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Bringing together key researchers in disciplines ranging from visualization and image processing to applications in structural mechanics, fluid dynamics, elastography, and numerical mathematics, the workshop that generated this edited volume was the third in the successful Dagstuhl series. Its aim, reflected in the quality and relevance of the papers presented, was to foster collaboration and fresh lines of inquiry in the analysis and visualization of tensor fields, which offer a concise model for numerous physical phenomena. Despite their utility, there remains a dearth of methods for studying all but the simplest ones, a shortage the workshops aim to address.

Documenting the latest progress and open research questions in tensor field analysis, the chapters reflect the excitement and inspiration generated by this latest Dagstuhl workshop, held in July 2009. The topics they address range from applications of the analysis of tensor fields to purer research into their mathematical and analytical properties. They show how cooperation and the sharing of ideas and data between those engaged in pure and applied research can open new vistas in the study of tensor fields.



Structure-Tensor Computation


Structure Tensor Estimation: Introducing Monomial Quadrature Filter Sets

Description and estimation of local spatial structure has a long history and numerous analysis tools have been developed. A concept that is widely recognized as fundamental in the analysis is the structure tensor. It has, however, a fairly broad and unspecific meaning. This chapter is intended to provide a framework for displaying the differences and similarities of existing structure estimation approaches. A new method for structure tensor estimation, which is a generalization of many of it’s predecessors, is presented. The method uses pairs of filter sets having Fourier directional responses in the form of monomials, one odd order set and one even order set. It is shown that such filter sets allow for a particularly simple way of attaining phase invariant, positive semi-definite, local structure tensor estimates. In addition, we show that the chosen filter sets directly links order, scale and the gradient operator. We continue to compare a number of known structure tensor algorithms by formulating them in monomial filter set terms. In conclusion we show how higher order tensors can be estimated using a generalization of the same simple formulation.
Hans Knutsson, Carl-Fredrik Westin, Mats Andersson

Adaptation of Tensor Voting to Image Structure Estimation

Tensor voting is a well-known robust technique for extracting perceptual information from clouds of points. This chapter proposes a general methodology to adapt tensor voting to different types of images in the specific context of image structure estimation. This methodology is based on the structural relationships between tensor voting and the so-called structure tensor, which is the most popular technique for image structure estimation. The problematic Gaussian convolution used by the structure tensor is replaced by tensor voting. Afterwards, the results are appropriately rescaled. This methodology is adapted to gray-valued, color, vector- and tensor-valued images. Results show that tensor voting can estimate image structure more appropriately than the structure tensor and also more robustly.
Rodrigo Moreno, Luis Pizarro, Bernhard Burgeth, Joachim Weickert, Miguel Angel Garcia, Domenec Puig

Edge-Enhancing Diffusion Filtering for Matrix Fields

The elimination of noise and small details from an image while simultaneously preserving or enhancing the edge structures in an image is a ever-lasting task in image processing. Edge-enhancing anisotropic diffusion is known to tackle this problem successfully. The problem of noise removal and edge enhancement is also a major concern in diffusion tensor magnetic resonance imaging (DT-MRI). This medical image acquisition technique outputs a 3D matrix field of symmetric 3 ×3-matrices, and it helps to visualise, for example, the nerve fibres in brain tissue. As any physical measurement DT-MRI is subjected to errors causing faulty representations of the tissue structure corrupted by noise. In this paper we address that problem by proposing a edge-enhancing diffusion filtering methodology for matrix fields. The approach is based on a generic structure tensor concept for matrix fields that relies on the operator-algebraic properties of symmetric matrices, rather than their channel-wise treatment of earlier proposals. Numerical experiments with artificial and real DT-MRI data confirm the noise-removing and edge-enhancing qualities of the technique presented.
Bernhard Burgeth, Luis Pizarro, Stephan Didas

Tensor-Field Visualization


Fabric-Like Visualization of Tensor Field Data on Arbitrary Surfaces in Image Space

