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About this book

Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties.

Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs.

The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI.

Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI).

A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research.

The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.

Table of Contents


Tensor Data Visualization


Top Challenges in the Visualization of Engineering Tensor Fields

In this chapter we summarize the top research challenges in creating successful visualization tools for tensor fields in engineering. The analysis is based on our collective experiences and on discussions with both domain experts and visualization practitioners. We find that creating visualization tools for engineering tensors often involves solving multiple different technical problems at the same time—including visual intuitiveness, scalability, interactivity, providing both detail and context, integration with modeling and simulation, representing uncertainty and managing multi-fields; as well as overcoming terminology barriers and advancing research in the mathematical aspects of tensor field processing. We further note the need for tools and data repositories to encourage faster advances in the field. Our interest in creating and proposing this list is to initiate a discussion about important research issues within the visualization of engineering tensor fields.
Mario Hlawitschka, Ingrid Hotz, Andrea Kratz, G. Elisabeta Marai, Rodrigo Moreno, Gerik Scheuermann, Markus Stommel, Alexander Wiebel, Eugene Zhang

Tensor Invariants and Glyph Design

Tensors provide a mathematical language for the description of many physical phenomena. They appear everywhere where the dependence of multiple vector fields is approximated as linear. Due to this generality they occur in various application areas, either as result or intermediate product of simulations. As different as these applications, is the physical meaning and relevance of particular mathematical properties. In this context, domain specific tensor invariants that describe the entities of interest play a crucial role. Due to their importance, we propose to build any tensor visualization upon a set of carefully chosen tensor invariants. In this chapter we focus on glyph-based representations, which still belong to the most frequently used tensor visualization methods. For the effectiveness of such visualizations the right choice of glyphs is essential. This chapter summarizes some common glyphs, mostly with origin in mechanical engineering, and link their interpretation to specific tensor invariants.
Andrea Kratz, Cornelia Auer, Ingrid Hotz

Representation and Processing of Higher-Order Descriptors


Monomial Phase: A Matrix Representation of Local Phase

Local phase is a powerful concept which has been successfully used in many image processing applications. For multidimensional signals the concept of phase is complex and there is no consensus on the precise meaning of phase. It is, however, accepted by all that a measure of phase implicitly carries a directional reference. We present a novel matrix representation of multidimensional phase that has a number of advantages. In contrast to previously suggested phase representations it is shown to be globally isometric for the simple signal class. The proposed phase estimation approach uses spherically separable monomial filter of orders 0, 1 and 2 which extends naturally to N dimensions. For 2-dimensional simple signals the representation has the topology of a Klein bottle. For 1-dimensional signals the new phase representation reduces to the original definition of amplitude and phase for analytic signals. Traditional phase estimation using quadrature filter pairs is based on the analytic signal concept and requires a pre-defined filter direction. The new monomial local phase representation removes this requirement by implicitly incorporating local orientation. We continue to define a phase matrix product which retains the structure of the phase matrix representation. The conjugate product gives a phase difference matrix in a manner similar to the complex conjugate product of complex numbers. Two motion estimation examples are given to demonstrate the advantages of this approach.
Hans Knutsson, Carl-Fredrik Westin

Order Based Morphology for Color Images via Matrix Fields

Mathematical morphology is a successful branch of image processing with a history of more than four decades. Its fundamental operations are dilation and erosion, which are based on the notion of supremum and infimum with respect to an order. From dilation and erosion one can build readily other useful elementary morphological operators and filters, such as opening, closing, morphological top-hats, derivatives, and shock filters. Such operators are available for grey value images, and recently useful analogs of these processes for matrix-valued images have been introduced by taking advantage of the so-called Loewner order. There is a number of approaches to morphology for vector-valued images, that is, color images based on various orders, however, each with its merits and shortcomings. In this chapter we propose an approach to (elementary) morphology for color images that relies on the existing order based morphology for matrix fields of symmetric 2 × 2-matrices. An RGB-image is embedded into a field of those 2 × 2-matrices by exploiting the geometrical properties of the order cone associated with the Loewner order. To this end a modification of the HSL-color model and a relativistic addition of matrices is introduced. The experiments performed with various morphological elementary operators on synthetic and real images provide results promising enough to serve as a proof-of-concept.
Bernhard Burgeth, Andreas Kleefeld

Sharpening Fibers in Diffusion Weighted MRI via Erosion

In this chapter erosion is generalized to the space of diffusion weighted MRI data. This is done effectively by solving a Hamilton-Jacobi-Bellman (HJB) system (erosion) on the coupled space of three dimensional positions and orientations, embedded as a quotient in the group of three dimensional rigid body motions. The solution to the HJB equations is given by a well-posed morphological convolution. We present two numerical approaches to solve the HJB equations: analytical kernels, and finite differences. Proof of concept is given by showing improved visibility of major fiber bundles in both artificial and human data. Furthermore, the method is shown to significantly improve the output of a probabilistic tractography algorithm used to extract the optic radiation.
Thomas C. J. Dela Haije, Remco Duits, Chantal M. W. Tax

