Stripe Parameterization of Tubular Surfaces

  • Felix Kälberer
  • Matthias Nieser
  • Konrad Polthier
Part of the Mathematics and Visualization book series (MATHVISUAL)


We present a novel algorithm for automatic parameterization oftube-like surfaces of arbitrarygenus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistentstripe texture. Mathematically these surfaces can be described as thickened graphs, and the calculatedparameterizationstripe will follow either around thetube, along the underlying graph, a spiraling combination of both, or obey an arbitrary texture map whosecharts have a 180 degree symmetry.We use the principalcurvature frame field of the underlyingtube-like surface to guide the creation of a global, topologically consistentstripeparameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projectivevector fields instead of frame fields, and second, on different types ofbranch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.


Riemann Surface Branch Point Texture Image Parameter Line Topological Disk 
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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The authors are grateful to Christian Hansen of Fraunhofer MEVIS (Bremen, Germany) for providing clinical 3D models of vascular structures and fruitful discussions concerning this work.

Many thanks to Sabine Krofczik and Jürgen Rybak, Department of Neurobiology at Freie Universität Berlin, as well as Steffen Prohaska and Anja Ku, Zuse Institute Berlin (ZIB) for supplying the neuron geometry.

This research was supported by the DFG Research Center MATHEON and by mental images.


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Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  • Felix Kälberer
    • 1
  • Matthias Nieser
    • 1
  • Konrad Polthier
    • 1
  1. 1.Freie Universität BerlinBerlinGermany

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