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Path-Based Mathematical Morphology on Tensor Fields

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Visualization and Processing of Higher Order Descriptors for Multi-Valued Data

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

Abstract

Traditional path-based morphology allows finding long, approximately straight, paths in images. Although originally applied only to scalar images, we show how this can be a very good fit for tensor fields. We do this by constructing directed graphs representing such data, and then modifying the traditional path opening algorithm to work on these graphs. Cycles are dealt with by finding strongly connected components in the graph. Some examples of potential applications are given, including path openings that are not limited to a specific set of orientations.

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Notes

  1. 1.

    A tangent vector can be considered an element in the tangent bundle, and is a combination of a position and a vector describing an orientation/direction. Physically, a tangent vector can be considered to describe the position and velocity of a particle, for example.

  2. 2.

    Code available at http://bit.ly/1zpfIXf.

  3. 3.

    This makes some intuitive sense, as every voxel contains the same amount of space, but at a crossing it is shared by fibres of multiple orientations.

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Acknowledgements

The first and third author were funded by the Netherlands Organisation for Scientific Research (NWO), project no. 612.001.001. Also, we thank Remco Renken and Jelmer Kok from the NeuroImaging Center Groningen for supplying us with the diffusion MRI data (including regions of interest and tractography results), as well as for some inspiring discussions.

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Authors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, 407, 9700 AK, Groningen, The Netherlands

    Jasper J. van de Gronde & Jos B. T. M. Roerdink

  2. Department of Computer Sciences, University of Wisconsin - Madison, 1210 W. Dayton St., Madison, WI, 53706-1613, USA

    Mikola Lysenko

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  1. Jasper J. van de Gronde
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  2. Mikola Lysenko
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  3. Jos B. T. M. Roerdink
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Correspondence to Jos B. T. M. Roerdink .

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Editors and Affiliations

  1. Linköping University, Norrköping, Sweden

    Ingrid Hotz

  2. Visualization and Medical Image Analysis, University of Bonn, Bonn, Germany

    Thomas Schultz

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van de Gronde, J.J., Lysenko, M., Roerdink, J.B.T.M. (2015). Path-Based Mathematical Morphology on Tensor Fields. In: Hotz, I., Schultz, T. (eds) Visualization and Processing of Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-15090-1_6

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