Journal of Mathematical Imaging and Vision

, Volume 26, Issue 1, pp 127–147

Alternative 2D Shape Representations using the Symmetry Set

  • Arjan Kuijper
  • Ole Fogh Olsen
  • Peter Giblin
  • Mads Nielsen
Article

DOI: 10.1007/s10851-006-8372-2

Cite this article as:
Kuijper, A., Olsen, O.F., Giblin, P. et al. J Math Imaging Vis (2006) 26: 127. doi:10.1007/s10851-006-8372-2

Abstract

Among the many attempts made to represent families of 2D shapes in a simpler way, the Medial Axis \(\mathcal{MA}\) takes a prominent place. Its graphical representation is intuitively appealing and can be computed efficiently. Small perturbations of the shape can have large impact on the \(\mathcal{MA}\) and are regarded as instabilities, although these changes are mathematically known from the investigations on a super set, the Symmetry Set \(\mathcal{SS}\). This set has mainly been in a mathematical research stage, partially due to computational aspects, and partially due to its unattractive representation in the plane.

In this paper novel methods are introduced to overcome both aspects. As a result, it is possible to represent the \(\mathcal{SS}\) as a string is presented. The advantage of such a structure is that it allows fast and simple query algorithms for comparisons.

Second, alternative ways to visualize the \(\mathcal{SS}\) are presented. They use the distances from the shape to the set as extra dimension as well as the so-called pre-Symmetry Set and anti-Symmetry Set. Information revealed by these representations can be used to calculate the linear string representation structure.

Example shapes from a data base are shown and their data structures derived.

Keywords

shapes symmetry set pre-symmetry set anti-symmetry set geometry skeletons medial axis 

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Arjan Kuijper
    • 1
  • Ole Fogh Olsen
    • 2
  • Peter Giblin
    • 3
  • Mads Nielsen
    • 2
  1. 1.Radon Institute for Computational and Applied MathematicsLinzAustria
  2. 2.IT University of CopenhagenDenmark
  3. 3.The University of LiverpoolUnited Kingdom