Superquadric glyphs for symmetric second-order tensors

Electromechanical Coupling in Electroactive Polymers - a Visual Analysis of a Third-Order Tensor Field

Electroactive polymers are frequently used in engineering applications due to their ability to change their shape and properties under the influence of an electric field. This process also works vice versa, such that mechanical deformation of the material induces an electric field in the EAP device. This specific behavior makes such materials highly attractive for the construction of actuators and sensors in various application areas. The electromechanical behaviour of electroactive polymers can be described by a third-order coupling tensor, which represents the sensitivity of mechanical stresses concerning the electric field, i.e., it establishes a relation between a second-order and a first-order tensor field. Due to this coupling tensor's complexity and the lack of meaningful visualization methods for third-order tensors in general, an interpretation of the tensor is rather difficult. Thus, the central engineering research question that this contribution deals with is a deeper understanding of electromechanical coupling by analyzing the third-order coupling tensor with the help of specific visualization methods. Starting with a deviatoric decomposition of the tensor, the multipoles of each deviator are visualized, which allows a first insight into this highly complex third-order tensor. In the present contribution, four examples, including electromechanical coupling, are simulated within a finite element framework and subsequently analyzed using the tensor visualization method.

BeauVis: A Validated Scale for Measuring the Aesthetic Pleasure of Visual Representations

We developed and validated a rating scale to assess the aesthetic pleasure (or beauty) of a visual data representation: the BeauVis scale. With our work we offer researchers and practitioners a simple instrument to compare the visual appearance of different visualizations, unrelated to data or context of use. Our rating scale can, for example, be used to accompany results from controlled experiments or be used as informative data points during in-depth qualitative studies. Given the lack of an aesthetic pleasure scale dedicated to visualizations, researchers have mostly chosen their own terms to study or compare the aesthetic pleasure of visualizations. Yet, many terms are possible and currently no clear guidance on their effectiveness regarding the judgment of aesthetic pleasure exists. To solve this problem, we engaged in a multi-step research process to develop the first validated rating scale specifically for judging the aesthetic pleasure of a visualization (osf.io/fxs76). Our final BeauVis scale consists of five items, "enjoyable," "likable," "pleasing," "nice," and "appealing." Beyond this scale itself, we contribute (a) a systematic review of the terms used in past research to capture aesthetics, (b) an investigation with visualization experts who suggested terms to use for judging the aesthetic pleasure of a visualization, and (c) a confirmatory survey in which we used our terms to study the aesthetic pleasure of a set of 3 visualizations.

A Review of Three-Dimensional Medical Image Visualization

Importance. Medical images are essential for modern medicine and an important research subject in visualization. However, medical experts are often not aware of the many advanced three-dimensional (3D) medical image visualization techniques that could increase their capabilities in data analysis and assist the decision-making process for specific medical problems. Our paper provides a review of 3D visualization techniques for medical images, intending to bridge the gap between medical experts and visualization researchers.Highlights. Fundamental visualization techniques are revisited for various medical imaging modalities, from computational tomography to diffusion tensor imaging, featuring techniques that enhance spatial perception, which is critical for medical practices. The state-of-the-art of medical visualization is reviewed based on a procedure-oriented classification of medical problems for studies of individuals and populations. This paper summarizes free software tools for different modalities of medical images designed for various purposes, including visualization, analysis, and segmentation, and it provides respective Internet links.Conclusions. Visualization techniques are a useful tool for medical experts to tackle specific medical problems in their daily work. Our review provides a quick reference to such techniques given the medical problem and modalities of associated medical images. We summarize fundamental techniques and readily available visualization tools to help medical experts to better understand and utilize medical imaging data. This paper could contribute to the joint effort of the medical and visualization communities to advance precision medicine.

An Analytical Method for Tensor Visualization in a Plane

The rendering of tensor glyphs is a progressive process of visualizing the vector space both in fluid dynamics and the latest medical scanning. Nowadays, the rendering accuracy is ensured by numerical methods based on interpolation of tensor functions. The tensor glyph functions to visualize significant properties of the vector space. Not all these properties are visualized at all times. The number of properties and their unambiguity depend on the method chosen. This work presents a direct analytical expression covering rank two tensors in a plane. Unlike the methods used so far, this method is accurate and unambiguous one for tensor visualization. The method was applied to the simplest tensor type, which presented an advantage for the method’s analytical approach. The analytical approach to the planar case is significant also because it provides instruction on how to expand analytical calculations to cover higher spatial dimensions. In this way, numerical methods for tensor rendering can be replaced with an accurate analytical method.

