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Issue No.12 - Dec. (2011 vol.17)
pp: 1902-1911
Bei Wang , SCI Inst., Univ. of Utah, Salt Lake City, UT, USA
B. Summa , SCI Inst., Univ. of Utah, Salt Lake City, UT, USA
V. Pascucci , SCI Inst., Univ. of Utah, Salt Lake City, UT, USA
M. Vejdemo-Johansson , Dept. of Math., Stanford Univ., Stanford, CA, USA
Large observations and simulations in scientific research give rise to high-dimensional data sets that present many challenges and opportunities in data analysis and visualization. Researchers in application domains such as engineering, computational biology, climate study, imaging and motion capture are faced with the problem of how to discover compact representations of highdimensional data while preserving their intrinsic structure. In many applications, the original data is projected onto low-dimensional space via dimensionality reduction techniques prior to modeling. One problem with this approach is that the projection step in the process can fail to preserve structure in the data that is only apparent in high dimensions. Conversely, such techniques may create structural illusions in the projection, implying structure not present in the original high-dimensional data. Our solution is to utilize topological techniques to recover important structures in high-dimensional data that contains non-trivial topology. Specifically, we are interested in high-dimensional branching structures. We construct local circle-valued coordinate functions to represent such features. Subsequently, we perform dimensionality reduction on the data while ensuring such structures are visually preserved. Additionally, we study the effects of global circular structures on visualizations. Our results reveal never-before-seen structures on real-world data sets from a variety of applications.
Data visualization, Topology, Algorithm design and analysis, Feature extraction, Approximation methods, topological analysis., Dimensionality reduction, circular coordinates, visualization
Bei Wang, B. Summa, V. Pascucci, M. Vejdemo-Johansson, "Branching and Circular Features in High Dimensional Data", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 12, pp. 1902-1911, Dec. 2011, doi:10.1109/TVCG.2011.177
[1] Free motion capture. http:/
[2] D. Attali, A. Lieutier, and D. Salinas, Efficient data structure for representing and simplifying simplicial complexes in high dimensions. Proceedings 27th Annual ACM Symposium on Computational Geometry, pages 501–509, 2011
[3] D. Attali, A. Lieutier, and D. Salinas, Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes. Proceedings 27th Annual ACM Symposium on Computational Geometry, pages 491–500, 2011
[4] M. Belkin and P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15: 1373–1396, 2003
[5] P. Bendich, D. Cohen-Steiner, H. Edelsbrunner, J. Harer, and D. Moro-zov, Inferring local homology from sampled stratified spaces. Proceedings 48th Annual IEEE Symposium on Foundations of Computer Science, pages 536–546, 2007
[6] P. Bendich, H. Edelsbrunner, and M. Kerber, Computing robustness and persistence for images. IEEE Transactions on Visualization and Computer Graphics, 16: 1251–1260, 2010
[7] P. Bendich, B. Wang, and S. Mukherjee, Towards stratification learning through homology inference. AAAI 2010 Fall Symposium on Manifold Learning and its Applications, 2010
[8] D. Burghelea and T. K. Dey, Defining and computing topological persistence for 1-cocycles. Manuscript, December 2010
[9] O. Busaryev, T. K. Dey, and Y. Wang, Tracking a generator by persistence. Discrete Mathematics, Algorithms and Applications, 2 (4): 539– 552, 2010
[10] G. Carlsson, A. J. Zomorodian, A. Collins, and L. J. Guibas, Persistence barcodes for shapes. Proceedings Eurographs/ACM SIGGRAPH Symposium on Geometry Processing, pages 124–135, 2004
[11] F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot, Proximity of persistence modules and their diagrams. Proceedings 25th Annual Symposium on Computational Geometry, pages 237–246, 2009
[12] C. Chen and D. Freedman, Quantifying homology classes. Proceedings 25th International Symposium on Theoritical Aspects of Computer Science, 1: 169–180, 2008
[13] C. Chen and M. Kerber, Persistent homology computation with a twist. Proceedings 27th European Workshop on Computational Geometry, 2011
[14] V. de Silva and G. Carlsson, Topological estimation using witness complexes. Symposium on Point-Based Graphics, pages 157–166, 2004
[15] V. de Silva, D. Morozov, and M. Vejdemo-Johansson, Persistent coho-mology and circular coordinates. Proceedings 25th Annual Symposium on Computational Geometry, pages 227–236, 2009
[16] V. de Silva, D. Morozov, and M. Vejdemo-Johansson, Dualities in persistent (co)homology. Manuscript, 2010
[17] V. de Silva, D. Morozov, and M. Vejdemo-Johansson, Persistent coho-mology and circular coordinates. Discrete & Computational Geometry, pages 1–23, 2011
