Summary
The Contour Tree of a scalar field is the graph obtained by contracting all the con- nected components of the level sets of the field into points. This is a powerful ab- straction for representing the structure of the field with explicit description of the topological changes of its level sets. It has proven effective as a data-structure for fast extraction of isosurfaces and its application has been advocated as a user inter- face component guiding interactive data exploration sessions. In practice, this use has been limited to trivial examples due to the problem of presenting a graph that may be overwhelming in size and in which a planar embedding may have self-intersections. We propose a new metaphor for visualizing the Contour Tree borrowed from the classical design of a mechanical orrery – see Fig. 1a – reproducing a hierarchy of orbits of the planets around the sun or moons around a planet. In the toporrery – see Fig. 1b – the hierarchy of stars, planets and moons is replaced with a hierarchy of maxima, minima and saddles that can be interactively filtered, both uniformly and adaptively, by importance with respect to a given metric.
The implementation of the system is based on (1) a hierarchical graph model al- lowing coarse-to-fine traversal for selective refinements and (2) a new algorithm for constructing a multiresolution Contour Tree with guaranteed topological correctness independently of the simplification metric. We have tested the approach using topo- logical persistence as the main metric for constructing the tree hierarchy, and using geometric position as a secondary metric for adaptive refinements. The result is pre- sented in linked views of the abstract toporrery and the geometric embedding of the input data.
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References
C. L. Bajaj, V. Pascucci, and D. R. Schikore. The contour spectrum. In Roni Yagel and Hans Hagen, editors, IEEE Visualization '97, pages 167–175. IEEE, November 1997.
P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci. A multi-resolution data structure for two-dimensional Morse functions. In Proceeding of IEEE Conference on Visualization, pages 139–146, October 2003.
H. Carr and J. Snoeyink. Path seeds and flexible isosurfaces – using topology for exploratory visualization. In Proceeding of IEEE TCVG Symposium on Visualization (Vis-Sym '03), pages 49–58, Grenoble, Fr, May 2003.
H. Carr, J. Snoeyink, and U. Axen. Computing contour trees in all dimensions. In Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, pages 918–926, January 2000.
H. Carr, J. Snoeyink, and U. Axen. Computing contour trees in all dimensions. Computational Geometry Theory and Applications, 2001. To appear (extended abstract appeared at SODA 2000) .
H. Carr, J. Snoeyink, and M. van de Panne. Simplifying flexible isosurfaces using local geometric measures. In IEEE Visualization, pages 497–504, October 2004.
K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Loops in reeb graphs of 2-manifolds. In ACM Symposium on Computational Geometry, pages 344–350, July 2003.
K. Cole-McLaughlin and V. Pascucci. Multiresolution representation of topology. In Proceedings of the 4th IASTED International Conference on Visualization, Imaging, And Image Processing (VIIP 2004), pages 282–289, Marbella, Sapin, September 2004.
G. di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Upper Saddle River, NJ, 1999.
H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. In Proceeding of The 41st Annual Symposium on Foundations of Computer Science. IEEE, November 2000.
A. T. Fomenko and T. L. Kunii, editors. Topological Modeling for Visualization. Springer, Tokyo, 1997.
M. Hilaga, Y. Shinagawa, T. Komura, and T. L. Kunii. Topology matching for full automatic similarity estimation of 3d shapes. In Proceedings of ACM SIGGRAPH 2001, pages 203–212, August 2001.
E. Kleiberg, H. van de Wetering, and J. J. van Wijk. Botanical visualization of huge hierarchies. In Proceedings IEEE Symposium on Information Visualization (InfoVis'2001), pages 87–94, 2001.
Y. Matsumoto. An Introduction to Morse Theory. AMS, 1997.
V. Pascucci and K. Cole-McLaughlin. Efficient computation of the topology of the level sets. In IEEE Visualization, pages 187–194, October 2002.
G. Reeb. Sur les points singuliers d'une forme de pfaff completement integrable ou d'une fonction numerique. Comptes Rendus Acad. Sciences Paris, 222:847–849, 1946.
P. J. Scott. An Application of Surface Networks in Surface Texture, chapter 11: Efficient contour tree and minimum seed set construction, pages 157–166. John Wiley & Sons, May 2004.
Y. Shinagawa, T. L. Kunii, H. Sato, and M. Ibusuki. Modeling the contact of two complex objects: With an application to characterizing dental articulations. Computers and Graphics, 19:21–28, 1995.
Y. Shinagawa and T. L. Kunii. Constructing a Reeb graph automatically from cross sections. IEEE Computer Graphics and Applications, 11:44–51, November 1991.
S. Takahashi, G. M. Nielson, Y. Takeshima, and I. Fujishiro. Topological volume skeletonization using adaptive tetrahedralization. In Proceeding of Geometric Modeling and Processing, pages 227–236, 2004.
S. Takahashi, Y. Takeshima, and I. Fujishiro. Topological volume skeletonization and its application to transfer function design. Graphical Models, 66(1):24–49, 2004.
A. Telea and J. J. van Wijk. Simplified representation of vector fields. In IEEE Computer Society Press, editor, Proceedings of the IEEE conference on Visualization '99, pages 35–42, 1999.
X. Tricoche, G. Scheuermann, and H. Hagen. A topology simplification method for 2d vector fields. In Proceedings of the IEEE conference on Visualization, pages 359–366. IEEE Computer Society Press, 2000.
E. R. Tufte. Envisioning Information. Graphics Press LLC, Cheshire, CT, 1990.
M. van Kreveld, R. van Oostrum, C. Bajaj, V. Pascucci, and D. Schikore. Contour trees and small seed sets for isosurface traversal. In Proceedings of the 13th International Annual Symposium on Computational Geometry (SCG-97), pages 212–220, June 1997. Extended version. Technical report UCRL-JC-132016 Lawrence Livermore National Laboratory.
G. H. Weber and G. Scheuermann. Automating Transfer Function Design Based on Topology Analysis, chapter IV:5, pages 293–308. Mathematics and Visualization. Springer, Berlin, 2004.
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Pascucci, V., Cole-McLaughlin, K., Scorzelli, G. (2009). The TOPORRERY: computation and presentation of multi-resolution topology. In: Möller, T., Hamann, B., Russell, R.D. (eds) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/b106657_2
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