Visualization of 3D Stress Tensor Fields Using Superquadric Glyphs on Displacement Streamlines

Stress tensor fields play a central role in solid mechanics studies, but their visualization in 3D space remains challenging as the information-dense multi-variate tensor needs to be sampled in 3D space while avoiding clutter. Taking cues from current tensor visualizations, we adapted glyph-based visualization for stress tensors in 3D space. We also developed a testing framework and performed user studies to evaluate the various glyph-based tensor visualizations for objective accuracy measures, and subjective user feedback for each visualization method. To represent the stress tensor, we color encoded the original superquadric glyph, and in the user study, we compared it to superquadric glyphs developed for second-order symmetric tensors. We found that color encoding improved the user accuracy measures, while the users also rated our method the highest. We compared our method of placing stress tensor glyphs on displacement streamlines to the glyph placement on a 3D grid. In the visualization, we modified the glyph to show both the stress tensor and the displacement vector at each sample point. The participants preferred our method of glyph placement on displacement streamlines as it highlighted the underlying continuous structure in the tensor field.

Visualization of the Anisotropy of the Velocity Dispersion and Characteristics of the Multi-Velocity Continuum in the Regions of Multi-Stream Flows of Gas-Dust Media with Polydisperse Dust

Collisionless media devoid of intrinsic stresses, for example, a dispersed phase in a multiphase medium, have a much wider variety of space-time structures and features formed in them than collisional media, for example, a carrier, gas, or liquid phase. This is a consequence of the fact that evolution in such media occurs in phase space, i.e., in a space of greater dimensions than the usual coordinate space. As a consequence, the process of the formation of features in collisionless media (clustering or vice versa, a loss of continuity) can occur primarily in the velocity space, which, in contrast to the features in the coordinate space (folds, caustics, or voids), is poorly observed directly. To identify such features, it is necessary to use visualization methods that allow us to consider, in detail, the evolution of the medium in the velocity space. This article is devoted to the development of techniques that allow visualizing the degree of anisotropy of the velocity fields of collisionless interpenetrating media. Simultaneously tracking the behavior of different fractions in such media is important, as their behavior can be significantly different. We propose three different techniques for visualizing the anisotropy of velocity fields using the example of two- and three-continuum dispersed media models. We proposed the construction of spatial distributions of eccentricity fields (scalar fields), or fields of principal directions of the velocity dispersion tensor (tensor fields). In the first case, we used some simple eccentricity functions for dispersion tensors for two fractions simultaneously, which we call surrogate entropy. In the second case, to visualize the anisotropy of the velocity fields of three fractions simultaneously, we used an ordered array (3-vector) of eccentricities for the color representation through decomposition in three basic colors. In the case of a multi-stream flow, we used cluster analysis methods to identify sections of a multi-stream flow (beams) and used glyphs to visualize the entire set of beams (vector-tensor fields).

Multi-Scale Topological Analysis of Asymmetric Tensor Fields on Surfaces

Asymmetric tensor fields have found applications in many science and engineering domains, such as fluid dynamics. Recent advances in the visualization and analysis of 2D asymmetric tensor fields focus on pointwise analysis of the tensor field and effective visualization metaphors such as colors, glyphs, and hyperstreamlines. In this paper, we provide a novel multi-scale topological analysis framework for asymmetric tensor fields on surfaces. Our multi-scale framework is based on the notions of eigenvalue and eigenvector graphs. At the core of our framework are the identification of atomic operations that modify the graphs and the scale definition that guides the order in which the graphs are simplified to enable clarity and focus for the visualization of topological analysis on data of different sizes. We also provide efficient algorithms to realize these operations. Furthermore, we provide physical interpretation of these graphs. To demonstrate the utility of our system, we apply our multi-scale analysis to data in computational fluid dynamics.

Cardiac Diffusion: Technique and Practical Applications

The 3D microarchitecture of the cardiac muscle underlies the mechanical and electrical properties of the heart. Cardiomyocytes are arranged helically through the depth of the wall, and their shortening leads to macroscopic torsion, twist, and shortening during cardiac contraction. Furthermore, cardiomyocytes are organized in sheetlets separated by shear layers, which reorientate, slip, and shear during macroscopic left ventricle (LV) wall thickening. Cardiac diffusion provides a means for noninvasive interrogation of the 3D microarchitecture of the myocardium. The fundamental principle of MR diffusion is that an MRI signal is attenuated by the self-diffusion of water in the presence of large diffusion-encoding gradients. Since water molecules are constrained by the boundaries in biological tissue (cell membranes, collagen layers, etc.), depicting their diffusion behavior elucidates the shape of the myocardial microarchitecture they are embedded in. Cardiac diffusion therefore provides a noninvasive means to understand not only the dynamic changes in cardiac microstructure of healthy myocardium during cardiac contraction but also the pathophysiological changes in the presence of disease. This unique and innovative technology offers tremendous potential to enable improved clinical diagnosis through novel microstructural and functional assessment. in vivo cardiac diffusion methods are immediately translatable to patients, opening new avenues for diagnostic investigation and treatment evaluation in a range of clinically important cardiac pathologies. This review article describes the 3D microstructure of the LV, explains in vivo and ex vivo cardiac MR diffusion acquisition and postprocessing techniques, as well as clinical applications to date. Level of Evidence: 1 Technical Efficacy: Stage 3 J. Magn. Reson. Imaging 2019. J. Magn. Reson. Imaging 2020;52:348-368.

