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Mathematical Foundations in Visualization

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Foundations of Data Visualization

Abstract

Mathematical concepts and tools have shaped the field of visualization in fundamental ways and played a key role in the development of a large variety of visualization techniques. In this chapter, we sample the visualization literature to provide a taxonomy of the usage of mathematics in visualization and to identify a fundamental set of mathematics that should be taught to students as part of an introduction to contemporary visualization research. Within the scope of this chapter, we are unable to provide a full review of all mathematical foundations of visualization; rather, we identify a number of concepts that are useful in visualization, explain their significance, and provide references for further reading. We assume the reader has basic knowledge of linear algebra [90], multivariate calculus [89], statistics, combinatorics, and stochastics [39]. Other topics not covered in this chapter, such as image analysis [88], computer graphics [86], signal processing [41], computational geometry [2], geometric modeling, mesh generation, computer-aided geometric design [35, 106], and numerics [76] can be found in well-established textbooks. More advanced topics such as information theory, dimension reduction, and kernel methods are discussed in other parts of the book.

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Acknowledgements

The authors would like to thank the organizers of the Dagstuhl Seminar 18041 in January 2018, entitled “Foundations of Data Visualization”. Bei Wang is partially supported by NSF IIS-1910733, DBI-1661375, and IIS-1513616. Roxana Bujack is partially supported by the Laboratory Directed Research and Development (LDRD) program of the Los Alamos National Laboratory (LANL) under project number 20190143ER. Ingrid Hotz is supported through Swedish e-Science Research Center (SeRC) and the ELLIIT environment for strategic research in Sweden.

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Authors and Affiliations

  1. Linköping University, Norrköping, Sweden

    Ingrid Hotz

  2. Los Alamos National Laboratory, Santa Fe, NM, USA

    Roxana Bujack

  3. Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Christoph Garth

  4. University of Utah, Salt Lake City, UT, USA

    Bei Wang

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  1. Ingrid Hotz
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  1. University of Oxford, Oxford, UK

    Min Chen

  2. Department of Informatics, University of Bergen, Bergen, Norway

    Helwig Hauser

  3. University of Maryland, Baltimore County, Baltimore, MD, USA

    Penny Rheingans

  4. Institut für Informatik, Universität Leipzig, Leipzig, Germany

    Gerik Scheuermann

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Hotz, I., Bujack, R., Garth, C., Wang, B. (2020). Mathematical Foundations in Visualization. In: Chen, M., Hauser, H., Rheingans, P., Scheuermann, G. (eds) Foundations of Data Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-34444-3_5

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