Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
L. Antiga and DA Steinman. Robust and objective decomposition and mapping of bifurcating vessels. IEEE Trans. on Medical Imaging, 23(6):704–713, 2004.
David Cohen-Steiner and Jean-Marie Morvan. Restricted delaunay triangulations and normal cycle. In Proc. of Symp. on Comp. Geom., pages 312–321. ACM Press, 2003.
S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart. Spectral surface quadrangulation. ACM SIGGRAPH, 2006.
J. Erickson and K. Whittlesey. Greedy optimal homotopy and homology generators. In Proc. 16th ACM-SIAM Symp. on Discrete Algorithms, pages 1038–1046, 2005.
Hershel M. Farkas and Irwin Kra. Riemann Surfaces. Springer Verlag, 1980.
Michael S. Floater and Kai Hormann. Surface parameterization: a tutorial and survey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in multiresolution for geometric modelling, pages 157–186. Springer Verlag, 2005.
William Fulton. Algebraic Topology, A first course. Springer Verlag, 1995.
Xianfeng Gu and Shing-Tung Yau. Global conformal surface parameterization. In Symp. on Geom. Proc., pages 127–137, 2003.
Steven Haker, Sigurd Angenent, Allen Tannenbaum, Ron Kikinis, Guillermo Sapiro, and Michael Halle. Conformal surface parameterization for texture mapping. IEEE Trans. on Visualization and Computer Graphics, 6(2):181–189, 2000.
Klaus Hildebrandt and Konrad Polthier. Anisotropic filtering of non-linear surface features. Computer Graphics Forum, 23(3):391–400, 2004.
K. Hormann, K. Polthier, and A. Sheffer. Mesh parameterization: Theory and practice. In SIGGRAPH Asia 2008, Course Notes, number 11, 2008.
Toon Huysmans, Jan Sijbers, and Brigitte Verdonk. Parametrization of tubular surfaces on the cylinder. In WSCG (Journal Papers), pages 97–104, 2005.
Miao Jin, Yalin Wang, Shing-Tung Yau, and Xianfeng Gu. Optimal global conformal surface parameterization. In In IEEE Visualization, pages 267–274, 2004.
Jürgen Jost. Compact Riemann Surfaces. Springer, 2002.
Felix Kälberer, Matthias Nieser, and Konrad Polthier. Quadcover - surface parameterization using branched coverings. Comput. Graph. Forum, 26(3):375–384, 2007.
Liliya Kharevych, Boris Springborn, and Peter Schröder. Discrete conformal mappings via circle patterns. ACM Trans. on Graphics, 25(2), 2006.
Yu-Kun Lai, Miao Jin, Xuexiang Xie, Ying He, Jonathan Palacios, Eugene Zhang, Shi-Min Hu, and Xianfeng David Gu. Metric-driven rosy fields design. Technical report, Tsinghua Univ., Beijing, 2008.
Jonathan Palacios and Eugene Zhang. Rotational symmetry field design on surfaces. ACM Trans. on Graphics, 26(3):55:1–55, 2007.
Konrad Polthier and Eike Preuss. Identifying vector field singularities using a discrete Hodge decomposition. In Visualization and Mathematics III, pages 113–134. Springer, 2003.
Emil Praun, Hugues Hoppe, Matthew Webb, and Adam Finkelstein. Real-time hatching. In SIGGRAPH, page 581, 2001.
Nicolas Ray, Wan Chiu Li, Bruno Lévy, Alla Sheffer, and Pierre Alliez. Periodic global parameterization. ACM Trans. Graph., 25(4):1460–1485, 2006.
Nicolas Ray, Bruno Vallet, Wan Chiu Li, and Bruno Lévy. N-symmetry direction field design. ACM Trans. Graph., 27(2):1–13, 2008.
Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun. Designing quadrangulations with discrete harmonic forms. In Eurographics Symp. on Geom. Proc., 2006.
William T. Tutte. How to draw a graph. Proc. London Math. Soc., s3-13(1):743–767, 1963.
E. Zhang, J. Hays, and G. Turk. Interactive Tensor Field Design and Visualization on Surfaces. IEEE Trans. on Visualization and Computer Graphics, pages 94–107, 2007.
L. Zhu, S. Haker, and A. Tannenbaum. Flattening maps for the visualization of multi-branched vessels, 2005.
Malte Zöckler, Detlev Stalling, and Hans-Christian Hege. Fast and intuitive generation of geometric shape transitions. The Visual Computer, 16(5):241–253, 2000.
Acknowledgements
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Berlin Heidelberg
About this chapter
Cite this chapter
Kälberer, F., Nieser, M., Polthier, K. (2011). Stripe Parameterization of Tubular Surfaces. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds) Topological Methods in Data Analysis and Visualization. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15014-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-15014-2_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15013-5
Online ISBN: 978-3-642-15014-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)