This paper presents the state of the art in the area of topology-based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.

(a) (b) (c) Fig. 1. Three views of the tornado data set. (a) Arrows pointing along streamlines of velocity reveal the basic structure of the tornado. However, we are primarily interested in its vortex core, and specifically, where the vortex core resembles a Gaussian vortex. (b) The Q-criterion for the tornado data set, as expected, shows a vortical interior surrounded by a strain cell, where the spinning air of the tornado shears against the calmer air outside. The vortex core is brought out in the strong negative domain of Q, but an additional vortical funnel appears in the upper half of the tornado. (c) For our method, we compute the similarity of different Q thresholds to an idealized Gaussian vortex. The similarity holds well on the interior, where the flow of air resembles an idealized vortex, but quickly decays outside the core. Additionally, the funnel on top is not modeled well by Gaussian vorticity (except near the interface between the funnel and the main core), so it also has a low fit value and is excluded.Abstract-We consider the problem of extracting discrete two-dimensional vortices from a turbulent flow. In our approach we use a reference model describing the expected physics and geometry of an idealized vortex. The model allows us to derive a novel correlation between the size of the vortex and its strength, measured as the square of its strain minus the square of its vorticity. For vortex detection in real models we use the strength parameter to locate potential vortex cores, then measure the similarity of our ideal analytical vortex and the real vortex core for different strength thresholds. This approach provides a metric for how well a vortex core is modeled by an ideal vortex. Moreover, this provides insight into the problem of choosing the thresholds that identify a vortex. By selecting a target coefficient of determination (i.e., statistical confidence), we determine on a per-vortex basis what threshold of the strength parameter would be required to extract that vortex at the chosen confidence. We validate our approach on real data from a global ocean simulation and derive from it a map of expected vortex strengths over the global ocean.

The coordination of cell movements across spatio-temporal scales ensures precise positioning of organs during vertebrate gastrulation. Mechanisms governing such morphogenetic movements have been studied only within a local region, a single germlayer or in whole embryos without cell identity. Scale-bridging imaging and automated analysis of cell dynamics are needed for a deeper understanding of tissue formation during gastrulation. Here, we report pan-embryo analyses of formation and dynamics of all three germlayers simultaneously within a developing zebrafish embryo. We show that a distinct distribution of cells in each germlayer is established during early gastrulation via cell movement characteristics that are predominantly determined by their position in the embryo. The differences in initial germlayer distributions are subsequently amplified by a global movement, which organizes the organ precursors along the embryonic body axis, giving rise to the blueprint of organ formation. The tools and data are available as a resource for the community.

Understanding fluid flow data, especially vortices, is still a challenging task. Sophisticated visualization tools help to gain insight. In this paper, we present a novel approach for the interactive comparison of scalar fields using isosurfaces, and its application to fluid flow datasets. Features in two scalar fields are defined by largest contour segmentation after topological simplification. These features are matched using a volumetric similarity measure based on spatial overlap of individual features. The relationships defined by this similarity measure are ranked and presented in a thumbnail gallery of feature pairs and a graph representation showing all relationships between individual contours. Additionally, linked views of the contour trees are provided to ease navigation. The main render view shows the selected features overlapping each other. Thus, by displaying individual features and their relationships in a structured fashion, we enable exploratory visualization of correlations between similar structures in two scalar fields. We demonstrate the utility of our approach by applying it to a number of complex fluid flow datasets, where the emphasis is put on the comparison of vortex related scalar quantities.

Rendering large numbers of dense line bundles in three dimensions is a common need for many visualization techniques, including streamlines and fiber tractography. Unfortunately, depiction of spatial relations inside these line bundles is often difficult but critical for understanding the represented structures. Many approaches evolved for solving this problem by providing special illumination models or tube-like renderings. Although these methods improve spatial perception of individual lines or related sets of lines, they do not solve the problem for complex spatial relations between dense bundles of lines. In this paper, we present a novel approach that improves spatial and structural perception of line renderings by providing a novel ambient occlusion approach suited for line rendering in real time.

