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On the Local Form and Transitions of Pre-symmetry Sets

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Abstract

Symmetry is an important cue in shape analysis. It has lead to the definition of popular shape descriptors like the medial axis. Its properties have been analyzed with a superset, called the symmetry set which represents the midpoints of circles that are at least bitangent to a shape.

In this work we investigate the pre-symmetry set. This set considers the pairs of points at which the bitangent contact occurs. One thus obtains pairwise symmetric points of a 2D shape. A closed 2D shape has a parameterization P with finite length. Its pre-symmetry can therefore be represented by a symmetric diagram curves formed from the pairs of points (p i ,p j )∈S 1×S 1.

We discuss the properties of the pre-symmetry set visualized by this diagram. We firstly give the so-called transitions, changes caused by a perturbation of the shape and show the changes of the curves in the pre-symmetry set diagram. Secondly, we investigate curves that are spanned by all points on the shape. We name the curves essential loops and discuss their properties and transitions. As one important result we show that their are either zero or two essential loops. In the latter case a part of the medial axis is spanned by an essential loop and can therefore be considered as the main axis of the medial axis.

As application of pre-symmetry sets, we discuss two possibilities for shape matching based on representations of the pre-symmetry set. The first shape descriptor we present is given by a circular diagram representing the shape, with a set of points representing the extrema of the curvature in the order they appear on the shape. They are pairwise connected and endowed with a length measure. This descriptor is directly related to the curves and their lengths in the pre-symmetry set diagram. The second descriptor is given by a binary array representing the areas enclosed by the curves in the pre-symmetry set diagram. It is area based and relates to the geometric derivation of the symmetry set.

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Notes

  1. I would like to thank Peter Giblin from Liverpool University for providing this example, see also [40].

  2. http://www.lems.brown.edu/vision/researchAreas/SIID/.

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Acknowledgements

This work was partially supported by the Deep Structure, Singularities, and Computer Vision (DSSCV) project, an IST Programme of the European Union (IST-2001-35443), and carried out by the author while he was at the IT University of Copenhagen, Denmark. Parts of this work were presented at several conferences [19, 20, 22, 24]. I would like to thank my colleagues Peter Giblin at the University of Liverpool, UK, and Ole Fogh Olsen, Mads Nielsen, and Philip Bille at the (IT) University of Copenhagen, Denmark, for the stimulating discussions leading to those publications which formed the starting point for this paper, and the reviewers for suggestions to improve the readability of this paper.

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  1. Fraunhofer Institute for Computer Graphics Research IGD & Interactive Graphics Systems, CS, TU Darmstadt, Fraunhoferstrasse 5, 64283, Darmstadt, Germany

    Arjan Kuijper

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Kuijper, A. On the Local Form and Transitions of Pre-symmetry Sets. J Math Imaging Vis 45, 13–30 (2013). https://doi.org/10.1007/s10851-012-0341-3

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