[18] T. K. Dey, A. N. Hirani, and B. Krishnamoorthy, Optimal homologous cycles, total unimodularity, and linear programming. Proceedings 42nd ACM Symposium on Theory of Computing, pages 221–230, 2010
[19] T. K. Dey, K. Li, J. Sun, and D. Cohen-Steiner, Computing geometry-aware handle and tunnel loops in 3D models. SIGGRAPH, 45: 1–9, 2008
[20] T. K. Dey, J. Sun, and Y. Wang, Approximating loops in a shortest ho-mology basis from point data. Proceedings Annual Symposium on Computational Geometry, pages 166–175, 2010
[21] M. Dixon, N. Jacobs, and R. Pless, Finding minimal parametrizations of cylindrical image manifolds. Proceedings Conference on Computer Vision and Pattern Recognition Workshop, page 192, 2006
[22] P. Dlotko, A fast algorithm to compute cohomology group generators of orientable 2-manifolds 3rd International Workshop on Computational Topology in Image Context, 2010
[23] H. Edelsbrunner and J. Harer, Persistent homology - a survey. Contemporary Mathematics, 453:257–282, 2008
[24] H. Edelsbrunner and J. Harer, Computational Topology: An Introduction. American Mathematical Society, Providence, RI, USA, 2010
[25] H. Edelsbrunner, D. Letscher, and A. J. Zomorodian, Topological persistence and simplification. Discrete and Computational Geometry, 28: 511– 533, 2002
[26] Éric Colin de Verdière and F. Lazarus, Optimal system of loops on an ori-entable surface. Discrete Computational Geometry, 33: 627–636, 2005
[27] J. Erickson, E. W. Chambers, and A. Nayyeri, Homology flows, coho-mology cuts. Proceedings 41st Annual ACM Symposium on Theory of Computing, pages 273–282, 2009
[28] J. Erickson, E. W. Chambers, and A. Nayyeri, Minimum cuts and shortest homologous cycles. Proceedings 25th Annual ACM Symposium on Computational Geometry, pages 377–385, 2009
[29] J. Erickson and K. Whittlesey, Greedy optimal homotopy and homology generators. Proceedings 16th Annual ACM-SIAM symposium on Discrete Algorithms, pages 1038–1046, 2005
[30] J. Erickson and P. Worah, Computing the shortest essential cycle. Discrete and Computational Geometry, 2010
[31] R. K. Gabriel and R. R. Sokal, A new statistical approach to geographic variation analysis. Systematic Zoology, 18 (3): 259–278, 1969
[32] S. Gerber, P.-T. Bremer, V. Pascucci, and R. Whitaker, Visual exploration of high dimensional scalar functions. IEEE Transactions on Visualization and Computer Graphics, 16 (6): 1271–1280, 2010
[33] P. Giblin, Graphs, Surfaces and Homology. Cambridge University Press, New York, NY, USA, 2010
[34] A. Hatcher, Algebraic Topology. Cambridge University Press, 2002
[35] O. M. C. Lab Motion capture data sets.
[36] N. D. Lawrence, Probabilistic spectral dimensionality reduction. Manuscript, 2010
[37] J. A. Lee and M. Verleysen, Nonlinear dimensionality reduction of data manifolds with essential loops. Neurocomputing, 67: 29–53, 2005
[38] D. Morozov, Dionysus library for computing persistent homology.
[39] J. R. Munkres, Elements of algebraic topology. Addison-Wesley, Redwood City, CA, USA, 1984
[40] P. Oesterling, C. Heine, H. Jänicke, and G. Scheuermann, Visual analysis of high dimentional point clouds using topological landscapes. IEEE Pacific Visualization Symposium, pages 113–120, 2010
[41] P. Oesterling, G. Scheuermann, S. Teresniak, G. Heyer, S. Koch, T. Ertl, and G. H. Weber, Two-stage framework for a topology-based projection and visualization of classified document collections. Proceedings IEEE Visual Analytics Science and Technology, pages 91–98, 2010
[42] R. Pless and I. Simon, Embedding images in non-flat spaces. In Conference on Imaging Science Systems and Technology, pages 182–188, 2002
[43] S. T. Roweis and L. K. Saul, Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, 2000
[44] B. Schölkopf, A. Smola, and K.-R. Müller, Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10 (5): 1299–1319, 1998
[45] S. Takahashi, I. Fujishiro, and M. Okada, Applying manifold learning to plotting approximate contour trees. IEEE Transactions on Visualization and Computer Graphics, 15 (6): 1185–1192, 2009
[46] J. B. Tenenbaum, V. de Silva, and J. C. Langford, A global geometric framework for nonlinear dimensionality reduction. Science, 290 (5500): 2319–2323, 2000
[47] L. van der Maaten, Matlab toolbox for dimensionality reduction. http://homepage.tudelft.nl19j49/.
[48] G. Weber, P.-T. Bremer, and V. Pascucci, Topological landscapes: A terrain metaphor for scientific data. IEEE Transactions on Visualization and Computer Graphics, 13 (6): 1416–1423, 2007
[49] A. J. Zomorodian, Fast construction of the Vietoris-Rips complex. Computers & Graphics, 34 (3): 263–271, 2010
[50] A. J. Zomorodian and G. Carlsson, Computing persistent homology. Discrete and Computational Geometry, 33 (2): 249–274, 2005
[51] A. J. Zomorodian and G. Carlsson, Localized homology. Computational Geometry: Theory and Applications, 2008
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