Tensors are of great interest to many applications in engineering and in medical imaging, but a proper analysis and visualization remains challenging. It already has been shown that, by employing the metaphor of a fabric structure, tensor data can be visualized precisely on surfaces where the two eigendirections in the plane are illustrated as thread-like structures. This leads to a continuous visualization of most salient features of the tensor data set. We introduce a novel approach to compute such a visualization from tensor field data that is motivated by image space line integral convolution (LIC). Although our approach can be applied to arbitrary, non-self-intersecting surfaces, the main focus lies on special surfaces following important features, such as surfaces aligned to the neural pathways in the human brain. By adding a postprocessing step, we are able to enhance the visual quality of the results, which improves perception of the major patterns.
Sebastian Eichelbaum, Mario Hlawitschka, Bernd Hamann, Gerik Scheuermann

Beyond Topology: A Lagrangian Metaphor to Visualize the Structure of 3D Tensor Fields

Topology was introduced in the visualization literature some 15 years ago as a mathematical language to describe and capture the salient structures of symmetric second-order tensor fields. Yet, despite significant theoretical and algorithmic advances, this approach has failed to gain wide acceptance in visualization practice over the last decade. In fact, the very idea of a versatile visualization methodology for tensor fields that could transcend application domains has been virtually abandoned in favor of problem-specific feature definitions and visual representations. We propose to revisit the basic idea underlying topology from a different perspective. To do so, we introduce a Lagrangian metaphor that transposes to the structural analysis of eigenvector fields a perspective that is commonly used in the study of fluid flows. Indeed, one can view eigenvector fields as the local superimposition of two vector fields, from which a bidirectional flow field can be defined. This allows us to analyze the structure of a tensor field through the behavior of fictitious particles advected by this flow. Specifically, we show that the separatrices of 3D tensor field topology can in fact be captured in a fuzzy and numerically more robust setting as ridges of a trajectory coherence measure. As a result, we propose an alternative structure characterization strategy for the visual analysis of practical 3D tensor fields, which we demonstrate on several synthetic and computational datasets.
Xavier Tricoche, Mario Hlawitschka, Samer Barakat, Christoph Garth

Tensor Field Design: Algorithms and Applications

Tensor field design has found increasing applications in computer graphics, geometry processing, and scientific visualization. In this chapter, we review recent advances in tensor field design and discuss possible future research directions.
Eugene Zhang

Applications of Tensor-Field Analysis and Visualization


Interactive Exploration of Stress Tensors Used in Computational Turbulent Combustion

Simulation and modeling of turbulent flow, and of turbulent reacting flow in particular, involves solving for and analyzing time-dependent and spatially dense tensor quantities, such as turbulent stress tensors. The interactive visual exploration of these tensor quantities can effectively steer the computational modeling of combustion systems. In this chapter, we discuss the challenges in dense symmetric-tensor visualization applied to turbulent combustion calculation, and analyze the feasibility of using several established tensor visualization techniques in the context of exploring space-time relationships in computationally-simulated combustion tensor data. To tackle the pervasive problems of occlusion and clutter, we propose a solution combining techniques from information and scientific visualization. Specifically, the proposed solution combines a detailed 3D inspection view based on volume rendering with glyph-based representations—used as 2D probes—while leveraging interactive filtering and flow salience cues to clarify the structure of the tensor datasets. Side-by-side views of multiple timesteps facilitate the analysis of time-space relationships. The resulting prototype enables an analysis style based on the overview first, zoom and filter, then details on demand paradigm originally proposed in information visualization. The result is a visual analysis tool to be utilized in debugging, benchmarking, and verification of models and solutions in turbulent combustion. We demonstrate this analysis tool on three example configurations and report feedback from combustion researchers.
Adrian Maries, Abedul Haque, S. Levent Yilmaz, Mehdi B. Nik, G. Elisabeta Marai