Diffusion MRI visualization

Modern diffusion magnetic resonance imaging (dMRI) acquires intricate volume datasets and biological meaning can only be found in the relationship between its different measurements. Suitable strategies for visualizing these complicated data have been key to interpretation by physicians and neuroscientists, for drawing conclusions on brain connectivity and for quality control. This article provides an overview of visualization solutions that have been proposed to date, ranging from basic grayscale and color encodings to glyph representations and renderings of fiber tractography. A particular focus is on ongoing and possible future developments in dMRI visualization, including comparative, uncertainty, interactive and dense visualizations.

Revisiting Superquadric Fitting: A Numerically Stable Formulation

Superquadric surfaces play an important role in many different research fields due to their ability to model a variety of shapes with a small number of parameters. One of their core applications is the estimation of object shape characteristics from a set of discrete samples from the surface of an object. However, the corresponding optimization problem is prone to numerical instabilities in some regions of the parameter space. To mitigate this problem, lower bound constraints to the shape parameters are applied during the optimization thus limiting the range of shapes which can be accurately represented by superquadrics. Therefore, the exact modeling of very common shapes like cuboids and cylinders is error-prone in practice. In this article we investigate this problem and provide a numerically stable formulation for the evaluation of the superquadric surface function and for its gradient. This new formulation enables numerically stable fitting of superquadrics in the previously constrained region, i.e., in its full range including cuboids and cylinders. In addition, the new formulation also leads to faster convergence speed. The theoretical contributions are substantiated by experiments on synthetic as well as real data.

ENIGMA-Viewer: interactive visualization strategies for conveying effect sizes in meta-analysis

BACKGROUND

Global scale brain research collaborations such as the ENIGMA (Enhancing Neuro Imaging Genetics through Meta Analysis) consortium are beginning to collect data in large quantity and to conduct meta-analyses using uniformed protocols. It becomes strategically important that the results can be communicated among brain scientists effectively. Traditional graphs and charts failed to convey the complex shapes of brain structures which are essential to the understanding of the result statistics from the analyses. These problems could be addressed using interactive visualization strategies that can link those statistics with brain structures in order to provide a better interface to understand brain research results.

RESULTS

We present ENIGMA-Viewer, an interactive web-based visualization tool for brain scientists to compare statistics such as effect sizes from meta-analysis results on standardized ROIs (regions-of-interest) across multiple studies. The tool incorporates visualization design principles such as focus+context and visual data fusion to enable users to better understand the statistics on brain structures. To demonstrate the usability of the tool, three examples using recent research data are discussed via case studies.

CONCLUSIONS

ENIGMA-Viewer supports presentations and communications of brain research results through effective visualization designs. By linking visualizations of both statistics and structures, users can gain more insights into the presented data that are otherwise difficult to obtain. ENIGMA-Viewer is an open-source tool, the source code and sample data are publicly accessible through the NITRC website ( http://www.nitrc.org/projects/enigmaviewer_20 ). The tool can also be directly accessed online ( http://enigma-viewer.org ).

Diffusion MRI in the heart

Diffusion MRI provides unique information on the structure, organization, and integrity of the myocardium without the need for exogenous contrast agents. Diffusion MRI in the heart, however, has proven technically challenging because of the intrinsic non-rigid deformation during the cardiac cycle, displacement of the myocardium due to respiratory motion, signal inhomogeneity within the thorax, and short transverse relaxation times. Recently developed accelerated diffusion-weighted MR acquisition sequences combined with advanced post-processing techniques have improved the accuracy and efficiency of diffusion MRI in the myocardium. In this review, we describe the solutions and approaches that have been developed to enable diffusion MRI of the heart in vivo, including a dual-gated stimulated echo approach, a velocity- (M1 ) or an acceleration- (M2 ) compensated pulsed gradient spin echo approach, and the use of principal component analysis filtering. The structure of the myocardium and the application of these techniques in ischemic heart disease are also briefly reviewed. The advent of clinical MR systems with stronger gradients will likely facilitate the translation of cardiac diffusion MRI into clinical use. The addition of diffusion MRI to the well-established set of cardiovascular imaging techniques should lead to new and complementary approaches for the diagnosis and evaluation of patients with heart disease. © 2015 The Authors. NMR in Biomedicine published by John Wiley & Sons Ltd.

GlyphLens: View-Dependent Occlusion Management in the Interactive Glyph Visualization

Glyph as a powerful multivariate visualization technique is used to visualize data through its visual channels. To visualize 3D volumetric dataset, glyphs are usually placed on 2D surface, such as the slicing plane or the feature surface, to avoid occluding each other. However, the 3D spatial structure of some features may be missing. On the other hand, placing large number of glyphs over the entire 3D space results in occlusion and visual clutter that make the visualization ineffective. To avoid the occlusion, we propose a view-dependent interactive 3D lens that removes the occluding glyphs by pulling the glyphs aside through the animation. We provide two space deformation models and two lens shape models to displace the glyphs based on their spatial distributions. After the displacement, the glyphs around the user-interested region are still visible as the context information, and their spatial structures are preserved. Besides, we attenuate the brightness of the glyphs inside the lens based on their depths to provide more depth cue. Furthermore, we developed an interactive glyph visualization system to explore different glyph-based visualization applications. In the system, we provide a few lens utilities that allows users to pick a glyph or a feature and look at it from different view directions. We compare different display/interaction techniques to visualize/manipulate our lens and glyphs.

