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Alternative 2D Shape Representations using the Symmetry Set

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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.

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Author information

Authors and Affiliations

  1. Radon Institute for Computational and Applied Mathematics, Linz, Austria

    Arjan Kuijper

  2. IT University of Copenhagen, Denmark

    Ole Fogh Olsen & Mads Nielsen

  3. The University of Liverpool, United Kingdom

    Peter Giblin

Authors
  1. Arjan Kuijper
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  2. Ole Fogh Olsen
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  3. Peter Giblin
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  4. Mads Nielsen
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Corresponding author

Correspondence to Arjan Kuijper.

Additional information

Arjan Kuijper is Senior Researcher at the Johann Radon Institute (RICAM) of the Austrian Academy of Sciences in Linz, Austria. He received his M.Sc. degree in applied mathematics in 1995 from the University of Twente, The Netherlands. During the period 1996–1997 he worked at ELTRA Parkeergroep, Ede, The Netherlands. He has been a Ph.D. student (1997–2001), associate researcher (2001–2002), and postdoc (202) at the Institute of Information and Computing Sciences of Utrecht University. In 2003-2005 he served as assistant research professor at the IT University of Copenhagen in Denmark. His interest subtends all mathematical aspects of image and shape analysis, notably multi-scale representations (scale spaces), catastrophe and singularity theory, medial axes and symmetry sets, partial differential equations, singular theories, and their applications.

Ole Fogh Olsen is associate professor in the image group at the IT University of Copenhagen. He received the PhD degree in 2000 in computer science from University of Copenhagen, Denmark. Main research interest areas are image analysis, medical image analysis and computer vision with focus on scale space theory, differential geometry, singularity theory, statistics, segmentation, optic flow and shape modelling.

Peter Giblin is Professor of Mathematics at the University of Liverpool and a former Head of the Mathematical Sciences Department. He joined the staff there in 1967 and has been visiting professor at the University of North Carolina at Chapel Hill, Five Colleges in Amherst, Massachusetts, and Brown University. His research interests are in singularity theory and its applications to differential geometry and computer vision.

Mads Nielsen received a MSc in 1992 and a PhD in 1995 both in computer science from DIKU, Department of Computer Science, University of Copenhagen, Denmark. During his PhD studies he spent one year 93–94 at the Robotvis lab at INRIA, Sophia-Antipolis, France. In the second half of 1995 he was post-doc at the Image Sciences Institute of Utrecht University, The Netherlands. In 1996 he was joint post-doc at DIKU and 3D-Lab, School of Dentistry, University of Copenhagen, where he served as assistant professor 1997–99. In 1998–99 he served as external associate professor at Institute of Mathematical Modelling, Technical University of Denmark. April 1999 he became the first associate professor at the new IT University of Copenhagen. Since June 2002 he has been professor the same place heading the Image Analysis Group. He is head of the PhD-studies at ITU, member of the Academical Council of ITU, General chair of MICCAI 2006, member of the editorial board of IJCV and JMIV. His research interests are in the mathematical foundation of image analysis, the computational aspects, and the applications, especially in the medical area.

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Kuijper, A., Olsen, O.F., Giblin, P. et al. Alternative 2D Shape Representations using the Symmetry Set. J Math Imaging Vis 26, 127–147 (2006). https://doi.org/10.1007/s10851-006-8372-2

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