Higher Order Tensors and Riemannian-Finsler Geometry


Higher-Order Tensors in Diffusion Imaging

Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathematical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging. This survey gives a careful introduction to the foundations of higher-order tensor algebra, and explains how some concepts from linear algebra generalize to the higher-order case. From the application side, it reviews a variety of distinct higher-order tensor models that arise in the context of diffusion imaging, such as higher-order diffusion tensors, q-ball or fiber Orientation Distribution Functions (ODFs), and fourth-order covariance and kurtosis tensors. By bridging the gap between mathematical foundations and application, it provides an introduction that is suitable for practitioners and applied mathematicians alike, and propels the field by stimulating further exchange between the two.
Thomas Schultz, Andrea Fuster, Aurobrata Ghosh, Rachid Deriche, Luc Florack, Lek-Heng Lim

Fourth Order Symmetric Tensors and Positive ADC Modeling

High Order Cartesian Tensors (HOTs) were introduced in Generalized DTI (GDTI) to overcome the limitations of DTI. HOTs can model the apparent diffusion coefficient (ADC) with greater accuracy than DTI in regions with fiber heterogeneity. Although GDTI HOTs were designed to model positive diffusion, the straightforward least square (LS) estimation of HOTs doesn’t guarantee positivity. In this chapter we address the problem of estimating 4th order tensors with positive diffusion profiles.Two known methods exist that broach this problem, namely a Riemannian approach based on the algebra of 4th order tensors, and a polynomial approach based on Hilbert’s theorem on non-negative ternary quartics. In this chapter, we review the technicalities of these two approaches, compare them theoretically to show their pros and cons, and compare them against the Euclidean LS estimation on synthetic, phantom and real data to motivate the relevance of the positive diffusion profile constraint.
Aurobrata Ghosh, Rachid Deriche

Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

We consider Riemann-Finsler geometry as a potentially powerful mathematical framework in the context of diffusion weighted magnetic resonance imaging. We explain its basic features in heuristic terms, but also provide mathematical details that are essential for practical applications, such as tractography and voxel-based classification. We stipulate a connection between the (dual) Finsler function and signal attenuation observed in the MRI scanner, which directly generalizes Stejskal-Tanner’s solution of the Bloch-Torrey equations and the diffusion tensor imaging (DTI) model inspired by this. The proposed model can therefore be regarded as an extension of DTI. Technically, reconstruction of the (dual) Finsler function from diffusion weighted measurements is a fairly straightforward generalization of the DTI case. The extension of the Riemann differential geometric paradigm for DTI analysis is, however, nontrivial.
Luc Florack, Andrea Fuster

Riemann-Finsler Multi-valued Geodesic Tractography for HARDI

We introduce a geodesic based tractography method for High Angular Resolution Diffusion Imaging (HARDI). The concepts used are similar to the ones in geodesic based tractography for Diffusion Tensor Imaging (DTI). In DTI, the inverse of the second-order diffusion tensor is used to define the manifold where the geodesics are traced. HARDI models have been developed to resolve complex fiber populations within a voxel, and higher order tensors (HOT) are possible representations for HARDI data. In our framework, we apply Finsler geometry, which extends Riemannian geometry to a directionally dependent metric. A Finsler metric is defined in terms of HARDI higher order tensors. Furthermore, the Euler-Lagrange geodesic equations are derived based on the Finsler geometry. In contrast to other geodesic based tractography algorithms, the multi-valued numerical solution of the geodesic equations can be obtained. This gives the possibility to capture all geodesics arriving at a single voxel instead of only computing the shortest one. Results are analyzed to show the potential and characteristics of our algorithm.
Neda Sepasian, Jan H. M. ten Thije Boonkkamp, Luc M. J. Florack, Bart M. Ter Haar Romeny, Anna Vilanova

Tensor Signal Processing


Kernel-Based Morphometry of Diffusion Tensor Images

Voxel-based group-wise statistical analysis of diffusion tensor imaging (DTI) is an integral component in most population-based neuroimaging studies such as those studying brain development during infancy or aging, or those investigating structural differences between healthy and diseased populations. The majority of studies using DTI limit themselves by testing only certain properties of the tensor that mainly include anisotropy and diffusivity. However, the pathology under study may affect other aspects like the orientation information provided by the tensors. Therefore, for detecting subtle pathological changes it is important to perform group-wise testing on the whole tensor, which encompasses the changes in anisotropy, diffusivity and orientation. This is rendered challenging by the fact that conventional linear statistics cannot be applied to tensors. Moreover, the pathology over the population is unknown and could be non-linear, further complicating the group-based statistical analysis. This chapter gives a perspective on performing voxel-wise morphometry of tensor data using kernel-based approach. The method is referred as Kernel-based morphometry (KBM) as it models the tensor distribution using kernel principal component analysis (kPCA), which linearizes the data in high dimensional space. Subsequently a Hotelling T 2 test is performed on the high dimensional kernelized data to determine statistical group differences. We apply this method on simulated and real datasets and show that KBM can effectively identify the underlying tensorial distribution. Thus it can potentially elucidate pathology-induced population differences, thereby establishing a kernelized full tensor framework for population studies.
Madhura Ingalhalikar, Parmeshwar Khurd, Ragini Verma

