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A discrete morse-based approach to multivariate data analysis

Authors:

Federico Iuricich,

Sara Scaramuccia,

Leila De FlorianiAuthors Info & Claims

SA '16: SIGGRAPH ASIA 2016 Symposium on Visualization

Article No.: 5, Pages 1 - 8

https://doi.org/10.1145/3002151.3002166

Published: 28 November 2016 Publication History

Abstract

Multivariate data are becoming more and more popular in several applications, including physics, chemistry, medicine, geography, etc. A multivariate dataset is represented by a cell complex and a vector-valued function defined on the complex vertices. The major challenge arising when dealing with multivariate data is to obtain concise and effective visualizations. The usability of common visual elements (e.g., color, shape, size) deteriorates when the number of variables increases. Here, we consider Discrete Morse Theory (DMT) [Forman 1998] for computing a discrete gradient field on a multivariate dataset. We propose a new algorithm, well suited for parallel and distribute implementations. We discuss the importance of obtaining the discrete gradient as a compact representation of the original complex to be involved in the computation of multidimensional persistent homology. Moreover, the discrete gradient field that we obtain is at the basis of a visualization tool for capturing the mutual relationships among the different functions of the dataset.

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References

[1]

Allili, M., Kaczynski, T., Landi, C., and Masoni, F., 2015. A new matching algorithm for multidimensional persistence. ArXiv, Id:1511.05427, Nov.

[2]

Allili, M., Kaczynski, T., and Landi, C. 2016. Reducing complexes in multidimensional persistent homology theory. Journal of Symbolic Computation.

Digital Library

[3]

Carlsson, G., and Zomorodian, A. 2007. The theory of multidimensional persistence. In SoCG '07 Proceedings of the twenty-third annual symposium on Computational geometry, ACM New York, Gyeongju, South-Korea, vol. 392, 184--193.

Digital Library

[4]

Carlsson, G., Singh, G., and Zomorodian, A. 2009. Computing multidimensional persistence. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 5878 LNCS, 1, 730--739.

Digital Library

[5]

Carr, H., and Duke, D. 2013. Joint contour nets: Computation and properties. In Visualization Symposium (PacificVis), 2013 IEEE Pacific, 161--168.

[6]

Carr, H., and Duke, D. 2014. Joint contour nets. In IEEE Transactions on Visualization and Computer Graphics, IEEE Computer Society, vol. 20, 1100--1113.

Digital Library

[7]

Cerri, A., Fabio, B. D., Jablonski, G., and Medri, F. 2014. Comparing shapes through multi-scale approximations of the matching distance. Computer Vision and Image Understanding 121, 43--56.

Digital Library

[8]

De Floriani, L., Fugacci, U., Iuricich, F., and Magillo, P. 2015. Morse Complexes for Shape Segmentation and Homological Analysis: Discrete Models and Algorithms. Computer Graphics Forum 34, 2, 761--785.

Digital Library

[9]

Edelsbrunner, H., and Harer, J. 2004. Jacobi sets of multiple Morse functions. In Foundations of Computational Mathematics, Minneapolis 2002, vol. 312 of London Mathematical Society Lecture Note Series. Cambridge University Press, 35--57.

[10]

Edelsbrunner, H., and Harer, J. 2008. Persistent homology-a survey. Contemporary mathematics 453, 257--282.

[11]

Edelsbrunner, H., and Mücke, E. P. 1990. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9, 1, 66--104.

Digital Library

[12]

Edelsbrunner, H., Harer, J., and Patel, A. K. 2008. Reeb spaces of piecewise linear mappings. In Proceedings of the Twenty-fourth Annual Symposium on Computational Geometry, ACM, New York, NY, USA, SoCG '08, 242--250.

Digital Library

[13]

Fellegara, R., luricich, F., De Floriani, L., and Weiss, K. 2014. Efficient computation and simplification of discrete morse decompositions on triangulated terrains. In Proceedings of the 22Nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM, New York, NY, USA, SIGSPATIAL '14, 223--232.

