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The choice of knot vector has immense influence on the resulting accuracy of a B-spline approximation of a curve. However, despite the significance of this problem and the various solutions that were proposed in the literature, optimizing the number and placement of knots remains a difficult task. This paper presents a novel method for the approximation of a curve by a B-spline of arbitrary order, which automatically determines a knot vector that achieves high approximation quality. At the core of our approach is a feature function that characterizes the amount and spatial distribution of geometric details in the input curve by estimating its derivatives. Knots are then selected in such a way as to evenly distribute the feature contents across their intervals. A comparison to the state of the art for a wide variety of curves shows that our method is faster and achieves more accurate reconstruction results, while typically reducing the number of necessary knots.

We report our findings on the evolution of solids fraction in a tapped system of inelastic, frictional spheres as a function of the applied acceleration obtained via discrete element simulations. Animations of the simulation data reveal the propagation of a wave initiated from the base that causes local rearrangements of the particles ultimately leading to the development of a dense microstructure. We also describe the analysis of dynamical models capable of predicting the simulated behavior, and advanced visualization techniques for revealing the dynamics.

The dynamics of a vertical stack of particles subject to gravity and a sequence of small, periodically applied taps is considered. First, the motion of the particles, assumed to be identical, is modeled as a system of ordinary differential equations, which is analyzed with an eye to observing connections with finite-dimensional Hamiltonian systems. Then, two approaches to obtaining approximate continuum models for large numbers of particles are described: the long-wave approximation that yields partial differential equations and the BSR method that employs integro-partial differential models. These approximate continuum models, which comprise infinite-dimensional dynamical systems, are studied with a focus on nonlinear wave type behavior, which naturally leads to investigating links to infinite-dimensional Hamiltonian systems. Several examples are solved numerically to show similarities among the solution properties of the finite-dimensional (lattice-dynamics), and the approximate long-wave and BSR continuum models. Extensions to higher dimensions and more general dynamically driven particle configurations are also sketched.

Ridge and valley manifolds are receiving a growing attention in visualization research due to their ability to reveal the shapes of salient structures in numerical datasets across scientific, engineering, and medical applications. However, the methods proposed to date for their extraction in the visualization and image analysis literature are computationally expensive and typically applied in an offline setting. This setup does not properly support a userdriven exploration, which often requires control over various parameters tuned to filter false positives and spurious artifacts and highlight the most significant structures. This paper presents a GPU-based adaptive technique for crease extraction and visualization across scales. Our method combines a scale-space analysis of the data in pre-processing with a ray casting approach supporting a robust and efficient one-dimensional numerical search, and an image-based rendering strategy. This general framework achieves high-quality crease surface representations at interactive frame rates. Results are proposed for analytical, medical, and computational datasets.

We introduce a versatile framework for characterizing and extracting salient structures in three-dimensional symmetric second-order tensor fields. The key insight is that degenerate lines in tensor fields, as defined by the standard topological approach, are exactly crease (ridge and valley) lines of a particular tensor invariant called mode. This reformulation allows us to apply well-studied approaches from scientific visualization or computer vision to the extraction of topological lines in tensor fields. More generally, this main result suggests that other tensor invariants, such as anisotropy measures like fractional anisotropy (FA), can be used in the same framework in lieu of mode to identify important structural properties in tensor fields. Our implementation addresses the specific challenge posed by the non-linearity of the considered scalar measures and by the smoothness requirement of the crease manifold computation. We use a combination of smooth reconstruction kernels and adaptive refinement strategy that automatically adjust the resolution of the analysis to the spatial variation of the considered quantities. Together, these improvements allow for the robust application of existing ridge line extraction algorithms in the tensor context of our problem. Results are proposed for a diffusion tensor MRI dataset, and for a benchmark stress tensor field used in engineering research.

