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Tensor field visualization aims either at depiction of the full information contained in the field or at extraction and display of specific features. Here, we focus on the first task and evaluate integral and glyph based methods with regard to their power of providing an intuitive visual representation. Tensor fields are considered in a differential geometric context, using a coordinate-free notation when possible. An overview and classification of glyph-based methods is given and selected innovative visualization techniques are presented in more detail. The techniques are demonstrated for applications from medicine and relativity theory.

Second-order tensors may be described in terms of and . Shape is quantified by tensor , which are fixed with respect to coordinate system changes. This chapter describes an anatomically-motivated method of detecting edges in diffusion tensor fields based on the of invariants. Three particular invariants (the mean, variance, and skewness of the tensor eigenvalues) are described in two ways: first, as the geometric parameters of an intuitive graphical device for representing tensor shape (the ), and second, in terms of their physical and anatomical significance in diffusion tensor MRI. Tensor-valued gradients of these invariants lead to an orthonormal basis for describing changes in tensor shape. The spatial gradient of the diffusion tensor field may be projected onto this basis, producing three different measures of edge strength, selective for different kinds of anatomical boundaries. The gradient measures are grounded in standard tensor analysis, and are demonstrated on synthetic data.

We introduce the underlying theory behind degenerate points in 2D tensor fields to study the local field properties in the vicinity of linear and nonlinear singularities. The structural stability of these features and their corresponding separatrices are also analyzed. From here, we highlight the main techniques for visualizing and simplifying the topology of both static and time-varying 2D tensor fields.

Topological analysis of 3D tensor fields starts with the identification of degeneracies in the tensor field. In this chapter, we present a new, intuitive and numerically stable method for finding degenerate tensors in symmetric second order 3D tensor fields. This method is based on a description of a tensor having an isotropic spherical component and a linear or planar component. As such, we refer to this formulation as the . In this chapter, we also show that the stable degenerate features in 3D tensor fields form lines. On the other hand, degenerate features that form points, surfaces or volumes are not stable and either disappear or turn into lines when noise is introduced into the system. These topological feature lines provide a compact representation of the 3D tensor field and are useful in helping scientists and engineers understand their complex nature.

The analysis and visualization of tensor fields is an advancing area in scientific visualization. Topology-based methods that investigate the eigenvector fields of second order tensor fields have gained increasing interest in recent years. Most algorithms focus on features known from vector fields, such as saddle points and attracting or repelling nodes. However, more complex features, such as closed hyperstreamlines are usually neglected. In this chapter, a method for detecting closed hyperstreamlines in tensor fields as a topological feature is presented. The method is based on a special treatment of cases where a hyperstreamline reenters a cell and prevents infinite cycling during hyperstreamline calculation. The algorithm checks for possible exits of a loop of crossed edges and detects structurally stable closed hyperstreamlines. These global features cannot be detected by conventional topology and feature detection algorithms used for the visualization of second order tensor fields.

This chapter introduces a visualization method specifically tailored to the class of tensor fields with properties similar to stress and strain tensors. Such tensor fields play an important role in many application areas such as structure mechanics or solid state physics. The presented technique is a global method that represents the physical meaning of these tensor fields with their central features: regions of compression or expansion. The method consists of two steps: first, the tensor field is interpreted as a distortion of a flat metric with the same topological structure; second, the resulting metric is visualized using a texture-based approach. The method supports an intuitive distinction between positive and negative eigenvalues.

In many engineering applications that use tensor analysis, such as tensor imaging, the underlying tensors have the characteristic of being positive definite. It might therefore be more appropriate to use techniques specially adapted to such tensors. We will describe the geometry and calculus on the Riemannian symmetric space of positive-definite tensors. First, we will explain why the geometry, constructed by Emile Cartan, is a natural geometry on that space. Then, we will use this framework to present formulas for means and interpolations specific to positive-definite tensors.

Diffusion Tensor MRI (DT-MRI) measurements are a discrete noisy sample of an underlying macroscopic effective diffusion tensor field, (x), of water. This field is presumed to be piecewise continuous/smooth at a gross anatomical length scale. Here we describe a mathematical framework for obtaining an estimate of this tensor field from the measured DT-MRI data using a spline-based continuous approximation. This methodology facilitates calculation of new structural quantities and provides a framework for applying differential geometric methods to DT-MRI data. A B-spline approximation has already been used to improve robustness of DT-MRI fiber tractography. Here we propose a piecewise continuous approximation based on Non-Uniform Rational B-Splines (NURBS), which addresses some of the shortcomings of the previous implementation.

We present a unified framework for interpolation and regularisation of scalar- and tensor-valued images. This framework is based on elliptic partial differential equations (PDEs) and allows rotationally invariant models. Since it does not require a regular grid, it can also be used for tensor-valued scattered data interpolation and for tensor field inpainting. By choosing suitable differential operators, interpolation methods using radial basis functions are covered. Our experiments show that a novel interpolation technique based on anisotropic diffusion with a diffusion tensor should be favoured: It outperforms interpolants with radial basis functions, it allows discontinuity-preserving interpolation with no additional oscillations, and it respects positive semidefiniteness of the input tensor data.

In this chapter, we introduce the problem of registering diffusion tensor magnetic resonance (DT-MR) images. The registration task for these images is made challenging by the orientational information they contain, which is affected by the registration transformation. This information about orientation and other aspects of the diffusion tensor are exploited in the development of similarity measures with which to guide DT-MR image registration, and the current state-of-the-art is reviewed. The chapter concludes with a discussion of some outstanding issues and future avenues for research in diffusion tensor registration.