In this work, a one-group reduced population balance model based on the one primary and one secondary particle method (OPOSPM) developed recently by Attarakih et al. ( Jezowski J. Thullie J. Proceedings of the 19th European Symposium on Computer Aided Process Engineering, ESCAPE-19, Cracow, Poland, June 14−17, 2009New York2009978-0-444-53433-0) is implemented in the commercial computational fluid dynamics (CFD) package FLUENT 6.3 for solving the population balance equation in a combined CFD−population balance model (PBM). The one-group reduced population balance conserves the total number (N) and volume (α) concentrations of the population by directly solving two transport equations for N and α and provides a one-quadrature point for closing the unclosed integrals in the population balance equation. Unlike the published two-equation models, the present method offers accuracy improvement and internal consistency (with respect to the continuous population balance equation) by increasing the number of primary particles (sections). The one-group reduced population balance provides the possibility of a one-equation model for the solution of the PBM in CFD based on the mathematically consistent d 30 instead of the classical d 32 mean droplet diameter. Droplet breakage and coalescence are considered in the PBM, which is coupled to the fluid dynamics in order to describe real droplet behavior in a stirred liquid−liquid extraction column. As a case study, a full pilot-plant extraction column of a rotating disk contactor (RDC) type consisting of 50 compartments was simulated with the new model. The predicted results for the mean droplet diameter and the dispersed-phase volume fraction (holdup) agree well with literature data. The results show that the new CFD−PBM model is very efficient from a computational point of view (a factor of 2 less than the QMOM and a factor of 5 less than the method of classes). This is because the one-group reduced population balance requires the solution of only one equation (the total number concentration) when coupled to the CFD solver. It is therefore suitable for fast and efficient simulations of small-scale devices and even large-scale industrial processes.

Electrical activity of neuronal populations is a crucial aspect of brain activity. This activity is not measured directly but recorded as electrical potential changes using head surface electrodes (electroencephalogram - EEG). Head surface electrodes can also be deployed to inject electrical currents in order to modulate brain activity (transcranial electric stimulation techniques) for therapeutic and neuroscientific purposes. In electroencephalography and noninvasive electric brain stimulation, electrical fields mediate between electrical signal sources and regions of interest (ROI). These fields can be very complicated in structure, and are influenced in a complex way by the conductivity profile of the human head. Visualization techniques play a central role to grasp the nature of those fields because such techniques allow for an effective conveyance of complex data and enable quick qualitative and quantitative assessments. The examination of volume conduction effects of particular head model parameterizations (e.g., skull thickness and layering), of brain anomalies (e.g., holes in the skull, tumors), location and extent of active brain areas (e.g., high concentrations of current densities) and around current injecting electrodes can be investigated using visualization. Here, we evaluate a number of widely used visualization techniques, based on either the potential distribution or on the current-flow. In particular, we focus on the extractability of quantitative and qualitative information from the obtained images, their effective integration of anatomical context information, and their interaction. We present illustrative examples from clinically and neuroscientifically relevant cases and discuss the pros and cons of the various visualization techniques.

Abstract. While tensors occur in many areas of science and engineering, little has been done to visualize tensors with order higher than two. Tensors of higher orders can be used for example to describe complex diffusion patterns in magnetic resonance imaging (MRI). Recently, we presented a method for tracking lines in higher order tensor fields that is a generalization of methods known from first order tensor fields (vector fields) and symmetric second order tensor fields. Here, this method is applied to magnetic resonance imaging where tensor fields are used to describe diffusion patterns for example of hydrogen in the human brain. These patterns align to the internal structure and can be used to analyze interconnections between different areas of the brain, the so called tractography problem. The advantage of using higher order tensor lines is the ability to detect crossings locally, which is not possible in second order tensor fields. In this paper, the theoretical details will be extended and tangible results will be given on MRI data sets.