Glyphs for General Second-Order 2D and 3D Tensors

Glyphs are a powerful tool for visualizing second-order tensors in a variety of scientic data as they allow to encode physical behavior in geometric properties. Most existing techniques focus on symmetric tensors and exclude non-symmetric tensors where the eigenvectors can be non-orthogonal or complex. We present a new construction of 2d and 3d tensor glyphs based on piecewise rational curves and surfaces with the following properties: invariance to (a) isometries and (b) scaling, (c) direct encoding of all real eigenvalues and eigenvectors, (d) one-to-one relation between the tensors and glyphs, (e) glyph continuity under changing the tensor. We apply the glyphs to visualize the Jacobian matrix fields of a number of 2d and 3d vector fields.

Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.

Diffusion Tensor Imaging in Patients with Glioblastoma Multiforme Using the Supertoroidal Model

Purpose

Diffusion Tensor Imaging (DTI) is a powerful imaging technique that has led to improvements in the diagnosis and prognosis of cerebral lesions and neurosurgical guidance for tumor resection. Traditional tensor modeling, however, has difficulties in differentiating tumor-infiltrated regions and peritumoral edema. Here, we describe the supertoroidal model, which incorporates an increase in surface genus and a continuum of toroidal shapes to improve upon the characterization of Glioblastoma multiforme (GBM).

Materials and Methods

DTI brain datasets of 18 individuals with GBM and 18 normal subjects were acquired using a 3T scanner. A supertoroidal model of the diffusion tensor and two new diffusion tensor invariants, one to evaluate diffusivity, the toroidal volume (TV), and one to evaluate anisotropy, the toroidal curvature (TC), were applied and evaluated in the characterization of GBM brain tumors. TV and TC were compared with the mean diffusivity (MD) and fractional anisotropy (FA) indices inside the tumor, surrounding edema, as well as contralateral to the lesions, in the white matter (WM) and gray matter (GM).

Results

The supertoroidal model enhanced the borders between tumors and surrounding structures, refined the boundaries between WM and GM, and revealed the heterogeneity inherent to tumor-infiltrated tissue. Both MD and TV demonstrated high intensities in the tumor, with lower values in the surrounding edema, which in turn were higher than those of unaffected brain parenchyma. Both TC and FA were effective in revealing the structural degradation of WM tracts.

Conclusions

Our findings indicate that the supertoroidal model enables effective tensor visualization as well as quantitative scalar maps that improve the understanding of the underlying tissue structure properties. Hence, this approach has the potential to enhance diagnosis, preoperative planning, and intraoperative image guidance during surgical management of brain lesions.

Visualizing Tensor Normal Distributions at Multiple Levels of Detail

Despite the widely recognized importance of symmetric second order tensor fields in medicine and engineering, the visualization of data uncertainty in tensor fields is still in its infancy. A recently proposed tensorial normal distribution, involving a fourth order covariance tensor, provides a mathematical description of how different aspects of the tensor field, such as trace, anisotropy, or orientation, vary and covary at each point. However, this wealth of information is far too rich for a human analyst to take in at a single glance, and no suitable visualization tools are available. We propose a novel approach that facilitates visual analysis of tensor covariance at multiple levels of detail. We start with a visual abstraction that uses slice views and direct volume rendering to indicate large-scale changes in the covariance structure, and locations with high overall variance. We then provide tools for interactive exploration, making it possible to drill down into different types of variability, such as in shape or orientation. Finally, we allow the analyst to focus on specific locations of the field, and provide tensor glyph animations and overlays that intuitively depict confidence intervals at those points. Our system is demonstrated by investigating the effects of measurement noise on diffusion tensor MRI, and by analyzing two ensembles of stress tensor fields from solid mechanics.

Glyph-Based Comparative Visualization for Diffusion Tensor Fields

Diffusion Tensor Imaging (DTI) is a magnetic resonance imaging modality that enables the in-vivo reconstruction and visualization of fibrous structures. To inspect the local and individual diffusion tensors, glyph-based visualizations are commonly used since they are able to effectively convey full aspects of the diffusion tensor. For several applications it is necessary to compare tensor fields, e.g., to study the effects of acquisition parameters, or to investigate the influence of pathologies on white matter structures. This comparison is commonly done by extracting scalar information out of the tensor fields and then comparing these scalar fields, which leads to a loss of information. If the glyph representation is kept, simple juxtaposition or superposition can be used. However, neither facilitates the identification and interpretation of the differences between the tensor fields. Inspired by the checkerboard style visualization and the superquadric tensor glyph, we design a new glyph to locally visualize differences between two diffusion tensors by combining juxtaposition and explicit encoding. Because tensor scale, anisotropy type, and orientation are related to anatomical information relevant for DTI applications, we focus on visualizing tensor differences in these three aspects. As demonstrated in a user study, our new glyph design allows users to efficiently and effectively identify the tensor differences. We also apply our new glyphs to investigate the differences between DTI datasets of the human brain in two different contexts using different b-values, and to compare datasets from a healthy and HIV-infected subject.