Digital Library

[14]

Forman, R. 1998. Morse theory for cell complexes. Advances in mathematics 134, 1, 90--145.

[15]

Fugacci, U., Iuricich, F., and De Floriani, L. 2014. Efficient computation of simplicial homology through acyclic matching. In 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2014, Timisoara, Romania, September 22-25, 2014, IEEE, F. Winkler, V. Negru, T. Ida, T. Jebelean, D. Petcu, S. M. Watt, and D. Zaharie, Eds., 587--593.

[16]

Günther, D., Reininghaus, J., Wagner, H., and Hotz, I. 2012. Efficient computation of 3d morse-smale complexes and persistent homology using discrete morse theory. The Visual Computer 28, 10, 959--969.

[17]

Gyulassy, A., Kotava, N., Kim, M., Hansen, C., Hagen, H., and Pascucci, V. 2012. Direct Feature Visualization Using Morse-Smale Complexes. IEEE transactions on visualization and computer graphics 18, 9, 1549--62.

Digital Library

[18]

Hatcher, A. 2002. Algebraic topology. Cambridge UP, Cambridge.

[19]

Huettenberger, L., and Garth, C. 2015. A Comparison of Pareto Sets and Jacobi Sets. Springer Berlin Heidelberg, Berlin, Heidelberg, 125--141.

[20]

Huettenberger, L., Heine, C., Carr, H., Scheuermann, G., and Garth, C. 2013. Towards multifield scalar topology based on Pareto optimality. Computer Graphics Forum 32, 3 (Jun), 341--350.

Digital Library

[21]

King, H., Knudson, K., and Mramor, N. 2005. Generating Discrete Morse Functions from Point Data. Experimental Mathematics 14, 4, 435--444.

[22]

Lesnick, M., and Wright, M. 2015. Interactive Visualization of 2-D Persistence Modules. Preprint ArXiv, 1--75.

[23]

Milnor, J. W. 1963. Morse Theory. Annals of Mathematics Studies. Princeton University Press.

[24]

Mischaikow, K., and Nanda, V. 2013. Morse Theory for Filtrations and Efficient Computation of Persistent Homology. Discrete & Computational Geometry 50, 2, 330--353.

Digital Library

[25]

Nagaraj, S., Natarajan, V., and Nanjundiah, R. S. 2011. A gradient-based comparison measure for visual analysis of multifield data. Computer Graphics Forum 30, 3, 1101--1110.

Digital Library

[26]

Nielson, G. M. 1997. Tools for triangulations and tetrahedralizations and constructing functions defined over them. In Scientific Visualization: overviews, Methodologies and Techniques, G. M. Nielson, H. Hagen, and H. Müller, Eds. IEEE Computer Society, Silver Spring, MD, ch. 20, 429--525.

Digital Library

[27]

Robins, V., Wood, P. J., and Sheppard, A. P. 2011. Theory and algorithms for constructing discrete morse complexes from grayscale digital images. IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 8, 1646--1658.

Digital Library

[28]

Shivashankar, N., and Natarajan, V. 2012. Parallel computation of 3d morse-smale complexes. Comput. Graph. Forum 31, 3, 965--974.

Digital Library

[29]

Shivashankar, N., Maadasamy, S., and Natarajan, V. 2012. Parallel computation of 2d morse-smale complexes. IEEE Trans. Vis. Comput. Graph. 18, 10, 1757--1770.

Digital Library

[30]

Weiss, K., Iuricich, F., Fellegara, R., and De Floriani, L. 2013. A primal/dual representation for discrete Morse complexes on tetrahedral meshes. Computer Graphics Forum 32, 3pt3, 361--370.