We introduce a novel flow visualization method called Flow Charts, which uses a texture atlas approach for the visualization of flows defined over curved surfaces. In this scheme, the surface and its associated flow are segmented into overlapping patches, which are then parameterized and packed in the texture domain. This scheme allows accurate particle advection across multiple charts in the texture domain, providing a flexible framework that supports various flow visualization techniques. The use of surface parameterization enables flow visualization techniques requiring the global view of the surface over long time spans, such as Unsteady Flow LIC (UFLIC), particle-based Unsteady Flow Advection Convolution (UFAC), or dye advection. It also prevents visual artifacts normally associated with view-dependent methods. Represented as textures, Flow Charts can be naturally integrated into hardware accelerated flow visualization techniques for interactive performance.

Current unsteady multi-field simulation data-sets consist of millions of data-points. To efficiently reduce this enormous amount of information, local statistical complexity was recently introduced as a method that identifies distinctive structures using concepts from information theory. Due to high computational costs this method was so far limited to 2D data. In this paper we propose a new strategy for the computation that is substantially faster and allows for a more precise analysis. The bottleneck of the original method is the division of spatio-temporal configurations in the field (light-cones) into different classes of behavior. The new algorithm uses a density-driven Voronoi tessellation for this task that more accurately captures the distribution of configurations in the sparsely sampled high-dimensional space. The efficient computation is achieved using structures and algorithms from graph theory. The ability of the method to detect distinctive regions in 3D is illustrated using flow and weather simulations.

Flow visualization is a classic sub-field of scientific visualization. The task of flow visualization algorithms is to depict vector data, i.e., data with magnitude and direction. An important cate ...

Geometric models of white matter architecture play an increasing role in neuroscientific applications of diffusion tensor imaging, and the most popular method for building them is fiber tractography. For some analysis tasks, however, a compelling alternative may be found in the first and second derivatives of diffusion anisotropy. We extend to tensor fields the notion from classical computer vision of ridges and valleys, and define anisotropy creases as features of locally extremal values of tensor anisotropy. Mathematically, these are the loci where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of the major white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. The crease extraction algorithm we present generates high-quality polygonal models of crease surfaces, which are further simplified by connected-component analysis. We demonstrate anisotropy creases on measured diffusion MRI data, and visualize them in combination with tractography to confirm their anatomic relevance.

The paper presents a topology-based visualization method for time-dependent two-dimensional vector elds. A time interpolation enables the accurate tracking of critical points and closed orbits as well as the detection and identication of structural changes. This completely characterizes the topology of the unsteady ow. Bifurcation theory provides the theoretical framework. The results are conveyed by surfaces that separate subvolumes of uniform ow behavior in a three-dimensional space-time domain.

Topological methods produce simple and meaningful depicti ons of symmetric, second order two-dimensional tensor fields. Extending previous work dealing with vector fields, we propose here a scheme for the visualization of time-dependent tensor fields. Basic notions of unsteady t topology are discussed. Topological changes known as bifurcations - are precisely detected and identifie d by our method which permits an accurate tracking of degenerate points and related structures.

We present an approximate dynamical systems model for the mass center trajectory of a tapped column of N uniform, inelastic, spheres (diameter d), in which collisional energy loss is governed by the Walton-Braun linear loading-unloading soft interaction. Rigorous analysis of the model, akin to the equations for the motion of a single bouncing ball on a vibrating plate, reveals a parameter γ≔2aω2(1+e)/g that gauges the dynamical regimes and their transitions. In particular, we find bifurcations from periodic to chaotic dynamics that depend on frequency ω, amplitude a/d of the tap. Dynamics predicted by the model are also qualitatively observed in discrete element simulations carried out over a broad range of the tap parameters.

The paper describes the application of topological methods to the visualization of vortical flow patterns that arise in simulations from Computational Fluid Dynamics. Two techniques are presented: the first is concerned with the exploration of complicated, instanta- neous flow structures while the second one permits the visualization of their temporal evolu- tion in large-scale transient simulations. In both cases the mathematical framework is derived from the notion of parametric topology. This yields a unified formalism that permits to effi- ciently address the challenges raised by typical flow problems. The benefits of this approach are demonstrated in the analysis and visualization of transient vortical flows that undergo the phenomenon of vortex breakdown.