Shear Wave Diffusion Observed by Magnetic Resonance Elastography

Dynamic elastography is a noninvasive imaging-based modality for the measurement of viscoelastic constants of living soft tissue. The method employs propagating shear waves to induce elastic deformations inside the target organ. Using magnetic resonance elastography (MRE), components of harmonic shear wave fields can be measured inside gel phantoms or in vivo soft tissues. Soft tissues have heterogeneous elastic properties, giving rise to scattering of propagating shear waves. However, to date little attention has been paid to an analysis of shear wave scattering as a possible means to resolve local elastic heterogeneities in dynamic elastography. In this article we present an analysis of shear wave scattering based on a statistical analysis of shear wave intensity speckles. Experiments on soft gel phantoms with cylindrical glass inclusions are presented where the polarization of the shear wave field was adjusted relative to the orientation of the scatterers. A quantitative analysis of the resulting fields of shear horizontal (SH) waves and shear vertical (SV) waves demonstrates that the distribution of wave intensities in both modes obeys restricted diffusion in a similar order. This observation sets the background for quantification of shear wave scattering in MRE of body tissue where SH and SV wave scattering occur simultaneously.
Sebastian Papazoglou, Jürgen Braun, Dieter Klatt, Ingolf Sack

Diffusion Weighted MRI Visualization


A Comparative Analysis of Dimension Reduction Techniques for Representing DTI Fibers as 2D/3D Points

Dimension Reduction is the process of transfering high-dimensional data into lower dimensions while maintaining the original intrinsic structures. This technique of finding low-dimensional embedding from high-dimensional data is important for visualizing dense 3D DTI fibers because it is hard to visualize and analyze the fiber tracts with high geometric, spatial, and anatomical complexity. Color-mapping, selection, and abstraction are widely used in DTI fiber visualization to depict the properties of fiber models. Nonetheless, visual clutters and occlusion in 3D space make it hard to grasp even a few thousand fibers. In addition, real time interaction (exploring and navigating) on such complex 3D models consumes large amount of CPU/GPU power. Converting DTI fiber to 2D or 3D points with dimension reduction techniques provides a complimentary visualization for these fibers. This chapter analyzes and compares dimension reduction methods for DTI fiber models. An interaction interface augments the 3D visualization with a 2D representation that contains a low-dimensional embedding of the DTI fibers. To achieve real-time interaction, the framework is implemented with GPU programming.
Xiaoyong Yang, Ruiyi Wu, Ziáng Ding, Wei Chen, Song Zhang

Exploring Brain Connectivity with Two-Dimensional Maps

We present and compare two low-dimensional visual representations, 2D point and 2D path, for studying tractography datasets. The goal is to facilitate the exploration of dense tractograms by reducing visual complexity both in static representations and during interaction. The proposed planar maps have several desirable properties, including visual clarity, easy tract-of-interest selection, and multiscale hierarchy. The 2D path representations convey the anatomical familiarity of 3D brain models and cross-sectional views. We demonstrate the utility of both types of representation in two interactive systems where the views and interactions of the standard 3D streamtube representation are linked to those of the planar representations. We also demonstrate a web interface that integrates precomputed neural-path representations into a geographical digital-maps framework with associated labels, metrics, statistics, and linkouts. We compare the two representations both anecdotally and quantitatively via expert input. Results indicate that the planar path representation is more intuitive and easier to use and learn. Similarly, users are faster and more accurate in selecting bundles using the path representation than the 2D point representation. Finally, expert feedback on the web interface suggests that it can be useful for collaboration as well as quick exploration of data.
Çağatay Demiralp, Radu Jianu, David H. Laidlaw

Uncertainty Propagation in DT-MRI Anisotropy Isosurface Extraction

Scalar anisotropy indices are important means for the analysis and visualization of diffusion tensor fields. While the propagation of uncertainty and errors has been studied for a variety measures, this chapter additionally considers the extraction of isosurfaces from anisotropy fields. We use the numerical condition to estimate the uncertainty propagation from the diffusion tensor eigenvalues via fractional (FA) and relative anisotropy (RA) to the position and shape of isosurfaces. Using level crossing probabilities we quantify and visualize the spatial distribution of uncertain isosurfaces. The superiority of FA to RA in terms of uncertainty propagation that was shown for anisotropy images in the literature does not hold for isosurfaces extracted from these images. Instead, our results indicate that for the purpose of isosurface extraction both measures perform approximately equally well.
Kai Pöthkow, Hans-Christian Hege

Beyond Second-Order Diffusion Tensor MRI


Classification Study of DTI and HARDI Anisotropy Measures for HARDI Data Simplification

