Geometry, Imaging and Computing

Volume 2 (2015)

Number 4

Metric curvatures and their applications I

Pages: 257 – 334

DOI: http://dx.doi.org/10.4310/GIC.2015.v2.n4.a2

Author

Emil Saucan (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; and Department of Electrical Engineering, Technion City, Haifa, Israel)

Abstract

We present, in a natural, developmental manner, the main types of metric curvatures and investigate their relationship with the notions of Hausdorff and Gromov–Hausdorff distances, which by now have been widely adopted in the fields of Imaging, Vision and Graphics. In addition we present a number of applications to the fields above, as well as to Communication Networks and Regge Calculus. Further connections with such established notions as excess and fatness are also investigated. While the present paper represents essentially a survey, a number of possible applications are presented here for the first time, for instance to a numerically feasible quantification of a notion of quasi-flatness of manifolds, with applications in Imaging; and to the introduction of curvatures, in particular the Lipschitz–Killing curvature measures, for almost Riemannian manifolds, with a view to their usage in Regge Calculus, as well as in Graphics. Further directions of study are also suggested.

Full Text (PDF format)