James Damon

Multiscale Medial Loci and Their Properties

Blums medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasi-hierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has led to some definitions of multiscale medial axes different from the Blum medial axis that considerably overcame the weakness. Three major multiscale medial axes have been proposed: iteratively pruned trees of Voronoi edges (Ogniewicz, 1993; Székely, 1996; Näf, 1996), shock loci of reaction-diffusion equations (Kimia et al., 1995; Siddiqi and Kimia, 1996), and height ridges of medialness (cores) (Fritsch et al., 1994; Morse et al., 1993; Pizer et al., 1998). These are different from the Blum medial axis, and each has different mathematical properties of generic branching and ending properties, singular transitions, and geometry of implied boundary, and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries. These mathematical properties and computational abilities are laid out and compared and contrasted in this paper.

Flux invariants for shape

We consider the average outward flux through a Jordan curve of the gradient vector field of the Euclidean distance function to the boundary of a 2D shape. Using an alternate form of the divergence theorem, we show that in the limit as the area of the region enclosed by such a curve shrinks to zero, this measure has very different behaviors at medial points than at non-medial ones, providing a theoretical justification for its use in the Hamilton-Jacobi skeletonization algorithm of Siddiqi et al. (2002). We then specialize to the case of shrinking circular neighborhoods and show that the average outward flux measure also reveals the object angle at skeletal points. Hence, formulae for obtaining the boundary curves, their curvatures, and other geometric quantities of interest, can be written in terms of the average outward flux limit values at skeletal points. Thus this measure can be viewed as a Euclidean invariant for shape description: it can be used to both detect the skeleton from the Euclidean distance function, as well as to explicitly reconstruct the boundary from it. We illustrate our results with several numerical simulations.

Determining the Geometry of Boundaries of Objects from Medial Data

We consider a region Ω in R2 or R3 with generic smooth boundary B and Blum medial axis M, on which is defined a multivalued “radial vector field” U from points x on M to the points of tangency of the sphere at x with B. We introduce a “radial shape operator” Srad and an “edge shape operator” SE which measure how U bends along M. These are not traditional differential geometric shape operators, nonetheless we derive all local differential geometric invariants of B from these operators.This allows us to define from (M, U) a “geometric medial map” on M which corresponds, via a “radial map” from M to B, to the differential geometric properties of B. The geometric medial map also includes a description of the relative geometry of B. This is defined using the “relative critical set” of the radius function r on M. This set consists of a network of curves on M which describe where B is thickest and thinnest. It is computed using the covariant derivative of the tangential component of the unit radial vector field.We further determine how these invariants are related to the differential geometric invariants of M and how these invariants change under deforming diffeomorphisms of M.

\(A\) and the vanishing topology of discriminants

SummarySuppose thatf: ℂn, 0→ℂp, 0 is finitely

Properties of Ridges and Cores for Two-Dimensional Images

A

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

-determined withn≧p. We define a “Milnor fiber” for the discriminant off; it is the discriminant of a “stabilization” off. We prove that this “discriminant Milnor fiber” has the homotopy type of a wedge of spheres of dimensionp−1, whose number we denote byµΔ(f). One of the main theorems of the paper is a “μ=τ” type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then

Nested Sphere Statistics of Skeletal Models

\mu _\Delta (f) \geqq A_e