nonlinear mean shift algorithm to develop filtering methods for data lying on analytic manifolds. We work out the computational details of using mean shift on Symn+, the manifold of n times n symmetric positive definite matrices. We apply our algorithm to chromatic noise filtering, which requires mean shift over the Grassmann manifold G3,1, and obtain better results then standard mean shift filtering. We also use our method for DTI filtering, which requires smoothing over Sym3+.

We propose a new robust estimator for camera pose estimation based on a recently developed nonlinear mean shift algorithm. This allows us to treat pose estimation as a clustering problem in the presence of outliers. We compare our method to RANSAC, which is the standard robust estimator for computer vision problems. We also show that under fairly general assumptions our method is provably better than RANSAC. Synthetic and real examples to support our claims are provided

The mean shift algorithm is widely applied for nonparametric clustering in Euclidean spaces. Recently, mean shift was generalized for clustering on matrix Lie groups. We further extend the algorithm to a more general class of nonlinear spaces, the set of analytic manifolds. As examples, two specific classes of frequently occurring parameter spaces, Grassmann manifolds and Lie groups, are considered. When the algorithm proposed here is restricted to matrix Lie groups the previously proposed method is obtained. The algorithm is applied to a variety of robust motion segmentation problems and multibody factorization. The motion segmentation method is robust to outliers, does not require any prior specification of the number of independent motions and simultaneously estimates all the motions present.

We propose a new method to estimate multiple rigid motions from noisy 3D point correspondences in the presence of outliers. The method does not require prior specification of number of motion groups and estimates all the motion parameters simultaneously. We start with generating samples from the rigid motion distribution. The motion parameters are then estimated via mode finding operations on the sampled distribution. Since rigid motions do not lie on a vector space, classical statistical methods can not be used for mode finding. We develop a mean shift algorithm which estimates modes of the sampled distribution using the Lie group structure of the rigid motions. We also show that proposed mean shift algorithm is general and can be applied to any distribution having a matrix Lie group structure. Experimental results on synthetic and real image data demonstrate the superior performance of the algorithm.

Estimation of camera pose is an integral part of augmented reality systems. Vision-based methods offer a flexible and accurate method for this estimation. Current vision based methods rely on markers to reduce the computation and increase robustness of the pose estimation. However, this limits the algorithm's applicability while being expensive since the markers also require maintenance. Alternatively, reconstructed scene features can be used for pose estimation but this can lead to a loss of accuracy. To avoid this we propose a two-stage balanced tracking method which does not require any visual markers in the scene. The first stage of our method is based on the sequential recovery of structure from motion which allows the system to learn the scene from a few frames in which the markers are visible. In the next stage, the learned features are used for camera tracking. The system ensures greater accuracy and reduces error drift due to its use of the HEIV estimator which is provably unbiased to the first degree. We also make use of a novel method for the detection and removal of outliers which are unavoidable in such systems. The experiments show the superiority of our method when compared to a nonlinear method based on Levenberg-Marquardt minimization.