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A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality
November/December 2008 (vol. 14 no. 6)
pp. 1619-1626
ASCII Text
x 
 
"A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality," IEEE Transactions on Visualization and Computer Graphics, vol. 14, no. 6, pp. 1619-1626, November/December, 2008.
 
 
BibTex
x
 
@article{ 10.1109/TVCG.2008.110,
author = {},
title = {A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality},
journal ={IEEE Transactions on Visualization and Computer Graphics},
volume = {14},
number = {6},
issn = {1077-2626},
year = {2008},
pages = {1619-1626},
doi = {http://doi.ieeecomputersociety.org/10.1109/TVCG.2008.110},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Visualization and Computer Graphics
TI - A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality
IS - 6
SN - 1077-2626
SP1619
EP1626
EPD - 1619-1626
PY - 2008
KW - Scalability
KW - Grid computing
KW - Large-scale systems
KW - Topology
KW - Data mining
KW - Data visualization
KW - Feature extraction
KW - Adaptive mesh refinement
KW - Size control
KW - Data analysis
KW - large scale data.
KW - Index Terms&#8212
KW - Topology-based analysis
KW - Morse-Smale complex
VL - 14
JA - IEEE Transactions on Visualization and Computer Graphics
ER -
 
The Morse-Smale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalar-valued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closure-finite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divide-and-conquer strategy allows for memory-efficient computation of the MS complex and simplification on-the-fly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementation-specific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on off-the-shelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/10243 node grid on a laptop computer with 2 Gb memory.

[1] S. Beucher, Watershed, heirarchical segmentation and waterfall algorithm. In J. Serra and P. Soille, editors, Mathematical Morphology and its Applications to Image Processing, pages 69–76, 1994.
[2] P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci, A topological hierarchy for functions on triangulated surfaces. IEEE Transactions on Visualization and Computer Graphics, 10 (4): 385–396, 2004.
[3] H. Carr, J. Snoeyink, and U. Axen, Computing contour trees in all dimensions. In Symposium on Discrete Algorithms, pages 918–926, 2000.
[4] H. Carr, J. Snoeyink, and M. van de Panne, Simplifying flexible isosurfaces using local geometric measures. In Proc. IEEE Conf. Visualization, pages 497–504, 2004.
[5] A. Cayley, On contour and slope lines. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, XVII: 264–268, 1859.
[6] Y.-J. Chiang and X. Lu, Progressive simplification of tetrahedral meshes preserving all isosurface topologies. Computer Graphics Forum, 22 (3): 493–504, 2003.
[7] P. Cignoni, D. Constanza, C. Montani, C. Rocchini, and R. Scopigno, Simplification of tetrahedral meshes with accurate error evaluation. In Proc. IEEE Conf. Visualization, pages 85–92, 2000.
[8] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Morse-Smale complexes for piecewise linear 3-manifolds. In Proc. 19th Ann. Sympos. Comput. Geom., pages 361–370, 2003.
[9] H. Edelsbrunner, J. Harer, and A. Zomorodian, Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete and Computational Geometry, 30 (1): 87–107, 2003.
[10] R. Forman, A user's guide to discrete morse theory, 2001.
[11] M. Garland and P. S. Heckbert, Simplifying surfaces with color and texture using quadric error metrics. In Proc. IEEE Conf. Visualization, pages 263–269, 1998.
[12] M. Garland and Y. Zhou, Quadric-based simplification in any dimension. ACM Transactions on Graphics, 24 (2): 209–239, 2005.
[13] T. Gerstner and R. Pajarola, Topology preserving and controlled topology simplifying multiresolution isosurface extraction. In Proc. IEEE Conf. Visualization, pages 259–266, 2000.
[14] I. Guskov and Z. Wood, Topological noise removal. In Proc. Graphics Interface, pages 19–26, 2001.
[15] A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann, Topology-based simplification for feature extraction from 3d scalar fields. In Proc. IEEE Conf. Visualization, pages 535–542, 2005.
[16] A. Gyulassy, V. Natarajan, V. Pascucci, P. T. Bremer, and B. Hamann, A topological approach to simplification of three-dimensional scalar fields. IEEE Transactions on Visualization and Computer Graphics (special issue IEEE Visualization 2005), pages 474–484, 2006.
[17] A. Gyulassy, V. Natarajan, V. Pascucci, and B. Hamann, Efficient computation of morse-smale complexes for three-dimensional scalar functions. IEEE Transactions on Visualization and Computer Graphics, 13 (6): 1440–1447, 2007.
[18] H. Hoppe, Progressive meshes. In Proc. SIGGRAPH, pages 99–108, 1996.
[19] H. King, K. Knudson, and N. Mramor, Generating discrete morse functions from point data. Experimental Mathematics, 14 (4): 435–444, 2005.
[20] D. Laney, A. Mascarenhas, and P. Miller, Understanding the structure of the turbulent mixing layer in hydrodynamic instabilities. IEEE Transactions on Visualization and Computer Graphics, 12 (5): 1053–1060, 2006. Member-P. -T. Bremer and Member-V. Pascucci.
[21] T. Lewiner, H. Lopes, and G. Tavares, Applications of forman's discrete morse theory to topology visualization and mesh compression. IEEE Transactions on Visualization and Computer Graphics, 10 (5): 499–508, 2004.
[22] J. C. Maxwell, On hills and dales. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, XL: 421–427, 1870.
[23] V. Natarajan and H. Edelsbrunner, Simplification of three-dimensional density maps. IEEE Transactions on Visualization and Computer Graphics, 10 (5): 587–597, 2004.
[24] V. Pascucci, G. Scorzelli, P.-T. Bremer, and A. Mascarenhas, Robust on-line computation of reeb graphs: simplicity and speed. ACM Trans. Graph., 26 (3): 58, 2007.
[25] S. Rana, Topological Data Structures for Surfaces: An Introduction to Geographical Information Science. Wiley, 2004.
[26] G. Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique. Comptes Rendus de L'Académie ses Séances, Paris, 222: 847–849, 1946.
[27] J. Roerdink and A. Meijster, The watershed transform: Definitions, algorithms and parallelization techniques, 1999.
[28] S. Smale, Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math., 74: 391–406, 1961.
[29] S. Smale, On gradient dynamical systems. Ann. of Math., 74: 199–206, 1961.
[30] O. G. Staadt and M. H. Gross, Progressive tetrahedralizations. In Proc. IEEE Conf. Visualization, pages 397–402 1998.
[31] S. Takahashi, G. M. Nielson, Y. Takeshima, and I. Fujishiro, Topological volume skeletonization using adaptive tetrahedralization. In Proc. Geometric Modeling and Processing, pages 227–236, 2004.
[32] S. Takahashi, Y. Takeshima, and I. Fujishiro, Topological volume skeletonization and its application to transfer function design. Graphical Models, 66 (1): 24–49, 2004.
[33] Z. Wood, H. Hoppe, M. Desbrun, and P. Schröder, Removing excess topology from isosurfaces. ACM Transactions on Graphics, 23 (2): 190–208, 2004.

Index Terms:
Scalability,Grid computing,Large-scale systems,Topology,Data mining,Data visualization,Feature extraction,Adaptive mesh refinement,Size control,Data analysis,large scale data.,Index Terms&#8212,Topology-based analysis,Morse-Smale complex
Citation:
"A Practical Approach to Morse-Smale Complex Computation: Scalability and Generality," IEEE Transactions on Visualization and Computer Graphics, vol. 14, no. 6, pp. 1619-1626, Nov.-Dec. 2008, doi:10.1109/TVCG.2008.110
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