Visualization of Brain Microstructure Through Spherical Harmonics Illumination of High Fidelity Spatio-Angular Fields

Diffusion kurtosis imaging (DKI) is gaining rapid adoption in the medical imaging community due to its ability to measure the non-Gaussian property of water diffusion in biological tissues. Compared to traditional diffusion tensor imaging (DTI), DKI can provide additional details about the underlying microstructural characteristics of the neural tissues. It has shown promising results in studies on changes in gray matter and mild traumatic brain injury where DTI is often found to be inadequate. The DKI dataset, which has high-fidelity spatio-angular fields, is difficult to visualize. Glyph-based visualization techniques are commonly used for visualization of DTI datasets; however, due to the rapid changes in orientation, lighting, and occlusion, visually analyzing the much more higher fidelity DKI data is a challenge. In this paper, we provide a systematic way to manage, analyze, and visualize high-fidelity spatio-angular fields from DKI datasets, by using spherical harmonics lighting functions to facilitate insights into the brain microstructure.

An Algebraic Process for Visualization Design

We present a model of visualization design based on algebraic considerations of the visualization process. The model helps characterize visual encodings, guide their design, evaluate their effectiveness, and highlight their shortcomings. The model has three components: the underlying mathematical structure of the data or object being visualized, the concrete representation of the data in a computer, and (to the extent possible) a mathematical description of how humans perceive the visualization. Because we believe the value of our model lies in its practical application, we propose three general principles for good visualization design. We work through a collection of examples where our model helps explain the known properties of existing visualizations methods, both good and not-so-good, as well as suggesting some novel methods. We describe how to use the model alongside experimental user studies, since it can help frame experiment outcomes in an actionable manner. Exploring the implications and applications of our model and its design principles should provide many directions for future visualization research.

Connexel visualization: a software implementation of glyphs and edge-bundling for dense connectivity data using brainGL

The visualization of brain connectivity becomes progressively more challenging as analytic and computational advances begin to facilitate connexel-wise analyses, which include all connections between pairs of voxels. Drawing full connectivity graphs can result in depictions that, rather than illustrating connectivity patterns in more detail, obfuscate patterns owing to the data density. In an effort to expand the possibilities for visualization, we describe two approaches for presenting connexels: edge-bundling, which clarifies structure by grouping geometrically similar connections; and, connectivity glyphs, which depict a condensed connectivity map at each point on the cortical surface. These approaches can be applied in the native brain space, facilitating interpretation of the relation of connexels to brain anatomy. The tools have been implemented as part of brainGL, an extensive open-source software for the interactive exploration of structural and functional brain data.

Three-dimensional mean-shift edge bundling for the visualization of functional connectivity in the brain

Functional connectivity, a flourishing new area of research in human neuroscience, carries a substantial challenge for visualization: while the end points of connectivity are known, the precise path between them is not. Although a large body of work already exists on the visualization of anatomical connectivity, the functional counterpart lacks similar development. To optimize the clarity of whole-brain and complex connectivity patterns in three-dimensional brain space, we develop mean-shift edge bundling, which reveals the multitude of connections as derived from correlations in the brain activity of cortical regions.

Open-box spectral clustering: applications to medical image analysis

Spectral clustering is a powerful and versatile technique, whose broad range of applications includes 3D image analysis. However, its practical use often involves a tedious and time-consuming process of tuning parameters and making application-specific choices. In the absence of training data with labeled clusters, help from a human analyst is required to decide the number of clusters, to determine whether hierarchical clustering is needed, and to define the appropriate distance measures, parameters of the underlying graph, and type of graph Laplacian. We propose to simplify this process via an open-box approach, in which an interactive system visualizes the involved mathematical quantities, suggests parameter values, and provides immediate feedback to support the required decisions. Our framework focuses on applications in 3D image analysis, and links the abstract high-dimensional feature space used in spectral clustering to the three-dimensional data space. This provides a better understanding of the technique, and helps the analyst predict how well specific parameter settings will generalize to similar tasks. In addition, our system supports filtering outliers and labeling the final clusters in such a way that user actions can be recorded and transferred to different data in which the same structures are to be found. Our system supports a wide range of inputs, including triangular meshes, regular grids, and point clouds. We use our system to develop segmentation protocols in chest CT and brain MRI that are then successfully applied to other datasets in an automated manner.

