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Modern imaging techniques and computational simulations yield complex multi-valued data that require higher-order mathematical descriptors. This book addresses topics of importance when dealing with such data, including frameworks for image processing, visualization and statistical analysis of higher-order descriptors. It also provides examples of the successful use of higher-order descriptors in specific applications and a glimpse of the next generation of diffusion MRI. To do so, it combines contributions on new developments, current challenges in this area and state-of-the-art surveys.

Compared to the increasing importance of higher-order descriptors in a range of applications, tools for analysis and processing are still relatively hard to come by. Even though application areas such as medical imaging, fluid dynamics and structural mechanics are very different in nature they face many shared challenges. This book provides an interdisciplinary perspective on this topic with contributions from key researchers in disciplines ranging from visualization and image processing to applications. It is based on the 5th Dagstuhl seminar on Visualization and Processing of Higher Order Descriptors for Multi-Valued Data.

This book will appeal to scientists who are working to develop new analysis methods in the areas of image processing and visualization, as well as those who work with applications that generate higher-order data or could benefit from higher-order models and are searching for novel analytical tools.



Mathematical Foundations


Diffusion-Weighted Magnetic Resonance Signal for General Gradient Waveforms: Multiple Correlation Function Framework, Path Integrals, and Parallels Between Them

Effects of diffusion on the magnetic resonance (MR) signal carry a wealth of information regarding the microstructure of the medium. Characterizing such effects is immensely important for quantitative studies aiming to obtain microstructural parameters using diffusion MR acquisitions. Studies in recent years have demonstrated the potential of sophisticated gradient waveforms to provide novel information inaccessible by traditional measurements. There are mainly two approaches that can be used to incorporate the influence of restricted diffusion, particularly on experiments featuring general gradient waveforms . The multiple propagator framework essentially reduces the problem to a path integral , which can be evaluated analytically or approximated via a matrix representation . The multiple correlation function method tackles the Bloch–Torrey equation , and employs an alternative matrix formulation. In this work, we present the two techniques in a unified fashion and link the two approaches. We provide an explanation for why the multiple correlation function is computationally more efficient in the case of waveforms featuring piecewise constant gradients.
Cem Yolcu, Evren Özarslan

Finslerian Diffusion and the Bloch–Torrey Equation

By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that the metric tensor is proportional to the inverse of the diffusion tensor, and this opens up a range of geometric analysis techniques. Unfortunately a number of these methods have limited applicability, as the Riemannian framework is not rich enough to capture key aspects of the tissue structure, such as fiber crossings.An extension of the Riemannian framework to the more general Finsler manifolds has been proposed in the literature as a possible alternative. The main contribution of this work is the conclusion that simply considering Brownian motion on the Finsler base manifold does not reproduce the signal model proposed in the Finslerian framework, nor lead to a model that allows the extraction of the Finslerian metric structure from the signal.
T. C. J. Dela Haije, A. Fuster, L. M. J. Florack

Fiber Orientation Distribution Functions and Orientation Tensors for Different Material Symmetries

In this paper we give closed-form expressions of the orientation tensors up to the order four associated with some axially-symmetric orientation distribution functions (ODF), including the well-known von Mises-Fisher, Watson, and de la Vallée Poussin ODFs. Each is characterized by a mean direction and a concentration parameter. Then, we use these elementary ODFs as building blocks to construct new ones with a specified material symmetry and derive the corresponding orientation tensors. For a general ODF we present a systematic way of calculating the corresponding orientation tensors from certain coefficients of the expansion of the ODF in spherical harmonics.
Maher Moakher, Peter J. Basser

Topology of 3D Linear Symmetric Tensor Fields

There has been much research in 3D symmetric tensor fields, including recent work on tensor field topology. In this book chapter, we apply these research results to the most fundamental types of 3D tensor fields, i.e., linear tensor fields, and provide some novel insights on such fields. We also propose a number of hypotheses about linear tensor fields. We hope by studying linear tensor fields, we can gain more critical insights into the topology of general 3D tensor fields in the future.
Yue Zhang, Jonathan Palacios, Eugene Zhang