Digital Library

Cited By

Raith FScheuermann GHeine C(2024)Topological Simplifcation of Jacobi Sets for Piecewise-Linear Bivariate 2D Scalar Fields with Adjustment of the Underlying Data2024 IEEE Topological Data Analysis and Visualization (TopoInVis)10.1109/TopoInVis64104.2024.00007(23-33)Online publication date: 13-Oct-2024
https://doi.org/10.1109/TopoInVis64104.2024.00007
Fellegara RIuricich FSong YFloriani L(2023)Terrain trees: a framework for representing, analyzing and visualizing triangulated terrainsGeoinformatica10.1007/s10707-022-00472-327:3(525-564)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1007/s10707-022-00472-3
Landi CScaramuccia S(2022)Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homologyJournal of Combinatorial Optimization10.1007/s10878-021-00729-x44:4(2347-2374)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s10878-021-00729-x
Show More Cited By

Index Terms

A discrete morse-based approach to multivariate data analysis
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Shape analysis

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cover image ACM Conferences

SA '16: SIGGRAPH ASIA 2016 Symposium on Visualization

November 2016

129 pages

ISBN:9781450345477

DOI:10.1145/3002151

Conference Chairs:
Wei Chen
Zhejiang University, China
,
Daniel Weiskopf
University of Stuttgart, Germany

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 28 November 2016

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National Science Foundation

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SA '16

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SA '16: SIGGRAPH Asia 2016

December 5 - 8, 2016

Macau

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Overall Acceptance Rate 178 of 869 submissions, 20%

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Cited By

Raith FScheuermann GHeine C(2024)Topological Simplifcation of Jacobi Sets for Piecewise-Linear Bivariate 2D Scalar Fields with Adjustment of the Underlying Data2024 IEEE Topological Data Analysis and Visualization (TopoInVis)10.1109/TopoInVis64104.2024.00007(23-33)Online publication date: 13-Oct-2024
https://doi.org/10.1109/TopoInVis64104.2024.00007
Fellegara RIuricich FSong YFloriani L(2023)Terrain trees: a framework for representing, analyzing and visualizing triangulated terrainsGeoinformatica10.1007/s10707-022-00472-327:3(525-564)Online publication date: 1-Jul-2023
https://dl.acm.org/doi/10.1007/s10707-022-00472-3
Landi CScaramuccia S(2022)Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homologyJournal of Combinatorial Optimization10.1007/s10878-021-00729-x44:4(2347-2374)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s10878-021-00729-x
Klötzl DKrake TZhou YHotz IWang BWeiskopf D(2022)Local bilinear computation of Jacobi setsThe Visual Computer10.1007/s00371-022-02557-438:9-10(3435-3448)Online publication date: 30-Jun-2022
https://doi.org/10.1007/s00371-022-02557-4
Allili MKaczynski TLandi CMasoni F(2019)Acyclic Partial Matchings for Multidimensional PersistenceJournal of Mathematical Imaging and Vision10.1007/s10851-018-0843-861:2(174-192)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1007/s10851-018-0843-8
Armstrong RMcClure JRobins VLiu ZArns CSchlüter SBerg S(2018)Porous Media Characterization Using Minkowski Functionals: Theories, Applications and Future DirectionsTransport in Porous Media10.1007/s11242-018-1201-4130:1(305-335)Online publication date: 27-Nov-2018
https://doi.org/10.1007/s11242-018-1201-4
Fellegara RIuricich FDe Floriani L(2017)Efficient representation and analysis of triangulated terrainsProceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems10.1145/3139958.3140050(1-4)Online publication date: 7-Nov-2017
https://dl.acm.org/doi/10.1145/3139958.3140050
Allili MKaczynski TLandi CMasoni F(2017)Algorithmic Construction of Acyclic Partial Matchings for Multidimensional PersistenceDiscrete Geometry for Computer Imagery10.1007/978-3-319-66272-5_30(375-387)Online publication date: 22-Aug-2017
https://doi.org/10.1007/978-3-319-66272-5_30

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