A systematic review on the practice of evaluating visualization

We present an assessment of the state and historic development of evaluation practices as reported in papers published at the IEEE Visualization conference. Our goal is to reflect on a meta-level about evaluation in our community through a systematic understanding of the characteristics and goals of presented evaluations. For this purpose we conducted a systematic review of ten years of evaluations in the published papers using and extending a coding scheme previously established by Lam et al. [2012]. The results of our review include an overview of the most common evaluation goals in the community, how they evolved over time, and how they contrast or align to those of the IEEE Information Visualization conference. In particular, we found that evaluations specific to assessing resulting images and algorithm performance are the most prevalent (with consistently 80-90% of all papers since 1997). However, especially over the last six years there is a steady increase in evaluation methods that include participants, either by evaluating their performances and subjective feedback or by evaluating their work practices and their improved analysis and reasoning capabilities using visual tools. Up to 2010, this trend in the IEEE Visualization conference was much more pronounced than in the IEEE Information Visualization conference which only showed an increasing percentage of evaluation through user performance and experience testing. Since 2011, however, also papers in IEEE Information Visualization show such an increase of evaluations of work practices and analysis as well as reasoning using visual tools. Further, we found that generally the studies reporting requirements analyses and domain-specific work practices are too informally reported which hinders cross-comparison and lowers external validity.

Uncertainty Visualization in HARDI based on Ensembles of ODFs

In this paper, we propose a new and accurate technique for uncertainty analysis and uncertainty visualization based on fiber orientation distribution function (ODF) glyphs, associated with high angular resolution diffusion imaging (HARDI). Our visualization applies volume rendering techniques to an ensemble of 3D ODF glyphs, which we call SIP functions of diffusion shapes, to capture their variability due to underlying uncertainty. This rendering elucidates the complex heteroscedastic structural variation in these shapes. Furthermore, we quantify the extent of this variation by measuring the fraction of the volume of these shapes, which is consistent across all noise levels, the certain volume ratio. Our uncertainty analysis and visualization framework is then applied to synthetic data, as well as to HARDI human-brain data, to study the impact of various image acquisition parameters and background noise levels on the diffusion shapes.

Reproducibility of in-vivo diffusion tensor cardiovascular magnetic resonance in hypertrophic cardiomyopathy

BACKGROUND

Myocardial disarray is an important histological feature of hypertrophic cardiomyopathy (HCM) which has been studied post-mortem, but its in-vivo prevalence and extent is unknown. Cardiac Diffusion Tensor Imaging (cDTI) provides information on mean intravoxel myocyte orientation and potentially myocardial disarray. Recent technical advances have improved in-vivo cDTI, and the aim of this study was to assess the interstudy reproducibility of quantitative in-vivo cDTI in patients with HCM.

METHODS AND RESULTS

A stimulated-echo single-shot-EPI sequence with zonal excitation and parallel imaging was implemented. Ten patients with HCM were each scanned on 2 different days. For each scan 3 short axis mid-ventricular slices were acquired with cDTI at end systole. Fractional anisotropy (FA), mean diffusivity (MD), and helix angle (HA) maps were created using a cDTI post-processing platform developed in-house. The mean ± SD global FA was 0.613 ± 0.044, MD was 0.750 ± 0.154 × 10-3 mm2/s and HA was epicardium -34.3 ± 7.6°, mesocardium 3.5 ± 6.9° and endocardium 38.9 ± 8.1°. Comparison of initial and repeat studies showed global interstudy reproducibility for FA (SD = ± 0.045, Coefficient of Variation (CoV) = 7.2%), MD (SD = ± 0.135 × 10-3 mm2/s, CoV = 18.6%) and HA (epicardium SD = ± 4.8°; mesocardium SD = ± 3.4°; endocardium SD = ± 2.9°). Reproducibility of FA was superior to MD (p = 0.003). MD was significantly higher in the septum than the reference lateral wall (0.784 ±0.188 vs 0.714 ±0.155 ×10-3 mm2/s, p <0.001) [corrected]. Septal HA was significantly lower than the reference lateral wall in all 3 transmural layers (from -8.3° to -10.4°, all p < 0.001).

CONCLUSIONS

To the best of our knowledge, this is the first study to assess the interstudy reproducibility of DTI in the human HCM heart in-vivo and the largest cDTI study in HCM to date. Our results show good reproducibility of FA, MD and HA which indicates that current technology yields robust in-vivo measurements that have potential clinical value. The interpretation of regional differences in the septum requires further investigation.

Asymmetric tensor field visualization for surfaces

Asymmetric tensor field visualization can provide important insight into fluid flows and solid deformations. Existing techniques for asymmetric tensor fields focus on the analysis, and simply use evenly-spaced hyperstreamlines on surfaces following eigenvectors and dual-eigenvectors in the tensor field. In this paper, we describe a hybrid visualization technique in which hyperstreamlines and elliptical glyphs are used in real and complex domains, respectively. This enables a more faithful representation of flow behaviors inside complex domains. In addition, we encode tensor magnitude, an important quantity in tensor field analysis, using the density of hyperstreamlines and sizes of glyphs. This allows colors to be used to encode other important tensor quantities. To facilitate quick visual exploration of the data from different viewpoints and at different resolutions, we employ an efficient image-space approach in which hyperstreamlines and glyphs are generated quickly in the image plane. The combination of these techniques leads to an efficient tensor field visualization system for domain scientists. We demonstrate the effectiveness of our visualization technique through applications to complex simulated engine fluid flow and earthquake deformation data. Feedback from domain expert scientists, who are also co-authors, is provided.

