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Discrete calculus. Applied analysis on graphs for computational science. (English) Zbl 1195.68074

London: Springer (ISBN 978-1-84996-289-6/hbk; 978-1-84996-290-2/ebook). xvi, 366 p. (2010).
Publisher’s description: The field of discrete calculus, also known as “discrete exterior calculus”, focuses on finding a proper set of definitions and differential operators that make it possible to operate the machinery of multivariate calculus on a finite, discrete space. In contrast to traditional goals of finding an accurate discretization of conventional multivariate calculus, discrete calculus establishes a separate, equivalent calculus that operates purely in the discrete space without any reference to an underlying continuous process. This unique text brings together into a single framework current research in the three areas of discrete calculus, complex networks, and algorithmic content extraction. Although there have been a few intersections in the literature between these disciplines, they have developed largely independently of one another, yet researchers working in any one of these three areas can strongly benefit from the tools and techniques being used in the others. Many example applications from several fields of computational science are provided to demonstrate the usefulness of this framework to a broad range of problems. Readers are assumed to be familiar with the basics of vector calculus, graph theory, and linear algebra.
Topics and features: The book presents a thorough review of discrete calculus, with a focus on key concepts required for successful application. It unifies many standard image processing algorithms into a common framework for viewing a wide variety of standard algorithms in filtering, clustering, and manifold learning that may be applied to processing data associated with a graph or network. It explains how discrete calculus provides a natural definition of “low-frequency” on a graph, which then yields filtering and denoising algorithms. It discusses how filtering algorithms can give rise to clustering algorithms, which can be used to develop manifold learning and data discovery methods, and examines ranking algorithms, as well as algorithms for analyzing the structure of a network.
Graduate students and researchers interested in discrete calculus, complex networks, image processing and computer graphics will find this text/reference a clear introduction to the foundations of discrete calculus as well as a useful guide to have readily available for their work.

MSC:

68R10 Graph theory (including graph drawing) in computer science
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
68T05 Learning and adaptive systems in artificial intelligence
68U10 Computing methodologies for image processing
94A05 Communication theory
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science
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