Random Projections for Low Multilinear Rank Tensors

We propose two randomized tensor algorithms for reducing multilinear tensor rank. The basis of these randomized algorithms is from the work of Halko et al. (SIAM Rev 53(2):217–288, 2011). Here we provide some random versions of the higher order SVD and the higher order orthogonal iteration. Moreover, we provide a sharp probabilistic error bound for the matrix low rank approximation. In consequence, we provide an error bound for the tensor case. Moreover, we give several numerical examples which includes an implementation on a MRI dataset to test the efficacy of these randomized algorithms.
Carmeliza Navasca, Deonnia N. Pompey

Processing, Filtering and Interpolation


Path-Based Mathematical Morphology on Tensor Fields

Traditional path-based morphology allows finding long, approximately straight, paths in images. Although originally applied only to scalar images, we show how this can be a very good fit for tensor fields. We do this by constructing directed graphs representing such data, and then modifying the traditional path opening algorithm to work on these graphs. Cycles are dealt with by finding strongly connected components in the graph. Some examples of potential applications are given, including path openings that are not limited to a specific set of orientations.
Jasper J. van de Gronde, Mikola Lysenko, Jos B. T. M. Roerdink

Processing Multispectral Images via Mathematical Morphology

In this chapter, we illustrate how to process multispectral and hyperspectral images via mathematical morphology. First, according to the number of channels the data are embeded into a sufficiently high dimensional space. This transformation utilizes a special geometric structure, namely double hypersimplices, for further processing the data. For example, RGB-color images are transformed into points within a specific double hypersimplex. It is explained in detail how to calculate the supremum and infimum of samples of those transformed data to allow for the meaningful definition of morphological operations such as dilation and erosion and in a second step top hats, gradients, and morphological Laplacian. Finally, numerical results are presented to explore the advantages and shortcomings of the new proposed approach.
Andreas Kleefeld, Bernhard Burgeth

Direction-Controlled DTI Interpolation

Diffusion Tensor Imaging (DTI) is a popular model for representing diffusion weighted magnetic resonance images due to its simplicity and the fact that it strikes a good balance between signal fit and robustness. Nevertheless, problematic issues remain. One of these concerns the problem of interpolation. Because the DTI assumption forces Apparent Diffusion Coefficients (ADCs) to fit quadratic forms, destructive interference of diffusivity patterns tends to mask information on orientations. For some applications, notably tractography, one would like an interpolated DTI tensor to reflect not only some weighted average of its immediate grid neighbours, but also to preserve orientation information available at those points. This is possible if one declines from the quadratic restriction, considering general homogeneous functions of degree two instead. We show that one may interpret the interpolated ADC as a family of DTI tensors, parametrized by orientation. Any choice of a preferred direction—notably a stipulated fiber tangent—singles out a unique DTI tensor instance. Results are physically plausible and intuitive.
Luc Florack, Tom Dela Haije, Andrea Fuster

Tensor Voting: Current State, Challenges and New Trends in the Context of Medical Image Analysis

Perceptual organisation techniques aim at mimicking the human visual system for extracting salient information from noisy images. Tensor voting has been one of the most versatile of those methods, with many different applications both in computer vision and medical image analysis. Its strategy consists in propagating local information encoded through tensors by means of perception-inspired rules. Although it has been used for more than a decade, there are still many unsolved theoretical issues that have made it challenging to apply it to more problems, especially in analysis of medical images. The main aim of this chapter is to review the current state of the research in tensor voting, to summarise its present challenges, and to describe the new trends that we foresee will drive the research in this field in the next few years. Also, we discuss extensions of tensor voting that could lead to potential performance improvements and that could make it suitable for further medical applications.
Daniel Jörgens, Rodrigo Moreno



