Last update: Oct. 9, 20001

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Reading material :

• Local and Global Geometric Methods for Analysis Interrogation, Reconstruction, Modification and Design of Shape, 2000.
  • Authors: F. - E. Wolter and K. - I. Friese
  • Proceedings of Computer Graphics International 2000, Geneva, Switzerland, IEEE Computer Society, June 2000. Invited paper.
  • PDF

This talk gives an overview on some recent methods useful for local and global shape analysis and for the design of Solids. These methods include as new tools for global and local Shape analysis the Spectra of the Laplace and the Laplace Beltrami Operator and the Concept of stable Umbilical Points, i.e., stable singularities of the principal curvature line wireframe model of the solid's boundary surface. Most material in this paper deals with the Medial Axis transform as a tool for shape interrogation, reconstruction, modification and design. We show that it appears to be possible to construct an intuitive user interface that allows to mould shape employing the Medial Axis Transform.

In the past decade efforts of our lab could show for the first time that the Medial Axis and Voronoi diagrams can be defined and computed efficiently and precisely also on Free Form surfaces in a setting where the geodesic distance between two points p, q on a surface S is defined by the shortest surface path on S joining the two points p, q. This leads to the natural and computable generalized concepts of geodesic Medial Axis and geodesic Voronoi diagram on free form surfaces. Both can be computed with a reasonable speed and with a high accuracy.

Other results and papers available under :

• http://www.lems.brown.edu/vision/Presentations/Wolter/

Additional reading material :

This paper describes a method to extract the generic features of free-form parametric surfaces for shape interrogation. The umbilical points, which are the singular points of the orthogonal net of lines of curvature, have generic features and may act like fingerprints for shape recognition. We investigate the generic features of the umbilics and behavior of lines of curvature which pass through an umbilic on a parametric free-form surface. Our method is based on a coordinate transformation to set the parametric surface in Monge form and on a Taylor expansion to compute the angles of the tangent lines to the lines of curvatures at an umbilic. We also develop a novel and practical criterion which assures the existence of local extrema of principal curvature functions at umbilical points. Finally, numerical experiments illustrate how the generic features of the umbilics can be applied for surface recognition.

Author Keywords: Umbilics; Lines of curvature; Monge form; Shape recognition; Perturbation; CAGD; CAD; Vision

Index Terms: Computational geometry; Pattern recognition; Mathematical transformations; Computational methods; Functions; Perturbation techniques; Computer aided design; Computer vision; Umbilical points; Lines of curvature; Shape interrogation; Monge form; Taylor expansion; Principal curvature functions; Surface recognition

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Last Updated: Oct. 9, 2001