Mandatory Critical Points of 2D Uncertain Scalar Fields
This paper introduces a novel, non‐local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliab...
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| Vydané v: | Computer graphics forum Ročník 33; číslo 3; s. 31 - 40 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Oxford
Blackwell Publishing Ltd
01.06.2014
Wiley |
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | This paper introduces a novel, non‐local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi‐scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output‐sensitive. We illustrate, evaluate, and verify our method on synthetic and real‐world data. |
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| AbstractList | This paper introduces a novel, non‐local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies
mandatory critical points
: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct
mandatory join
and
split trees
. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi‐scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output‐sensitive. We illustrate, evaluate, and verify our method on synthetic and real‐world data. This paper introduces a novel, non-local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi-scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output-sensitive. We illustrate, evaluate, and verify our method on synthetic and real-world data. [PUBLICATION ABSTRACT] This paper introduces a novel, non-local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi-scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output-sensitive. We illustrate, evaluate, and verify our method on synthetic and real-world data. |
| Author | Salmon, Joseph Günther, David Tierny, Julien |
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| Cites_doi | 10.1111/j.1467-8659.2012.03097.x 10.1145/1276377.1276449 10.1111/cgf.12121 10.1109/TVCG.2012.228 10.1109/PacificVis.2013.6596141 10.1130/G19607.1 10.1615/Int.J.UncertaintyQuantification.2012003956 10.1016/S0021-9991(03)00298-5 10.1109/TVCG.2008.110 10.1111/cgf.12100 10.1145/364099.364331 10.1007/s00454-002-2885-2 10.2307/1970311 10.1090/coll/018 10.1559/1523040054738936 10.1145/1377676.1377720 10.1111/j.1467-8659.2009.01604.x 10.1145/77635.77639 10.1007/s00371-012-0726-8 10.1007/s003710050111 10.1007/978-3-642-18638-7_5 10.4310/HHA.2013.v15.n1.a3 10.1111/j.1467-8659.2011.01944.x 10.1007/978-3-642-32677-6_15 |
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| Copyright | 2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd. Distributed under a Creative Commons Attribution 4.0 International License |
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| References_xml | – reference: Edelsbrunner H., Mucke E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. on Graph. 9 (1990), 66-104. 3 – reference: Bendich P., Edelsbrunner H., Morozov D., Patel A.K.: Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications 15, 1 (2013), 51-72. 2 – reference: Cormen T., Leiserson C., Rivest R., Stein C.: Introduction to Algorithms, 2nd ed. McGraw-Hill Higher Education, 2001. 7 – reference: Reeb G.: Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique. Comptesrend. de l'Acad. des Scien. 222 (1946), 847-849. 2 – reference: Heine C., Schneider D., Carr H., Scheuermann G.: Drawing contour trees in the plane. IEEE TVCG 17 (2011), 1599-1611. 7 – reference: Bhatia H., Jadhav S., Bremer P., Chen G., Levine J., Nonato L., Pascucci V.: Flow visualization with quantified spatial and temporal errors using edge maps. IEEE TVCG 18, 9 (2012), 1383-1396. 2 – reference: Pang A.T., Wittenbrink C.M., Lodha S.K.: Approaches to uncertainty visualization. The Visual Computer 13, 8 (1997), 370-390. 2 – reference: Pöthkow K., Hege H.-C.: Positional uncertainty of isocontours: Condition analysis and probabilistic measures. IEEE TVCG 17, 10 (2011), 1393-1406. 2 – reference: Potter K., Gerber S., Anderson E.: Visualization of uncertainty without a mean. IEEE CGA 33 (2013), 75-79. 2 – reference: MacEachren A.M., Robinson A., Hopper S., Gardner S., Murray R., Gahegan M., Hetzler E.: Visualizing geospatial information uncertainty: What we know and what we need to know. Cartography and Geographic Information Science 32, 3 (2005), 139-160. 2 – reference: Tierny J., Gyulassy A., Simon E., Pascucci V.: Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees. IEEE TVCG 15 (2009), 1177-1184. 2 – reference: Smale S.: On gradient dynamical systems. The Annals of Mathematics 74 (1961), 199-206. 2 – reference: Hynek B.M., Phillips R.J.: New data reveal mature, integrated drainage systems on mars indicative of past precipitation. Geology 31, 9 (Sep. 2003), 757-760. 8 – reference: Pöthkow K., Hege H.-C.: Nonparametric models for uncertainty visualization. CGF 32 (2013), 131-140. 2 – reference: Morse M.: The Calculus of Variations in the Large. No. v. 18 in Colloquium Publications - American Mathematical Society. American Mathematical Society, 1934. 2 – reference: Galler B., Fischer M.: An improved equivalence algorithm. Communications of the ACM 7 (1964), 301-303. 7 – reference: Edelsbrunner H., Harer. J.: Jacobi Sets of Multiple Morse Functions. Cambridge Books Online, 2004. 2 – reference: Otto M., Germer T., Hege H.-C., Theisel H.: Uncertain 2D vector field topology. CGF 29 (2010), 347-356. 2 – reference: Edelsbrunner H., Letscher D., Zomorodian A.: Topological persistence and simplification. 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| SubjectTerms | Algorithms Analysis Categories and Subject Descriptors (according to ACM CCS) Combinatorial analysis Combinatorics Computer graphics Computer Science Critical point F.2.2 [Analysis Of Algorithms And Problem Complexity]: Nonnumerical Algorithms and Problems-Computations on discrete structures Image processing systems Mathematical models Probability density functions Scalars Studies Trees Two dimensional |
| Title | Mandatory Critical Points of 2D Uncertain Scalar Fields |
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