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The topology of symmetric, second-order tensor fields

Published: 17 October 1994 Publication History

Abstract

We study the topology of symmetric, second-order tensor fields. The goal is to represent their complex structure by a simple set of carefully chosen points and lines analogous to vector field topology. We extract topological skeletons of the eigenvector fields, and we track their evolution over time. We study tensor topological transitions and correlate tensor and vector data.The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. We identify two kinds of elementary degenerate points, which we call wedges and trisectors. They can combine to form more familiar singularities---such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields.Finally, we show a topological rule that puts a constraint on the topology of tensor fields defined across surfaces, extending to tensor fields the Pointcaré-Hopf theorem for vector fields.

References

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  • (2018)Extremal curves and surfaces in symmetric tensor fieldsThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-017-1450-134:10(1427-1442)Online publication date: 1-Oct-2018
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  1. The topology of symmetric, second-order tensor fields

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    cover image ACM Conferences
    VIS '94: Proceedings of the conference on Visualization '94
    October 1994
    455 pages
    ISBN:0780325214

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    IEEE Computer Society Press

    Washington, DC, United States

    Publication History

    Published: 17 October 1994

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    VIS94
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    VIS94: IEEE Visualization '94
    October 17 - 21, 1994
    Washinton, D.C.

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    • (2018)The parallel eigenvectors operatorProceedings of the Conference on Vision, Modeling, and Visualization10.2312/vmv.20181251(39-46)Online publication date: 10-Oct-2018
    • (2018)Extremal curves and surfaces in symmetric tensor fieldsThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-017-1450-134:10(1427-1442)Online publication date: 1-Oct-2018
    • (2017)Tensor field design in volumesACM Transactions on Graphics10.1145/3130800.313084436:6(1-15)Online publication date: 20-Nov-2017
    • (2017)Autonomous reconstruction of unknown indoor scenes guided by time-varying tensor fieldsACM Transactions on Graphics10.1145/3130800.313081236:6(1-15)Online publication date: 20-Nov-2017
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    • (2016)Topological analysis for 3D real, symmetric second-order tensor fields using Deviatoric Eigenvalue WheelComputers and Graphics10.1016/j.cag.2015.07.00954:C(28-37)Online publication date: 1-Feb-2016
    • (2014)Discrete 2-tensor fields on triangulationsProceedings of the Symposium on Geometry Processing10.1111/cgf.12427(13-24)Online publication date: 9-Jul-2014
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    • (2011)Simple quad domains for field aligned mesh parametrizationACM Transactions on Graphics10.1145/2070781.202417630:6(1-12)Online publication date: 12-Dec-2011
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