# Topology driven algorithms for ridge extraction on meshes

1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Given a smooth surface, a ridge is a curve along which one of the principal curvatures has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. Surprisingly, no method developed so far to report ridges from a mesh approximating a smooth surface comes with a careful analysis, which entails that one does not know whether the ridges are reported in a coherent fashion. To bridge this gap, we make the following contributions. First, we present a careful analysis of the orientation issues arising when one wishes to report the ridges associated to the two principal curvatures separately. The analysis highlights the subtle interplay between ridges, umbilics, and curvature lines. Second, given a triangulation $T$ approximating a smooth generic surface $S$, we present sufficient conditions on $T$ together with a certified algorithm reporting ridges in a topologically coherent fashion. Third, we develop a heuristic algorithm to process a mesh when no information on an underlying smooth surface is known. Fourth, for coarse models, we provide a filtering mechanism retaining the most stable ridges only. Fifth, we present experimental results of the heuristic algorithm for smooth surfaces as well as coarse models. Our running times improve of at least one order of magnitude state-of-the-art methods. The common point of these contributions is to exploit the topological patterns of ridges on smooth generic surfaces. These contributions also pave the way to the first certified algorithm for ridge extraction on polynomial parametric surfaces, developed in a companion paper.
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Cited literature [28 references]

https://hal.inria.fr/inria-00070481
Contributor : Rapport de Recherche Inria <>
Submitted on : Friday, May 19, 2006 - 8:37:10 PM
Last modification on : Saturday, January 27, 2018 - 1:31:30 AM
Long-term archiving on: : Sunday, April 4, 2010 - 9:19:03 PM

### Identifiers

• HAL Id : inria-00070481, version 1

### Citation

Frédéric Cazals, Marc Pouget. Topology driven algorithms for ridge extraction on meshes. [Research Report] RR-5526, INRIA. 2005, pp.29. ⟨inria-00070481⟩

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