High angular resolution diffusion imaging (HARDI) captures the angular diffusion pattern of water molecules more accurately than diffusion tensor imaging (DTI). This is of importance mainly in areas of complex intra-voxel fiber configurations. However, the extra complexity of HARDI models has many disadvantages that make it unattractive for clinical applications. One of the main drawbacks is the long post-processing time for calculating the diffusion models. Also intuitive and fast visualization is not possible, and the memory requirements are far from modest. Separating the data into anisotropic-Gaussian (i.e., modeled by DTI) and non-Gaussian areas can alleviate some of the above mentioned issues, by using complex HARDI models only when necessary. This work presents a study of DTI and HARDI anisotropy measures applied as classification criteria for detecting non-Gaussian diffusion profiles. We quantify the classification power of these measures using a statistical test of receiver operation characteristic (ROC) curves applied on ex-vivo ground truth crossing phantoms. We show that some of the existing DTI and HARDI measures in the literature can be successfully applied for data classification to the diffusion tensor or different HARDI models respectively. The chosen measures provide fast data classification that can enable data simplification. We also show that increasing the b-value and number of diffusion measurements above clinically accepted settings does not significantly improve the classification power of the measures. Moreover, we show that a denoising pre-processing step improves the classification. This denoising enables better quality classifications even with low b-values and low sampling schemes. Finally, the findings of this study are qualitatively illustrated on real diffusion data under different acquisition schemes.
Vesna Prčkovska, Maxime Descoteaux, Cyril Poupon, Bart M. ter Haar Romeny, Anna Vilanova

Towards Resolving Fiber Crossings with Higher Order Tensor Inpainting

The use of second-order tensors for the modeling of data from Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is limited by their inability to represent more than one dominant direction in cases of crossing fiber bundles or partial voluming. Higher-order tensors have been used in High Angular Resolution Diffusion Imaging (HARDI) to overcome these problems, but their larger number of parameters leads to longer measurement times for data acquisition. In this work, we demonstrate that higher-order tensors that indicate likely fiber directions can be estimated from a small number of diffusion-weighted measurements by taking into account information from local neighborhoods. To this end, we generalize tensor voting, a method from computer vision, to higher-order tensors. We demonstrate that the resulting even-order tensor fields facilitate fiber reconstruction at crossings both in synthetic and in real DW-MRI data, and that the odd-order fields differentiate crossings from junctions.
Thomas Schultz

Representation and Estimation of Tensor-Pairs

Over the years, several powerful models have been developed to represent specific elementary signal patterns, e.g. locally linear and planar structures. However, in real world problems there is often a need for handling more than one elementary pattern simultaneously. The straightforward approach of adaptive model selection has proven to be difficult and fragile. At the core of this problem is the vicious intractable search space created by having to simultaneously select models and corresponding samples. This calls for higher order models where multiple patterns are represented as one more complex pattern. In this work, we illustrate the advantages of this approach on data that has bi-modal tensor-valued distributions.The method uses first and second order invariants as a representation, and an eigenvector based solution for recovering the elementary tensor components. We show that this method allows estimation of the two tensors that best represent a given tensor distribution. This distribution can for example be samples from a local neighborhood. A bi-modal distribution will produce the two tensors corresponding to the peaks of the distribution. In addition, numbers indicating the amount of samples belonging to each sub distribution are produced. We demonstrate the potential of the approach by processing a number of simple tensor image examples. The results clearly show that new valuable information regarding the local tensor structure is revealed.
Carl-Fredrik Westin, Hans Knutsson

