Abstract
For the scientific visualization and analysis of univariate (scalar) fields several topological approaches like contour trees and Reeb graphs were studied and compared to each other some time ago. In recent years, some of those approaches were generalized to multivariate fields. Among others, data structures like the joint contour net (JCN) and the Pareto set were introduced and improved in subsequent work. However, both methods utilized individual data sets as test cases for their proof-of-concept sections and partially lacked a complete comparison to other multivariate approaches. Hence, to better understand the relationship between those two data structures and to gain insights into general multivariate topology, we present a deeper comparison of JCNs and Pareto sets in which we integrate data sets applied in the original JCN and Pareto set papers.
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Carlsson, G.E., Singh, G., Zomorodian, A.: Computing multidimensional persistence. J. Comput. Geom. 1(1), 72–100 (2010)
Carr, H., Duke, D.J.: Joint contour nets. IEEE Trans. Vis. Comput. Graph. 20(8), 1100–1113 (2014)
Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom. 24(2), 75–94 (2003)
Carr, H., Snoeyink, J., van de Panne, M.: Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree. Comput. Geom. 43(1), 42–58 (2010)
Carr, H., Geng, Z., Tierny, J., Chattopadhyay, A., Knoll, A.: Fiber surfaces: Generalizing isosurfaces to bivariate data. Comput. Graph. Forum 34(3), 241–250 (2015)
Chen, C.H., Härdle, W.K., Unwin, A. (eds.): Handbook of Data Visualization. Handbooks of Computational Statistics. Springer, Berlin (2008)
Chen, G., Mischaikow, K., Laramee, R.S., Zhang, E.: Efficient Morse decompositions of vector fields. IEEE Trans. Vis. Comput. Graph. 14(4), 848–862 (2008)
Doraiswamy, H., Natarajan, V.: Efficient algorithms for computing Reeb graphs. Comput. Geom. 42(6-7), 606–616 (2009)
Doraiswamy, H., Natarajan, V.: Computing Reeb graphs as a union of contour trees. IEEE Trans. Vis. Comput. Graph. 19(2), 249–262 (2013)
Edelsbrunner, H., Harer, J.: Jacobi sets. In: Cucker, F., DeVore, R., Olver, P., Süli, E. (eds.) Foundations of Computational Mathematics: Minneapolis, 2002, pp. 37–57. Cambridge University Press, Cambridge (2004)
Edelsbrunner, H., Harer, J., Patel, A.K.: Reeb spaces of piecewise linear mappings. In: Proceedings of the Twenty-fourth Annual Symposium on Computational Geometry, SCG ’08, pp. 242–250. ACM, New York (2008)
Forman, R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)
Huettenberger, L., Garth, C.: A comparison of Pareto sets and Jacobi sets. In: Bennett, J., Vivodtzev, F., Pascucci, V. (eds.) Topological and Statistical Methods for Complex Data: Tackling Large-Scale, High-Dimensional, and Multivariate Data Spaces, pp. 125–141. Springer, Berlin (2015)
Huettenberger, L., Heine, C., Carr, H., Scheuermann, G., Garth, C.: Towards multifield scalar topology based on Pareto optimality. Comput. Graph. Forum 32(3), 341–350 (2013)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: Proceeding of STOC, pp. 302–311. ACM, New York (1984)
Kniss, J., Kindlmann, G.L., Hansen, C.D.: Interactive volume rendering using multi-dimensional transfer functions and direct manipulation widgets. In: Proceeding of IEEE Visualization, pp. 255–262. IEEE Computer Society, New York (2001)
Nagaraj, S., Natarajan, V., Nanjundiah, R.S.: A gradient-based comparison measure for visual analysis of multifield data. Comput. Graph. Forum 30(3), 1101–1110 (2011)
Nixon, M.S., Aguado, A.S.: Feature Extraction and Image Processing, 2nd edn. Academic Press, New York (2008)
Pascucci, V., Cole-McLaughlin, K.: Parallel computation of the topology of level sets. Algorithmica 38(1), 249–268 (2003)
Sauber, N., Theisel, H., Seidel, H.: Multifield-graphs: an approach to visualizing correlations in multifield scalar data. IEEE Trans. Vis. Comput. Graph. 12(5), 917–924 (2006)
Schneider, D., Heine, C., Carr, H., Scheuermann, G.: Interactive comparison of multifield scalar data based on largest contours. Comput. Aided Geom. Des. 30(6), 521–528 (2013)
Szymczak, A., Zhang, E.: Robust Morse decompositions of piecewise constant vector fields. IEEE Trans. Vis. Comput. Graph. 18(6), 938–951 (2012)
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Huettenberger, L., Heine, C., Garth, C. (2017). A Comparison of Joint Contour Nets and Pareto Sets. In: Carr, H., Garth, C., Weinkauf, T. (eds) Topological Methods in Data Analysis and Visualization IV. TopoInVis 2015. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-44684-4_3
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DOI: https://doi.org/10.1007/978-3-319-44684-4_3
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