Visualization of Diffusion Propagator and Multiple Parameter Diffusion Signal

New advances in MRI technology allow the acquisition of high resolution diffusion-weighted datasets for multiple parameters such as multiple q-values, multiple b-values, multiple orientations and multiple diffusion times. These new and demanding acquisitions go beyond classical diffusion tensor imaging (DTI) and single b-value high angular resolution diffusion imaging (HARDI) acquisitions. Recent studies show that such multiple parameter diffusion can be used to infer axonal diameter distribution and other biophysical features of the white matter, otherwise not possible. Hence, this calls for novel visualization techniques to interact with such complex high-dimensional and high-resolution datasets. To date, there are no existing visualization techniques to visualize full brain images or fields of diffusion signal profiles and diffusion propagators reconstructed from them. It is important to be able to scroll in these images beyond single voxels, just as one would navigate in a whole brain map of fractional anisotropy extracted from DTI. In this chapter, we give a review of the existing visualization techniques for the local diffusion phenomenon and propose alternative visualization techniques for fields of high-dimensional 3D diffusion profiles. We introduce: (i) a volume rendering approach and (ii) a diffusion propagator silhouette glyph as a complement to existing DTI and HARDI visualization techniques. We show that these visualization techniques allow the real-time exploration of high-dimensional multi-b-value and multi-direction data such as diffusion spectrum imaging (DSI). Our visualization technique therefore opens new perspectives for 3D diffusion MRI visualization and interaction.
Olivier Vaillancourt, Maxime Chamberland, Jean-Christophe Houde, Maxime Descoteaux

Visual Knowledge Discovery for Diffusion Kurtosis Datasets of the Human Brain

Classification and visualization of structures in the human brain provide vital information to physicians who examine patients suffering from brain diseases and injuries. In particular, this information is used to recommend treatment to prevent further degeneration of the brain. Diffusion kurtosis imaging (DKI) is a new magnetic resonance imaging technique that is rapidly gaining broad interest in the medical imaging community, due to its ability to provide intricate details on the underlying microstructural characteristics of the whole brain. DKI produces a fourth-order tensor at every voxel of the imaged volume; unlike traditional diffusion tensor imaging (DTI), DKI measures the non-Gaussian property of water diffusion in biological tissues. It has shown promising results in studies on changes in grey matter and mild traumatic brain injury, a particularly difficult form of TBI to diagnose. In this paper, we use DKI imaging and report our results of the classification and visualization of various tissue types, diseases, and injuries. We evaluate segmentation performed using various clustering algorithms on different segmentation strategies including fusion of diffusion and kurtosis tensors. We compare our result to the well-known MRI segmentation technique based on Magnetization-Prepared Rapid Acquisition with Gradient Echo (MPRAGE) imaging.
Sujal Bista, Jiachen Zhuo, Rao P. Gullapalli, Amitabh Varshney

A Survey of Illustrative Visualization Techniques for Diffusion-Weighted MRI Tractography

Fiber tracking is a common method for analyzing 3D tensor fields that arise from diffusion-weighted magnetic resonance imaging. This method can visualize, e.g., the structure of the brain’s white matter or that of muscle tissue. Fiber tracking results in dense, line-based datasets that are often too large to understand when shown directly. This chapter provides a survey of recent illustrative visualization approaches that address this problem. We group this work into techniques that improve the depth perception of fiber tracts, techniques that visualize additional data about the tracts, techniques that employ focus+context visualization, visualizations of fiber tract bundles, representations of uncertainty in the context of probabilistic fiber tracking, and techniques that rely on a spatially abstracted visualization of connectivity.
Tobias Isenberg

Visualizing Symmetric Indefinite 2D Tensor Fields Using the Heat Kernel Signature

The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.
Valentin Zobel, Jan Reininghaus, Ingrid Hotz

Statistical Analysis


A Framework for the Analysis of Diffusion Compartment Imaging (DCI)

The brain microstructure consists of the complex organization of cellular structures and extra-cellular space. Insights into this microstructure can be gained in vivo by means of diffusion-weighted imaging that is sensitive to the local patterns of diffusion of water molecules throughout the brain. Diffusion compartment imaging (DCI) provides a separate parameterization for the diffusion signal arising from each compartment of water molecules at each voxel. Their use in population studies and longitudinal monitoring of diseases hold promise for unraveling alterations of the brain microstructure in various disorders and conditions. Yet, to analyze multi-compartment models, high-level operations commonly used in scalar images need to be generalized. We present a framework that enables interpolation, averaging, filtering, spatial normalization and statistical analyses of multi-compartment data with a focus on multi-tensor representations. This framework is based on the generalization of linear combinations of voxel values through mixture simplification. We illustrate the impact of this framework in registration, atlas construction, tractography and population studies.
Maxime Taquet, Benoit Scherrer, Simon K. Warfield

Statistical and Machine Learning Methods for Neuroimaging: Examples, Challenges, and Extensions to Diffusion Imaging Data

