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Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness. (English) Zbl 1047.57014

The notion of a skeletal structure \((M,U)\) in \(\mathbb R^{n+1}\) is introduced. Here, \(M\) is a special type of Whitney stratified subset of \(\mathbb R^{n+1}\) and \(U\) is a special type of multivalued radial vector field on \(M\). Such a structure has an associated boundary \(\mathcal B\) and the main result gives necessary and sufficient conditions (in terms of shape operators) for \(\mathcal B\) to be smooth. This is accomplished via a detailed study of radial flows and tubular neighborhoods appropriate in the current context. This paper is related to the analysis of shapes in computer imaging and vision. In fact, the main example of a skeletal structure offered is the Blum medial axis of an object in \(\mathbb R^{n+1}\).

MSC:

57N80 Stratifications in topological manifolds
58A35 Stratified sets
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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