Many attempts have been made to represent families of 2D shapes in a simpler way. These approaches lead to so-called structures as the Symmetry Set (\(\mathcal{SS}\)) and a subset of it, the Medial Axes (\(\mathcal{MA}\)). While the latter is commonly used, the former is still in the mathematical research stage. One reason for this is that in contrast to the \(\mathcal{SS}\), the \(\mathcal{MA}\) can be computed efficiently and fast, and yields one connected component for a closed shape.

In this paper a natural complement of the symmetry set, called the Anti-Symmetry Set (\(\mathcal{ASS}\)), is used to connect components bearing the full richness of the symmetry set. Secondly, new ways are presented to visualize these sets. One uses the radius of the describing circle as extra dimension, the other, the so-called pre-Symmetry Set (pre-\(\mathcal{SS}\)), uses the parameter space. Example shapes show the extra information carried in the \(\mathcal{ASS}\) and the pre-\(\mathcal{SS}\) in determining the special points on the \(\mathcal{SS}\) as well as revealing the structure of the \(\mathcal{SS}\) in more detail. They are also capable of distinguishing between different shapes where the \(\mathcal{SS}\) and the \(\mathcal{MA}\) in some cases fail.


Extra Dimension Local Extremum Medial Axis Radius Function Circle Tangent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Arjan Kuijper
    • 1
  • Ole Fogh Olsen
    • 1
  1. 1.Image groupIT-University of CopenhagenCopenhagenDenmark

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