Feb. 8, 2004

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The European Singularities Network.

Publications by Peter J. Giblin, J.W. Bruce et al. :

• Mathematics of Geometric Singularities:
  • Folding Maps & Focal Sets, 1991.
  • Local Reflexional & Rotational Symmetry in the Plane, 1991.
  • Projections of Surfaces with Boundary, 1990.
  • Geometry of Singular Sets, 1989.
  • Global Theorems for Symmetry Sets of Smooth Curves & Polygons in the Plane, 1987.
  • Reconstruction of Surfaces From Profiles, 1987.
  • Growth, Motion & 1-Parameter Families of Symmetry Sets, 1986.
  • Symmetry Sets, 1985.
  • Outlines and their Duals, 1985.
  • Local Symmetry of Plane Curves, 1985.
  • Projections & Reflections of Generic Surfaces in R³, 1984.
  • Seeing - The Mathematical Viewpoint, 1984.
  • Caustics Through the Looking Glass, 1984.
  • Curves & Singularities: A Geometrical Introduction to Singularity Theory, 1984 & 1992.
  • Generic Isotopies of Plane Curves, 1983.
  • Generic Geometry, 1983.
  • Wavefronts & Parallels in Euclidean Space, 1983.
  • Graphs, Surfaces, & Homology: An Introduction to Algebraic Topology, 1981.

• Computational approaches to the extraction & characterisation of Geometric Singularities:
  • A Formal Classification of 3D Medial Axis Points and Their Local Geometry, 2004.
  • On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks, 2003.
  • Transitions of the 3D Medial Axis Under a One-Parameter Family of Deformations, 2002.
  • On the Local Form of Symmetry Sets, and Medial Axes, and Shocks in 3D, 2000.
  • On the Local Form and Transitions of Symmetry Sets, and Medial Axes, and Shocks in 2D, 1999.
  • On the Intrinsic Reconstruction of Shape from its Symmetries, 1999.
  • Two- & Three-Dimensional Patterns of the Face, 1999.
  • Ridges, Crests & Sub-Parabolic Lines of Evolving Surfaces, 1996.
  • Skeletonization using an Extended Euclidean Distance Transform, 1995.
  • Local Symmetry in the Plane: Experiment & Theory, 1989.

BibTeX references.

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, generically, the medial axis consists of five types of points, which are then organized into sheets, curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A_1^2 (A_k^n notation means n distinct k{\hbox{-}}{\rm{fold}} tangencies of the sphere of contact, as explained in the text); two types of curves, 2) the intersection curve of three sheets and the locus of centers of tritangent spheres, A_1^3, and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A_3 points; and two types of points, 4) centers of quad-tangent spheres, A_1^4, and 5) centers of spheres with one regular tangency and one higher order tangency, A_1A_3. The geometry of the 3D medial axis thus consists of sheets (A_1^2) bounded by one type of curve (A_3) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curve (A_1^3), which support a generalized cylinder description. The A_3 curves can only end in A_1A_3 points where they must meet an A_1^3 curve. The A_1^3 curves meet together in fours at an A_1^4 point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A_1^4 and A_1 A_3 points), links between pairs of nodes (A_1^3 and A_3 curves) and hyperlinks between groups of links (A_1^2 sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and manipulation of shapes.

Index Terms- 3D medial axis, skeleton, shocks, curve skeleton, order of contact, local form, medial topology, ridges, generalized axis.

In this paper we explore the local geometry of the medial axis (MA) and shocks (SH), and their structural changes under deformations, by viewing these symmetries as subsets of the symmetry set (SS) and present two results. First, we establish that the local form of the medial axis must generically be one of three cases, which we denote by the A notation explained below (here, it merely serves as a reference to sections of the paper): endpoints (A3), interior points (A12), and junctions (A13). The local form of shocks is then derived from a sub-classification of these points into six types. Second, we address the (classical) instabilities of the MA, i.e., abrupt changes in the representation arising from slight changes in shape, as when a new branch appears with slight protrusion. The identification of these `transitions' is clearly crucial in robust object recognition. We show that for the medial axis only two such instabilities are generically possible: (i) when four branches come together (A14), and (ii) when a new branch grows out of an existing one (A1A3). Similarly, there are six cases of shock instabilities, derived as sub-classifications of the MA instabilities. We give an explicit example of a dent forming in an ellipse where many of the transitions described in the paper can be seen to appear.

Keywords: symmetry set, medial axis, shock

A 3-parameters family F of folding maps conjugated by Euclidean motions has a bifurcation set which is the DUAL of the union of the focal and symmetry sets. The latter sets are also bifurcation sets of the family D of distance squared functions; the focal set being the catastrophe set of D, while the symmetry set is its Maxwell set. The family F picks up geometric features of a surface (X in E³ ) undected by D, and blows up the subtle geometry at umbilics.

Computation of local symmetries of planar curves on the basis of a smooth parametrization (tangent defined at all points). In the piecewise-linear case a definition of tangency between a circle and a polygonal vertex is required. Inverse stereographic projection of a plane curve Y to a space curve Y' on a unit sphere: multiple-contact circles to Y lift to circles on the sphere which are cut out by multiple-contact planes for Y'.

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