A Bose type formula for the internal medial axis of an embedded manifold
The authors are interested to give an algebraic topologic characterization of the so-called \textit{medial axis}. This has been introduced by Blum for image analysis, and it is a compact representation of the shape of a finite dimensional manifold, that should filter the same ``structural properties'' of such a shape. Nowadays it is recognized that the notion of medial axis is an important geometric concept in computational geometry. In particular, the authors generalize to \(n\)-dimensional manifolds, \(n>2\), the \textit{Bose formula}, just obtained for closed plane curves: \(e-y=2\), where \(e\), (resp. \(y\)), denotes the number of end points, (resp. triple points), on the medial axis. The main results are a generalization of the Bose formula to the case \(n=3\), (Theorem 1.2), and, in the cases \(n=4,5,6\), to some explicit relationships between Euler characteristics of manifolds and the ones for their internal medial axes. The paper after an introduction, where Theorem 1.2 is formulated, splits in to five more sections. 2. An elementary proof of Theorem 1.2. 3. Some preliminaries. 4. A relationship between the Euler characteristic of a generic manifold and its internal medial axis. 5. Relation with work of Sedykh. 6. Some applications.
scientific article; zbMATH DE number 5544189
scientific article