GPU-Based Ray-Casting of Spherical Functions Applied to High Angular Resolution Diffusion Imaging

Abstract-Any sufficiently smooth, positive, real-valued function ψ : S(2) → K+ on a sphere S(2) can be expanded by a Laplace expansion into a sum of spherical harmonics. Given the Laplace expansion coefficients, we provide a CPU and GPU-based algorithm that renders the radial graph of ψ in a fast and efficient way by ray-casting the glyph of ψ in the fragment shader of a GPU. The proposed rendering algorithm has proven highly useful in the visualization of high angular resolution diffusion imaging (HARDI) data. Our implementation of the rendering algorithm can display simultaneously thousands of glyphs depicting the local diffusivity of water. The rendering is fast enough to allow for interactive manipulation of large HARDI data sets.

Visualizing the semantic structure in classical music works

A major obstacle in the appreciation of classical music is that extensive training is required to understand musical structure and compositional techniques toward comprehending the thoughts behind the musical work. In this paper, we propose an innovative visualization solution to reveal the semantic structure in classical orchestral works such that users can gain insights into musical structure and appreciate the beauty of music. We formulate the semantic structure into macrolevel layer interactions, microlevel theme variations, and macro-micro relationships between themes and layers to abstract the complicated construction of a musical composition. The visualization has been applied with success in understanding some classical music works as supported by highly promising user study results with the general audience and very positive feedback from music students and experts, demonstrating its effectiveness in conveying the sophistication and beauty of classical music to novice users with informative and intuitive displays.

Sampling and visualizing creases with scale-space particles

Particle systems have gained importance as a methodology for sampling implicit surfaces and segmented objects to improve mesh generation and shape analysis. We propose that particle systems have a significantly more general role in sampling structure from unsegmented data. We describe a particle system that computes samplings of crease features (i.e. ridges and valleys, as lines or surfaces) that effectively represent many anatomical structures in scanned medical data. Because structure naturally exists at a range of sizes relative to the image resolution, computer vision has developed the theory of scale-space, which considers an n-D image as an (n+1)-D stack of images at different blurring levels. Our scale-space particles move through continuous four-dimensional scale-space according to spatial constraints imposed by the crease features, a particle-image energy that draws particles towards scales of maximal feature strength, and an inter-particle energy that controls sampling density in space and scale. To make scale-space practical for large three-dimensional data, we present a spline-based interpolation across scale from a small number of pre-computed blurrings at optimally selected scales. The configuration of the particle system is visualized with tensor glyphs that display information about the local Hessian of the image, and the scale of the particle. We use scale-space particles to sample the complex three-dimensional branching structure of airways in lung CT, and the major white matter structures in brain DTI.

Stress tensor field visualization for implant planning in orthopedics

We demonstrate the application of advanced 3D visualization techniques to determine the optimal implant design and position in hip joint replacement planning. Our methods take as input the physiological stress distribution inside a patient's bone under load and the stress distribution inside this bone under the same load after a simulated replacement surgery. The visualization aims at showing principal stress directions and magnitudes, as well as differences in both distributions. By visualizing changes of normal and shear stresses with respect to the principal stress directions of the physiological state, a comparative analysis of the physiological stress distribution and the stress distribution with implant is provided, and the implant parameters that most closely replicate the physiological stress state in order to avoid stress shielding can be determined. Our method combines volume rendering for the visualization of stress magnitudes with the tracing of short line segments for the visualization of stress directions. To improve depth perception, transparent, shaded, and antialiased lines are rendered in correct visibility order, and they are attenuated by the volume rendering. We use a focus+context approach to visually guide the user to relevant regions in the data, and to support a detailed stress analysis in these regions while preserving spatial context information. Since all of our techniques have been realized on the GPU, they can immediately react to changes in the simulated stress tensor field and thus provide an effective means for optimal implant selection and positioning in a computational steering environment.

Anisotropic noise samples

We present a practical approach to generate stochastic anisotropic samples with Poisson-disk characteristic over a two-dimensional domain. In contrast to isotropic samples, we understand anisotropic samples as non-overlapping ellipses whose size and density match a given anisotropic metric. Anisotropic noise samples are useful for many visualization and graphics applications. The spot samples can be used as input for texture generation, e.g., line integral convolution (LIC), but can also be used directly for visualization. The definition of the spot samples using a metric tensor makes them especially suitable for the visualization of tensor fields that can be translated into a metric. Our work combines ideas from sampling theory and mesh generation. To generate these samples with the desired properties we construct a first set of non-overlapping ellipses whose distribution closely matches the underlying metric. This set of samples is used as input for a generalized anisotropic Lloyd relaxation to distribute noise samples more evenly. Instead of computing the Voronoi tessellation explicitly, we introduce a discrete approach which combines the Voronoi cell and centroid computation in one step. Our method supports automatic packing of the elliptical samples, resulting in textures similar to those generated by anisotropic reaction-diffusion methods. We use Fourier analysis tools for quality measurement of uniformly distributed samples. The resulting samples have nice sampling properties, for example, they satisfy a blue noise property where low frequencies in the power spectrum are reduced to a minimum.