Tensor Metrics


On the Choice of a Tensor Distance for DTI White Matter Segmentation

The segmentation of anatomical structures within the white matter of the brain from DTI is an important task for white matter analysis, and has therefore received considerable attention in the literature during the last few years. Any segmentation method relies on the choice of a tensor dissimilarity measure, which should be small between tensors belonging to the same region and large between tensors belonging to different structures. Many different tensor distances have been proposed in the literature (Frobenius, Kullback-Leibler, Geodesic, Log-Euclidean, Hybrid) for segmentation or other purposes, and there exist reasons (either theoretical or empirical) to justify the choice of any of them. Thus, determining which is the most appropriate tensor distance for a specific segmentation problem has become an extremely difficult decision. In this chapter we present a study on different tensor dissimilarity measures and their performance for white matter segmentation. The study is based on the use of two different DTI atlases of human brain, which provide a ground truth upon which the distances can be fairly compared. In order for the comparison to be independent of the segmentation method employed, it has been performed in terms of the separability of the different clases. Results show the Hybrid distance to perform better than other traditional tensor dissimilarity measures in terms of separability between classes, while the Frobenius, Kullback-Leibler, Geodesic and Log-Euclidean distances perform similarly.
Rodrigo de Luis-García, Carlos Alberola-López, Carl-Fredrik Westin

Divergence Measures and Means of Symmetric Positive-Definite Matrices

The importance of symmetric positive-definite matrices can hardly be exaggerated as they play fundamental roles in many disciplines such as mathematics, numerical analysis, probability and statistics, engineering, and biological and social sciences. On the other hand, in the last few years there has been a renewed interest in developing the theory of means of symmetric positive-definite matrices. In this work we present several divergence functions for measuring closeness between symmetric positive-definite matrices. We then study the invariance properties of these divergence functions as well as the matrix means based on them. We show that the means based on the various divergence functions of a finite collection of symmetric positive-definite matrices are bounded below by their harmonic mean and above by their arithmetic mean. Furthermore, the means based on the studied divergence functions of two symmetric positive-definite matrices are given in closed forms. In particular, we show that the mean based on the Bhattacharyya divergence function of a pair of symmetric positive-definite matrices coincides with their geometric mean.
Maher Moakher

Metric Selection and Diffusion Tensor Swelling

The measurement of the distance between diffusion tensors is the foundation on which any subsequent analysis or processing of these quantities, such as registration, regularization, interpolation, or statistical inference is based. Euclidean metrics were first used in the context of diffusion tensors; then geometric metrics, having the practical advantage of reducing the “swelling effect,” were proposed instead. In this chapter we explore the physical roots of the swelling effect and relate it to acquisition noise. We find that Johnson noise causes shrinking of tensors, and suggest that in order to account for this shrinking, a metric should support swelling of tensors while averaging or interpolating. This interpretation of the swelling effect leads us to favor the Euclidean metric for diffusion tensor analysis. This is a surprising result considering the recent increase of interest in the geometric metrics.
Ofer Pasternak, Nir Sochen, Peter J. Basser

Tensor Analysis


$${\mathcal{H}}^{2}$$ -Matrix Compression

Representing a matrix in a hierarchical data structure instead of the standard two-dimensional array can offer significant advantages: submatrices can be compressed efficiently, different resolutions of a matrix can be handled easily, and even matrix operations like multiplication, factorization or inversion can be performed in the compressed representation, thus saving computation time and storage. \({\mathcal{H}}^{2}\)-matrices use a subdivision of the matrix into a hierarchy of submatrices in combination with a hierarchical basis, similar to a wavelet basis, to handle \(n \times n\) matrices in \(\mathcal{O}(nk)\) units of storage, where k is a parameter controlling the compression error. This chapters gives a short introduction into the basic concepts of the \({\mathcal{H}}^{2}\)-matrix method, particularly concerning the compression of arbitrary matrices.
Steffen Börm

Harmonic Field Analysis

Harmonic analysis techniques are established and successful tools in a variety of application areas, with the Fourier decomposition as one well-known example. In this chapter, we describe recent work on possible approaches to use Harmonic Analysis on fields of arbitrary type to facilitate global feature extraction and visualization. We find that a global approach is hampered by significant computational costs, and thus describe a local framework for harmonic vector field analysis to address this concern. In addition to a description of our approach, we provide a high-level overview of mathematical concepts underlying it and address practical modeling and calculation issues. As a potential application, we demonstrate the definition of empirical features based on local harmonic analysis of vector fields that reduce field data to low dimensional feature sets and offers possibilities for visualization and analysis.
Christian Wagner, Christoph Garth, Hans Hagen


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