Ridges and umbilics of a sampled smooth surface: a complete picture gearing toward topological coherence

Consider a smooth surface, and at each point which is not an umbilic, respectively paint in blue (red) anything related to the maximum (minimum) principal curvature. Given such a surface, a blue (red) ridge is a curve on the surface such that at each of its points, the principal blue (red) curvature has an extremum along its blue (red) 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. This paper aims at bridging this gap with the following contributions. First, a careful analysis of the Acute rule - an orientation procedure used in most algorithms - is presented. 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 an algorithm and a filtering procedure aiming at reporting the most salient features of a coarse mesh $T$.

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