In neuroimaging research, a wide variety of quantitative computational methods enable inference of results regarding the brain’s structure and function. In this chapter, we survey two broad families of approaches to quantitative analysis of neuroimaging data: statistical testing and machine learning. We discuss how methods developed for traditional scalar structural neuroimaging data have been extended to diffusion magnetic resonance imaging data. Diffusion MRI data have higher dimensionality and allow the study of the brain’s connection structure. The intended audience of this chapter includes students or researchers in neuroimage analysis who are interested in a high-level overview of methods for analyzing their data.
Lauren J. O’Donnell, Thomas Schultz



A Clustering Method for Identifying Regions of Interest in Turbulent Combustion Tensor Fields

Production of electricity and propulsion systems involve turbulent combustion. Computational modeling of turbulent combustion can improve the efficiency of these processes. However, large tensor datasets are the result of such simulations; these datasets are difficult to visualize and analyze. In this work we present an unsupervised statistical approach for the segmentation, visualization and potentially the tracking of regions of interest in large tensor data. The approach employs a machine learning clustering algorithm to locate and identify areas of interest based on specified parameters such as strain tensor value. Evaluation on two combustion datasets shows this approach can assist in the visual analysis of the combustion tensor field.
Adrian Maries, Timothy Luciani, P. H. Pisciuneri, Mehdi B. Nik, S. Levent Yilmaz, Peyman Givi, G. Elisabeta Marai

Tensor Lines in Engineering: Success, Failure, and Open Questions

Today, product development processes in mechanical engineering are almost entirely carried out via computer-aided simulations. One essential output of these simulations are stress tensors, which are the basis for the dimensioning of the technical parts. The tensors contain information about the strength of internal stresses as well as their principal directions. However, for the analysis they are mostly reduced to scalar key metrics. The motivation of this work is to put the tensorial data more into focus of the analysis and demonstrate its potential for the product development process. In this context we resume a visualization method that has been introduced many years ago, tensor lines. Since tensor lines have been rarely used in visualization applications, they are mostly considered as physically not relevant in the visualization community. In this paper we challenge this point of view by reporting two case studies where tensor lines have been applied in the process of the design of a technical part. While the first case was a real success, we could not reach similar results for the second case. It became clear that the first case cannot be fully generalized to arbitrary settings and there are many more questions to be answered before the full potential of tensor lines can be realized. In this chapter, we review our success story and our failure case and discuss some directions of further research.
Marc Schöneich, Andrea Kratz, Valentin Zobel, Gerik Scheuermann, Markus Stommel, Ingrid Hotz

Contextual Diffusion Image Post-processing Aids Clinical Applications

Diffusion weighted magnetic resonance imaging ( dMRI) and tractography have shown great potential for the investigation of the white mater architecture in-vivo, especially with the recent advancements by using higher order techniques to model the data. Many clinical applications ranging from neurodegenerative disorders, psychiatric disorders as well as pre-surgical planning employ diffusion imaging-based analysis as an addition to conventional MRI imaging. However, despite the promising outlook, dMRI tractography confronts many challenges that complicate its use in everyday clinical practice. Some of these challenges are low test-retest accuracy, poor quantification of tracts size, poor understanding of the biological basis of the dMRI parameters, inaccuracies in the geometry of the reconstructed streamlines (especially in complex areas with curvature, bifurcations, fanning, crossings), poor alignment with the neighboring diffusion profiles, among others. Recently developed contextual processing techniques including the one presented in this work, for enhancement and well-posed geometric sharpening, have shown to result in sharper and better aligned diffusion profiles. In this paper, we present a possibility in enabling HARDI tractography on the data acquired under limited diffusion tensor imaging (DTI) conditions and modeled by DTI. We enhance local features from the DTI field using operators that take ‘context’ information into account. Moreover, we demonstrate the potential of the contextual processing techniques in two important clinical applications: enhancing the streamlines in data acquired from patients with Multiple Sclerosis (MS) and pre-surgical planning for tumor resection. For the latter, we explore the possibilities of using this framework for more accurate neurosurgical planning and evaluate our findings with a feedback from a neurosurgeon.
Vesna Prčkovska, Magí Andorrà, Pablo Villoslada, Eloy Martinez-Heras, Remco Duits, David Fortin, Paulo Rodrigues, Maxime Descoteaux


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