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Discrete Calculus: History and Future

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Discrete Calculus

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Abstract

The goal of this book is to bring together three active areas of current research into a single framework and show how each area benefits from more exposure to the other two. The areas are: discrete calculus, complex networks, and algorithmic content extraction. Although there have been a few intersections in the literature between these areas, they have largely developed independently of one another. However, we believe that researchers working in any one of these three areas can strongly benefit from the tools and techniques being developed in the others. We begin this book by outlining each of these three areas, their history and their relationship to one another. Subsequently, we outline the structure of this work and help the reader navigate its contents.

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Notes

  1. 1.

    Throughout this work we use the terms network and graph interchangeably to mean exactly the same thing—a 1-complex comprised of nodes and edges. A complex will be defined in Chap. 2.

  2. 2.

    In many disciplines the network structure is itself considered the dataset (since this information must be collected). Our distinction here between data and network is simply due to the fact that in several of the disciplines we consider there is data associated with each node or edge in the network (as in the case of image processing), in which case the network defines the spatial domain upon which the data is processed. However, we address algorithms that can be used to process data associated with a network and algorithms that can be used to analyze the structure of the network itself. Consequently, algorithms that process data associated with networks or algorithms that analyze the network structure itself are both considered to be content extraction algorithms.

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Authors and Affiliations

  1. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540-6632, USA

    Dr. Leo J. Grady

  2. Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA

    Dr. Jonathan R. Polimeni

Authors
  1. Dr. Leo J. Grady
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  2. Dr. Jonathan R. Polimeni
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Correspondence to Leo J. Grady .

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Grady, L.J., Polimeni, J.R. (2010). Discrete Calculus: History and Future. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_1

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  • DOI: https://doi.org/10.1007/978-1-84996-290-2_1

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84996-289-6

  • Online ISBN: 978-1-84996-290-2

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