The Estimation of Free-Water Corrected Diffusion Tensors

Diffusion tensor imaging (DTI) is sensitive to micron scale displacement of water molecules, providing unique insight into microstructural tissue architecture. The tensors provide a compact way to describe the average of these displacements that occur within a voxel. However, current practical image resolution is in the millimeter scale, and thus diffusivities from many tissue compartments are averaged in each voxel, reducing the specificity of the measurement to subtle pathologies. In this chapter we review the free-water model, and use it to derive diffusion tensors following the elimination of the free-water component, that is assumed to originate from the extracellular space. Doing so, the resulting diffusion tensors and their derived indices measure the tissue itself, and are more sensitive to the geometry of the tissue, increasing the specificity to pathologies that affect brain tissue.
Ofer Pasternak, Klaus Maier-Hein, Christian Baumgartner, Martha E. Shenton, Yogesh Rathi, Carl-Fredrik Westin

Techniques for Computing Fabric Tensors: A Review

The aim of this chapter is to review different approaches that have been proposed to compute fabric tensors with emphasis on trabecular bone research. Fabric tensors aim at modeling through tensors both anisotropy and orientation of a material with respect to another one. Fabric tensors are widely used in fields such as trabecular bone research, mechanics of materials and geology. These tensors can be seen as semi-global measurements since they are computed in relatively large neighborhoods, which are assumed quasi-homogeneous. Many methods have been proposed to compute fabric tensors. We propose to classify fabric tensors into two categories: mechanics-based and morphology-based. The former computes fabric tensors from mechanical simulations, while the latter computes them by analyzing the morphology of the materials. In addition to pointing out advantages and drawbacks for each method, current trends and challenges in this field are also summarized.
Rodrigo Moreno, Magnus Borga, Örjan Smedby

Applications of Tensor Processing


Tensors in Geometry Processing

Tensor fields have a wide range of applications outside scientific visualization. In this chapter, we review various types of tensors used in geometry processing, including their properties, application requirements, as well as theoretical and practical results. We will focus on the metric tensor and the curvature tensor, two of the most studied tensors in geometry processing.
Eugene Zhang

Preliminary Findings in Diagnostic Prediction of Schizophrenia Using Diffusion Tensor Imaging

We describe a probabilistic technique for diagnostic prediction of first-episode schizophrenia patients based on their brain diffusion MRI data. The method begins by transforming each voxel from a high-dimensional diffusion weighted signal to a low-dimensional diffusion tensor representation. Three orthogonal diffusion measures (fractional anisotropy, norm, mode) that capture different aspects of the local tissue properties are derived from this diffusion tensor representation. Next, we compute a one-dimensional probability density function of each of the diffusion measures with values obtained from the entire brain. This representation is affine invariant, thus obviating the need for registration of the images. We then use a Parzen window classifier to estimate the likelihood of a new patient belonging to either group. To demonstrate the technique, we apply it to the analysis of 22 first-episode schizophrenic patients and 20 normal control subjects. With leave-one-out cross validation, we find a detection rate of 90.91 % (10 % false positives). We also provide several error bounds on the performance of the classifier.
Yogesh Rathi, Martha E. Shenton, Carl-Fredrik Westin

A System for Combined Visualization of EEG and Diffusion Tensor Imaging Tractography Data

In this paper we present an interactive system that integrates the visual analysis of nerve fiber pathway approximations from diffusion tensor imaging (DTI) with electroencephalography (EEG) data. The technique uses source reconstructions from EEG data to define certain regions of interest in the brain. These regions, in turn, are used to selectively display subsets of the approximated fiber pathways in the brain. The selected pathways highlight potential connections from activated areas to other parts of the brain and can thus help to understand networks on which most higher brain function relies. Users can explore the neuronal network and activity by navigating in an EEG curve view. The navigation is supported by optional mechanisms like snapping to time points with present reconstructed dipoles and visual cues highlighting such points. To the best of our knowledge, the presented combination of time navigation in EEG curves together with DTI pathway selection at the corresponding dipole positions is new and has not been described before. The presented methods are freely available in an open source system for visualization and analysis in neuroscience.
Alexander Wiebel, Cornelius Müller, Christoph Garth, Thomas R. Knösche


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