Skip to main content
SpringerLink
Log in

A Comparison of Joint Contour Nets and Pareto Sets

  • Conference paper
  • First Online:
Topological Methods in Data Analysis and Visualization IV (TopoInVis 2015)

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Carlsson, G.E., Singh, G., Zomorodian, A.: Computing multidimensional persistence. J. Comput. Geom. 1(1), 72–100 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Carr, H., Duke, D.J.: Joint contour nets. IEEE Trans. Vis. Comput. Graph. 20(8), 1100–1113 (2014)

    Article  Google Scholar 

  3. Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom. 24(2), 75–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. Chen, C.H., Härdle, W.K., Unwin, A. (eds.): Handbook of Data Visualization. Handbooks of Computational Statistics. Springer, Berlin (2008)

    MATH  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Doraiswamy, H., Natarajan, V.: Efficient algorithms for computing Reeb graphs. Comput. Geom. 42(6-7), 606–616 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Doraiswamy, H., Natarajan, V.: Computing Reeb graphs as a union of contour trees. IEEE Trans. Vis. Comput. Graph. 19(2), 249–262 (2013)

    Article  Google Scholar 

  10. 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)

    Chapter  Google Scholar 

  11. 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)

    Google Scholar 

  12. Forman, R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: Proceeding of STOC, pp. 302–311. ACM, New York (1984)

    Google Scholar 

  16. 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)

    Google Scholar 

  17. 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)

    Article  Google Scholar 

  18. Nixon, M.S., Aguado, A.S.: Feature Extraction and Image Processing, 2nd edn. Academic Press, New York (2008)

    Google Scholar 

  19. Pascucci, V., Cole-McLaughlin, K.: Parallel computation of the topology of level sets. Algorithmica 38(1), 249–268 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Szymczak, A., Zhang, E.: Robust Morse decompositions of piecewise constant vector fields. IEEE Trans. Vis. Comput. Graph. 18(6), 938–951 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Kaiserslautern, Kaiserslautern, Germany

    Lars Huettenberger & Christoph Garth

  2. ETH Zurich, Zurich, Switzerland

    Christian Heine

Authors
  1. Lars Huettenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Heine
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christoph Garth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lars Huettenberger .

Editor information

Editors and Affiliations

  1. University of Leeds, Leeds, United Kingdom

    Hamish Carr

  2. Department of Computer Science, Technical University of Kaiserslautern, Kaiserslautern, Germany

    Christoph Garth

  3. School of Computer Science and Communication, KTH Royal Institute of Technology, Stockholm, Sweden

    Tino Weinkauf

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

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

Download citation

Publish with us

Policies and ethics

Search

Navigation