Swept regions and surfaces: Modeling and volumetric properties

We consider “swept regions”Ω and “swept hypersurfaces”B in R n+1 (and especially R) which are a disjoint union of subspaces Ωt = Ω∩Πt or Bt = B∩Πt obtained from a varying family of affine subspaces {Πt : t ∈ Γ}. We concentrate on the case where Ω and B are obtained from a skeletal structure (M, U). This generalizes the Blum medial axis M of a region Ω, which consists of the centers of interior spheres tangent to the boundary B at two or more points, with U denoting the vectors from the centers of the spheres to the points of tangency. We extend methods developed for skeletal structures so they can be deduced from the properties of the individual intersections Ωt or Bt and a relative shape operator Srel, which we introduce to capture changes relative to the varing family {Πt}. We use these results to deduce modeling properties of the global B in terms of the individual Bt, and determine volumetric properties of regions Ω expressed as global integrals of functions g on Ω in terms of iterated integrals over the skeletal structure of Ωt then integrated over the parameter space Γ. Introduction Let Ω ⊂ R be a region with boundary B, or let B denote a hypersurface. Considerable recent work has made use of medial representations for Ω and B for solving a variety of computer imaging problems, see e.g. the survey [P] and the book [PS]. Skeletal structures provide a generalized form of medial structure, which includes both the Blum medial axis [BN] and generalized offset (hyper)surfaces, and can be used to analyze the chordal locus models of Brady and Asada, [BA] and arcsegment medial axis of Leyton [Le]. The Blum medial axis M of a region Ω with smooth generic boundary B, consists of the locus of centers of spheres contained in Ω and tangent at two or more points (or with degenerate tangency). OnM is a multivalued vector field U from points on M to the points of tangency. If we appropriately relax the conditions required for (M,U), we still obtain a “skeletal structure”(see [D1]). These skeletal structures have been used to analyze the smoothness of B and determine the local, relative, Partially supported by DARPA grant HR0011-05-1-0057 and National Science Foundation grants DMS-0706941 and CCR-0310546. 1