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A Review of Tensors and Tensor Signal Processing

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Tensors in Image Processing and Computer Vision

Part of the book series: Advances in Pattern Recognition ((ACVPR))

Abstract

Tensors have been broadly used in mathematics and physics, since they are a generalization of scalars or vectors and allow to represent more complex properties. In this chapter we present an overview of some tensor applications, especially those focused on the image processing field. From a mathematical point of view, a lot of work has been developed about tensor calculus, which obviously is more complex than scalar or vectorial calculus. Moreover, tensors can represent the metric of a vector space, which is very useful in the field of differential geometry. In physics, tensors have been used to describe several magnitudes, such as the strain or stress of materials. In solid mechanics, tensors are used to define the generalized Hooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dynamics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics and mathematics. They have been used, for instance, in medical imaging, where we can highlight two applications: the diffusion tensor image, which represents how molecules diffuse inside the tissues and is broadly used for brain imaging; and the tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide information about the local structure or to define anisotropic image filters.

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Author information

Authors and Affiliations

  1. Signal Processing Institute Ecole Polytechnique Fédérale de, Lausanne, Switzerland

    L. Cammoun & J. P. Thiran

  2. Center for Technology in Medicine. Department of Signals and Communciations, University of Las Palmas de Gran Canaria, Spain

    C.A. Castaño-Moraga, D. Sosa-Cabrera & M.A. Rodriguez-Florido

  3. Canary Islands Institute of Technology, Spain

    M.A. Rodriguez-Florido

  4. Laboratory of Image Processing, Univ. de Valladolid, Spain

    E. Muñoz-Moreno

  5. Electrical-Electronics Eng. Dept, Bogazici University, Istanbul, Turkey

    B. Acar

  6. Biomedical Engineering Dept, CMIV Linköpings Universitet, Sweden

    A. Brun & H. Knutsson

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  1. L. Cammoun
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  2. C.A. Castaño-Moraga
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Correspondence to L. Cammoun .

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Editors and Affiliations

  1. Escuela Técnica Superior de Ingenieros de Telecomunicacíon, Universidad de Valladolid, Campus Miguel Delibes, 47011 Valladolid, Spain

    Santiago Aja-Fernández  & Rodrigo de Luis García  & 

  2. Dept. Computing, HongKong Polytechnic University, Hong Kong

    Dacheng Tao

  3. University of London Birkbeck College School of Computer Science & Information Systems, Malet Street, London, WC1E 7HX, UK

    Xuelong Li

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Cammoun, L. et al. (2009). A Review of Tensors and Tensor Signal Processing. In: Aja-Fernández, S., de Luis García, R., Tao, D., Li, X. (eds) Tensors in Image Processing and Computer Vision. Advances in Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-84882-299-3_1

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  • DOI: https://doi.org/10.1007/978-1-84882-299-3_1

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84882-298-6

  • Online ISBN: 978-1-84882-299-3

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