In this thesis, we consider the problem of fast and efficient indexing techniques for time sequences which evolve on manifold-valued spaces. Using manifolds is a convenient way to work with complex features that often do not live in Euclidean spaces. However, computing standard notions of geodesic distance, mean etc. can get very involved due to the underlying non-linearity associated with the ...

The objective of this work is to show how geodesic propagation techniques known in mathematical morphology can be employed to reconstruct terrain surfaces, and more generally grid data, from contour maps. We present the so-called generalised geodesic distance that was briefly introduced in a previous article. The shortest paths defined by the generalised geodesic distance can be used to linearl...

Let P be a simple polygon of n vertices and let S be a set of n points called sites lying in the interior of P. The geodesic-distance between two sites x and y in P is defined as the length of the shortest polygonal path connecting x and y constrained to lie in P. It is a useful notion in graphics problems concerning visibility between objects, computer vision problems concerned with the descri...

We obtain an optimal deviation from the mean upper bound D(x) def = sup f∈F μ{f −Eμf ≥ x}, for x ∈ R (0.1) where F is the complete class of integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of (0.1) for Euclidean unit sphere Sn−1 with a geodesic distance function and a normalized Haar measure, for R equipped with a Gaussian measure and...

The purpose of this paper is to introduce a form of update based on the minimization of the geodesic distance on a graph. We provide a characterization of this class using settheoretic operators and show that such operators bijectively correspond to geodesic metrics. As distance is generated by distinguishability, our framework is appropriate in contexts where distance is generated by threshold...

Finding shortest paths and shortest distances between points on a surface S in three-dimensional space is a well-studied problem in differential geometry and computational geometry. The shortest path between two points on S is denoted a geodesic path on the surface and the shortest distance between two points on S is denoted a geodesic distance. In this survey, we consider the case where a disc...

Nonlinear diffusion processes can be found in many recent methods for image processing and computer vision. In this article, four applications are surveyed: nonlinear diffusion filtering, variational image regularization, optic flow estimation, and geodesic active contours. For each of these techniques we explain the main ideas, discuss theoretical properties and present an appropriate numerica...

Social Network Analysis has been historically applied to single networks, e.g., interaction networks between co-workers. However, the advent of on-line social network sites has emphasized the stratified structure of our social experience. Individuals usually spread their identities over multiple services, e.g., Facebook, Twitter, LinkedIn and Foursquare. As a result, the analysis of on-line soc...