Geometry and Medial Structure
This chapter explains methods for determining the geometric properties of a region (or object) Ω and its boundary ℬ in terms of the “radial geometry” on its medial axis M, defined using the multivalued “radial vector field” V (denoted by S in other chapters) formed from spokes from points on M to points of tangency with spheres on ℬ. The radial geometry is captured by “radial and edge shape operators” which measure how the radial vector field V varies on M. In fact, the radial geometry is defined for more general skeletal structures (M, V), which allow the Blum conditions to be relaxed.
We explain how one can use these operators to compute both the local and relative geometry of ℬ. As global properties of Ω and ℬ are represented by global integrals on these regions, we further explain how to use the radial shape operator to express such integrals as integrals over the medial axis with respect to an intrinsic medial measure on M.
Finally we explain how the global structure of the medial axis is related to the global structure of the region. We concentrate on contractible regions (without holes or cavities), giving a characterization of them in terms of the medial axis structure.
KeywordsPrincipal Curvature Multivalued Function Medial Axis Edge Point Shape Operator
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