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## Book chapters

1. A Review of Tensors and Tensor Signal Processing
Authors: Leila Cammoun (1), Carlos Alberto Castaño-Moraga (2), Emma Muñoz-Moreno (3), Dario Sosa-Cabrera (4), Burak Acar (5), Anders Brun, Hans Knutsson (6) and Jean-Philippe Thiran (1)
(1) Signal Processing Institute Ecole Polytechnique Fédérale de, Lausanne, Switzerland
(2) Dept. of Signals and Communciations, University of Las Palmas de Gran Canaria, Spain
(3) Laboratory of Image Processing, University de Valladolid, Spain
(4) Canary Islands Institute of Technology, Spain
(5) Electrical-Electronics Eng. Dept, Bogazici University, Istanbul, Turkey
(6) Dept. of medical engineering, Linköping University
Book: Tensors in Image Processing and Computer Vision, pp. 1-32
Publisher: Springer London
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.

2. Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential Maps
Authors: Anders Brun and Hans Knutsson (1)
(1) Dept. of medical engineering, Linköpings universitet
Book: Visualization and Processing of Tensor Fields: Advances and Perspectives, pp. 139-160
Publisher: Springer Berlin Heidelberg
Abstract: The Riemannian exponential map, and its inverse the Riemannian logarithm map, can be used to visualize metric tensor fields. In this chapter we first derive the well-known metric sphere glyph from the geodesic equation, where the tensor field to be visualized is regarded as the metric of a manifold. These glyphs capture the appearance of the tensors relative to the coordinate system of the human observer. We then introduce two new concepts for metric tensor field visualization: geodesic spheres and geodesically warped glyphs. These extensions make it possible not only to visualize tensor anisotropy, but also the curvature and change in tensor-shape in a local neighborhood. The framework is based on the exp p (v i ) and log p (q) maps, which can be computed by solving a second-order ordinary differential equation (ODE) or by manipulating the geodesic distance function. The latter can be found by solving the eikonal equation, a nonlinear partial differential equation (PDE), or it can be derived analytically for some manifolds. To avoid heavy calculations, we also include first- and second-order Taylor approximations to exp and log. In our experiments, these are shown to be sufficiently accurate to produce glyphs that visually characterize anisotropy, curvature, and shape-derivatives in sufficiently smooth tensor fields where most glyphs are relatively similar in size.

3. Similar Tensor Arrays: A Framework for Storage of Tensor Array Data
Authors: Leila Cammoun (1), Carlos Alberto Castaño-Moraga (2), Emma Muñoz-Moreno (3), Dario Sosa-Cabrera (4), Burak Acar (5), Anders Brun, Hans Knutsson (6) and Jean-Philippe Thiran(1)
(1) Signal Processing Institute Ecole Polytechnique Fédérale de, Lausanne, Switzerland
(2) Dept. of Signals and Communciations, University of Las Palmas de Gran Canaria, Spain
(3) Laboratory of Image Processing, University de Valladolid, Spain
(4) Canary Islands Institute of Technology, Spain
(5) Electrical-Electronics Eng. Dept, Bogazici University, Istanbul, Turkey
(6) Dept. of medical engineering, Linköpings universitet
Book: Tensors in Image Processing and Computer Vision, pp. 407-428
Abstract: This chapter describes a framework for storage of tensor array data, useful to describe regularly sampled tensor fields. The main component of the framework, called Similar Tensor Array Core (STAC), is the result of a collaboration between research groups within the SIMILAR network of excellence. It aims to capture the essence of regularly sampled tensor fields using a minimal set of attributes and can therefore be used as a greatest common divisor and interface between tensor array processing algorithms. This is potentially useful in applied fields like medical image analysis, in particular in Diffusion Tensor MRI, where misinterpretation of tensor array data is a common source of errors. By promoting a strictly geometric perspective on tensor arrays, with a close resemblance to the terminology used in differential geometry, (STAC) removes ambiguities and guides the user to define all necessary information. In contrast to existing tensor array file formats, it is minimalistic and based on an intrinsic and geometric interpretation of the array itself, without references to other coordinate systems.

4. On Geometric Transformations of Local Structure Tensors
Authors: Björn Svensson (1), Anders Brun, Mats Andersson (1) and Hans Knutsson (1)
(1) Dept. of biomedical engineering, Linköpings universitet
Book: Tensors in Image Processing and Computer Vision, pp. 179-193
Publisher: Springer London
Abstract: The structure of images has been studied for decades and the use of local structure tensor fields appeared during the eighties. Since then numerous varieties of tensors and estimation schemes have been developed. Tensors have for instance been used to represent orientation, velocity, curvature and diffusion with applications to adaptive filtering, motion analysis and segmentation. Even though sampling in non-Cartesian coordinate system are common, analysis and processing of local structure tensor fields in such systems is less developed. Previous work on local structure in non-Cartesian coordinate systems include.

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