Stripe Parameterization of Tubular Surfaces
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.
KeywordsRiemann Surface Branch Point Texture Image Parameter Line Topological Disk
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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.
- 1.L. Antiga and DA Steinman. Robust and objective decomposition and mapping of bifurcating vessels. IEEE Trans. on Medical Imaging, 23(6):704–713, 2004.Google Scholar
- 2.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.Google Scholar
- 3.S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart. Spectral surface quadrangulation. ACM SIGGRAPH, 2006.Google Scholar
- 4.J. Erickson and K. Whittlesey. Greedy optimal homotopy and homology generators. In Proc. 16th ACM-SIAM Symp. on Discrete Algorithms, pages 1038–1046, 2005.Google Scholar
- 5.Hershel M. Farkas and Irwin Kra. Riemann Surfaces. Springer Verlag, 1980.Google Scholar
- 6.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.Google Scholar
- 7.William Fulton. Algebraic Topology, A first course. Springer Verlag, 1995.Google Scholar
- 8.Xianfeng Gu and Shing-Tung Yau. Global conformal surface parameterization. In Symp. on Geom. Proc., pages 127–137, 2003.Google Scholar
- 9.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.Google Scholar
- 11.K. Hormann, K. Polthier, and A. Sheffer. Mesh parameterization: Theory and practice. In SIGGRAPH Asia 2008, Course Notes, number 11, 2008.Google Scholar
- 12.Toon Huysmans, Jan Sijbers, and Brigitte Verdonk. Parametrization of tubular surfaces on the cylinder. In WSCG (Journal Papers), pages 97–104, 2005.Google Scholar
- 13.Miao Jin, Yalin Wang, Shing-Tung Yau, and Xianfeng Gu. Optimal global conformal surface parameterization. In In IEEE Visualization, pages 267–274, 2004.Google Scholar
- 14.Jürgen Jost. Compact Riemann Surfaces. Springer, 2002.Google Scholar
- 15.Felix Kälberer, Matthias Nieser, and Konrad Polthier. Quadcover - surface parameterization using branched coverings. Comput. Graph. Forum, 26(3):375–384, 2007.Google Scholar
- 16.Liliya Kharevych, Boris Springborn, and Peter Schröder. Discrete conformal mappings via circle patterns. ACM Trans. on Graphics, 25(2), 2006.Google Scholar
- 17.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.Google Scholar
- 18.Jonathan Palacios and Eugene Zhang. Rotational symmetry field design on surfaces. ACM Trans. on Graphics, 26(3):55:1–55, 2007.Google Scholar
- 19.Konrad Polthier and Eike Preuss. Identifying vector field singularities using a discrete Hodge decomposition. In Visualization and Mathematics III, pages 113–134. Springer, 2003.Google Scholar
- 20.Emil Praun, Hugues Hoppe, Matthew Webb, and Adam Finkelstein. Real-time hatching. In SIGGRAPH, page 581, 2001.Google Scholar
- 21.Nicolas Ray, Wan Chiu Li, Bruno Lévy, Alla Sheffer, and Pierre Alliez. Periodic global parameterization. ACM Trans. Graph., 25(4):1460–1485, 2006.Google Scholar
- 22.Nicolas Ray, Bruno Vallet, Wan Chiu Li, and Bruno Lévy. N-symmetry direction field design. ACM Trans. Graph., 27(2):1–13, 2008.Google Scholar
- 23.Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun. Designing quadrangulations with discrete harmonic forms. In Eurographics Symp. on Geom. Proc., 2006.Google Scholar
- 24.William T. Tutte. How to draw a graph. Proc. London Math. Soc., s3-13(1):743–767, 1963.Google Scholar
- 25.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.Google Scholar
- 26.L. Zhu, S. Haker, and A. Tannenbaum. Flattening maps for the visualization of multi-branched vessels, 2005.Google Scholar