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Issue No.12 - Dec. (2012 vol.18)
pp: 2014-2022
A. Gyulassy , SCI Inst., Univ. of Utah, Salt Lake City, UT, USA
P. Bremer , Lawrence Livermore Nat. Lab., Lawrence, CA, USA
V. Pascucci , SCI Inst., Univ. of Utah, Salt Lake City, UT, USA
ABSTRACT
Topological techniques have proven highly successful in analyzing and visualizing scientific data. As a result, significant efforts have been made to compute structures like the Morse-Smale complex as robustly and efficiently as possible. However, the resulting algorithms, while topologically consistent, often produce incorrect connectivity as well as poor geometry. These problems may compromise or even invalidate any subsequent analysis. Moreover, such techniques may fail to improve even when the resolution of the domain mesh is increased, thus producing potentially incorrect results even for highly resolved functions. To address these problems we introduce two new algorithms: (i) a randomized algorithm to compute the discrete gradient of a scalar field that converges under refinement; and (ii) a deterministic variant which directly computes accurate geometry and thus correct connectivity of the MS complex. The first algorithm converges in the sense that on average it produces the correct result and its standard deviation approaches zero with increasing mesh resolution. The second algorithm uses two ordered traversals of the function to integrate the probabilities of the first to extract correct (near optimal) geometry and connectivity. We present an extensive empirical study using both synthetic and real-world data and demonstrates the advantages of our algorithms in comparison with several popular approaches.
INDEX TERMS
Geometry, Manifolds, Vectors, Algorithm design and analysis, Standards, Robustness, Topology, Morse-Smale complex, Topology, topological methods
CITATION
A. Gyulassy, P. Bremer, V. Pascucci, "Computing Morse-Smale Complexes with Accurate Geometry", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 12, pp. 2014-2022, Dec. 2012, doi:10.1109/TVCG.2012.209
REFERENCES
[1] E. Babson and P. Hersh, Discrete Morse functions from lexicographic orders Transactions of the American Mathematical Society, 3(457): pages 509-534, 2005.
[2] J. Bennett, V. Krishnamurthy, S. Liu, V. Pascucci, R. Grout, J. Chen, and P.-T. Bremer, Feature-based statistical analysis of combustion simulation data IEEE Transactions Visualization and Computer Graphics, 17(12): pages 1822-1831, 2011.
[3] 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): pages 385-396, 2004.
[4] P.-T. Bremer, G. Weber, V. Pascucci, M. Day, and J. Bell, Analyzing and tracking burning structures in lean premixed hydrogen flames IEEE Transactions on Visualization and Computer Graphics, 16(2): pages 248-260, 2010.
[5] P.-T. Bremer, G. Weber, J. Tierny, V. Pascucci, M. Day, and J. B. Bell, Interactive exploration and analysis of large scale simulations using topology-based data segmentation IEEE Transactions on Visualization and Computer Graphics. 17(99), 2010.
[6] A. Cayley, On contour and slope lines The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, XVIII: pages 264-268, 1859.
[7] F. Cazals, F. Chazal, and T. Lewiner, Molecular shape analysis based upon the morse-smale complex and the connolly function. In Proceedings of the 19th Symposium on Computational Geometry, SCG'03 pages 351-360, New York, NY, USA, 2003. ACM.
[8] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Morse-Smale complexes for piecewise linear 3-manifolds. In Proceedings of the 19th Symposium on Computational Geometry, pages 361-370, 2003.
[9] H. Edelsbrunner, J. Harer, and A. Zomorodian, Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds Discrete Computational Geometry, 30: pages 87-107, 2003.
[10] H. Edelsbrunner and E. P. Mücke, Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms ACM Transactions on Graphics (TOG), 9: pages 66-104, 1990.
[11] R. Forman, A user's guide to discrete Morse theory. In Seminate Lotharinen de Combinatore. volume 48, 2002.
[12] A. Gyulassy, Combinatorial Construction of Morse-Smale Complexes for Data Analysis and Visualization. PhD thesis, University of California Davis, 2008.
[13] A. Gyulassy, P.-T. Bremer, V. Pascucci, and B. Hamann, A practical approach to Morse-Smale complex computation: Scalability and generality. IEEE Transactions on Visualization and Computer Graphics, 14(6): pages 1619-1626, 2008.
[14] A. Gyulassy, P.-T. Bremer, V. Pascucci, and B. Hamann, Practical considerations in Morse-Smale complex computation. In Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, Mathematics and Visualization, pages 67-78. Springer, 2011.
[15] A. Gyulassy, M. Duchaineau, V. Natarajan, V. Pascucci, E. Bringa, A. Higginbotham, and B. Hamann, Topologically clean distance fields IEEE Transactions on Computer Graphics and Visualization, 13(6): 1432-1439 2007.
[16] A. Gyulassy, N. Kotava, M. Kim, C. D. Hansen, H. Hagen, and V. Pascucci, Direct feature visualization using Morse-Smale complexes IEEE Transactions on Visualization and Computer Graphics, 99 (PrePrints), 2011.
[17] A. Gyulassy, V. Natarajan, V. Pascucci, P.- T. Bremer, and B. Hamann, Topology-based simplification for feature extraction from 3D scalar fields. IEEE Transactions on Computer Graphics and Visualization, 12(4): pages 474-484, 2006.
[18] A. Gyulassy, T. Peterka, V. Pascucci, and R. Ross, Characterizing the parallel computation of Morse-Smale complexes. In Proceedings of IPDPS, 12, Shanghai, China, 2012.
[19] J. Kasten, J. Reininghaus, I. Hotz, and H.-C. Hege, Two-dimensional time-dependent vortex regions based on the acceleration magnitude. IEEE Transactions on Visualization and Computer Graphics, 17(12): 2080-2087. 2011.
[20] H. King, K. Knudson, and M. Neza, Generating discrete Morse functions from point data Experimental Mathematics, 14(4): pages 435-444, 2005.
[21] M. A. Koch, D. G. Norris, and M. Hund-Georgiadis, An investigation of functional and anatomical connectivity using magnetic resonance imaging. NeuroImage. 16(1): pages 241-250, 2002.
[22] D. Laney, P.-T. Bremer, A. Mascarenhas, P. Miller, and V. Pascucci, Un-derstanding the structure of the turbulent mixing layer in hydrodynamic instabilities IEEE Transactions Visualization and Computer Graphics, 12(5): pages 1052-1060, 2006.
[23] T. Lewiner, Constructing discrete Morse functions. Master's thesis De-partment of Mathematics, PUC-Rio, 2002.
[24] T. Lewiner, Critical sets in discrete morse theories: relating forman and piecewise-linear approaches Computer Aided Geometric Design, 2012.
[25] 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, 10:499508, 2004.
[26] J. C, Maxwell. On hills and dales The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, XL: pages 421-427, 1870.
[27] J. Reininghaus and I. Hotz, Combinatorial 2d vector field topology extraction and simplification. In Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, Mathematics and Visualization, Springer, pages 103-114. 2011.
[28] J. Reininghaus, C. Lowen, and I. Hotz, Fast combinatorial vector field topology IEEE Transactions on Visualization and Computer Graphics, 17: pages 1433-1443, 2011.
[29] V. Robins, P. Wood, and A. Sheppard, Theory and algorithms for constructing discrete Morse complexes from grayscale digital images IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(8): page 16461658, 2011.
[30] N. Shivashankar, M. Senthilnathan, and V. Natarajan, Parallel computation of 2d Morse-Smale complexes IEEE Transactions on Visualization and Computer Graphics, to appear, 2012.
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