Diffusion tensor analysis with invariant gradients and rotation tangents

Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.

Interactive visualization of volumetric white matter connectivity in DT-MRI using a parallel-hardware Hamilton-Jacobi solver

In this paper we present a method to compute and visualize volumetric white matter connectivity in diffusion tensor magnetic resonance imaging (DT-MRI) using a Hamilton-Jacobi (H-J) solver on the GPU (Graphics Processing Unit). Paths through the volume are assigned costs that are lower if they are consistent with the preferred diffusion directions. The proposed method finds a set of voxels in the DTI volume that contain paths between two regions whose costs are within a threshold of the optimal path. The result is a volumetric optimal path analysis, which is driven by clinical and scientific questions relating to the connectivity between various known anatomical regions of the brain. To solve the minimal path problem quickly, we introduce a novel numerical algorithm for solving H-J equations, which we call the Fast Iterative Method (FIM). This algorithm is well-adapted to parallel architectures, and we present a GPU-based implementation, which runs roughly 50-100 times faster than traditional CPU-based solvers for anisotropic H-J equations. The proposed system allows users to freely change the endpoints of interesting pathways and to visualize the optimal volumetric path between them at an interactive rate. We demonstrate the proposed method on some synthetic and real DT-MRI datasets and compare the performance with existing methods.

Stochastic DT-MRI connectivity mapping on the GPU

We present a method for stochastic fiber tract mapping from diffusion tensor MRI (DT-MRI) implemented on graphics hardware. From the simulated fibers we compute a connectivity map that gives an indication of the probability that two points in the dataset are connected by a neuronal fiber path. A Bayesian formulation of the fiber model is given and it is shown that the inversion method can be used to construct plausible connectivity. An implementation of this fiber model on the graphics processing unit (GPU) is presented. Since the fiber paths can be stochastically generated independently of one another, the algorithm is highly parallelizable. This allows us to exploit the data-parallel nature of the GPU fragment processors. We also present a framework for the connectivity computation on the GPU. Our implementation allows the user to interactively select regions of interest and observe the evolving connectivity results during computation. Results are presented from the stochastic generation of over 250,000 fiber steps per iteration at interactive frame rates on consumer-grade graphics hardware.

Bilateral filtering of diffusion tensor magnetic resonance images

We extend the well-known scalar image bilateral filtering technique to diffusion tensor magnetic resonance images (DTMRI). The scalar version of bilateral image filtering is extended to perform edge-preserving smoothing of DT field data. The bilateral DT filtering is performed in the Log-Euclidean framework which guarantees valid output tensors. Smoothing is achieved by weighted averaging of neighboring tensors. Analogous to bilateral filtering of scalar images, the weights are chosen to be inversely proportional to two distance measures: The geometrical Euclidean distance between the spatial locations of tensors and the dissimilarity of tensors. We describe the noniterative DT smoothing equation in closed form and show how interpolation of DT data is treated as a special case of bilateral filtering where only spatial distance is used. We evaluate different recent DT tensor dissimilarity metrics including the Log-Euclidean, the similarity-invariant Log-Euclidean, the square root of the J-divergence, and the distance scaled mutual diffusion coefficient. We present qualitative and quantitative smoothing and interpolation results and show their effect on segmentation, for both synthetic DT field data, as well as real cardiac and brain DTMRI data.

Diffusion tensor visualization with glyph packing

A common goal of multivariate visualization is to enable data inspection at discrete points, while also illustrating larger-scale continuous structures. In diffusion tensor visualization, glyphs are typically used to meet the first goal, and methods such as texture synthesis or fiber tractography can address the second. We adapt particle systems originally developed for surface modeling and anisotropic mesh generation to enhance the utility of glyph-based tensor visualizations. By carefully distributing glyphs throughout the field (either on a slice, or in the volume) into a dense packing, using potential energy profiles shaped by the local tensor value, we remove undue visual emphasis of the regular sampling grid of the data, and the underlying continuous features become more apparent. The method is demonstrated on a DT-MRI scan of a patient with a brain tumor.

Visualization of geologic stress perturbations using Mohr diagrams

Huge salt formations, trapping large untapped oil and gas reservoirs, lie in the deepwater region of the Gulf of Mexico. Drilling in this region is high-risk and drilling failures have led to well abandonments, with each costing tens of millions of dollars. Salt tectonics plays a central role in these failures. To explore the geomechanical interactions between salt and the surrounding sand and shale formations, scientists have simulated the stresses in and around salt diapirs in the Gulf of Mexico using nonlinear finite element geomechanical modeling. In this paper, we describe novel techniques developed to visualize the simulated subsurface stress field. We present an adaptation of the Mohr diagram, a traditional paper-and-pencil graphical method long used by the material mechanics community for estimating coordinate transformations for stress tensors, as a new tensor glyph for dynamically exploring tensor variables within three-dimensional finite element models. This interactive glyph can be used as either a probe or a filter through brushing and linking.