Geometric and Topological Inference in the Nonlinear Realm: On the Importance of Singularity Theory

Friday, June 1, 2007 - 1:30pm - 2:00pm
EE/CS 3-180
Frédéric Cazals (Institut National de Recherche en Informatique Automatique (INRIA))
In the non linear realm, a powerful way to describe complex shapes is
in terms of stratifications, that is topological disks together with
the proper incidences, so as to define some kind of complex (cell, CW,
etc). In particular, singularity theory provides a convenient way to
specify such incidences through local normal forms, the parametric
form of the topological disks depending on the particular problem
investigated. We shall review two examples illustrating this
framework in 3D : the detection of loci of extremal curvature of
smooth surfaces (the so-called ridges), and the calculation of the
Morse-Smale complex of the distance function to points in 3D space
(the so-called flow complex). We shall conclude with perspectives
arising in trying to generalize such approaches in higher dimension.