Publisher Summary This chapter presents an overview of the techniques devised by scientific visualization research to address the specific needs of biomedical applications. Computers have become indispensable to the research and the clinical practice of biomedicine. Their widespread use now permits the manipulation of massive amounts of measured data, as well as, the study of sophisticated models through numerical simulations. The rapidly growing size and complexity of the resulting information from experiments or simulations creates a challenging demand on the tools and techniques needed to derive knowledge and insight from the data. Scientific visualization offers a very powerful approach to tackle this data analysis challenge by creating visual representations that convey salient properties of large data sets and permit their effective interpretation and analysis. The field of visualization is focused on creating images that convey salient information about underlying data and processes. In the past three decades, there has been unprecedented growth in computational and acquisition technologies, a growth that has resulted in an increased ability both to sense the physical world in precise detail and to model and simulate complex physical phenomena. As such, visualization plays a crucial role in our ability to comprehend such large and complex data—data which, in two, three, or more dimensions, convey insight into such diverse biomedical applications as understanding the bioelectric currents within the heart, characterizing white matter tracts by diffusion tensor imaging, and understanding morphology differences between different genetic mice phenotypes, among many others.

The study of flow separation from walls or solid objects is still an active research area in the fluid dynamics and flow visualization communities and many open questions remain. This paper aims at introducing a new method for the extraction of separation manifolds originating from separation lines. We address the problem from the flow visualization side by investigating features in flow cross sections around separation lines. We use the topological signature of the separation in these sections, in particular the presence of saddle points and their separatrices, as a guide to initiate the construction of the separation manifolds. Having this first part we use well known stream surface construction methods to propagate the surface further into the flow. Additionally, we discuss some lessons learned in the course of our experimentation with well known and new ideas for the extraction of separation lines.

The structure of fiber tracts in DT-MRI data presents a challenging problem for visualization and analysis. We derive visualization of such traces from a local coherence measure and achieve much improved visual segmentation. We introduce a coherence measure defined for fiber tracts. This quantitative assessment is based on infinitesimal deviations of neighboring tracts and allows identification and segmentation of coherent fiber regions. We use a hardware-accelerated implementation to achieve interactive visualization on slices and provide several approaches to visualize coherence information. Furthermore, we enhance existing techniques by combining them with coherence. We demonstrate our method on both a canine heart, where the myocardial structure is visualized, and a human brain, where we achieve detailed visualization of major and minor fiber bundles in a quality similar to and exceeding fiber clustering approaches. Our approach allows detailed and fast visual- ization of important anatomical structures in DT-MRI data sets.

We present a novel approach for the direct computation of integral surfaces in time-dependent vector fields. As opposed to previous work, which we analyze in detail, our approach is based on a separation of integral surface computation into two stages: surface approximation and generation of a graphical representation. This allows us to overcome several limitations of existing techniques. We first describe an algorithm for surface integration that approximates a series of time lines using iterative refinement and computes a skeleton of the integral surface. In a second step, we generate a well-conditioned triangulation. Our approach allows a highly accurate treatment of very large time-varying vector fields in an efficient, streaming fashion. We examine the properties of the presented methods on several example datasets and perform a numerical study of its correctness and accuracy. Finally, we investigate some visualization aspects of integral surfaces.

This paper accompanies a poster that is being presented at the SciDAC 2006 meeting in Denver, CO. This project focuses on leveraging scientific visualization and analytics software technology as an enabling technology for increasing scientific productivity and insight. Advances incomputational technology have resulted in an "information big bang," which in turn has createda significant data understanding challenge. This challenge is widely acknowledged to be one of the primary bottlenecks in contemporary science. The vision for our Center is to respond directly to that challenge by adapting, extending, creating when necessary and deploying visualization and data understanding technologies for our science stakeholders. Using an organizational model as a Visualization and Analytics Center for Enabling Technologies (VACET), we are well positioned to be responsive to the needs of a diverse set of scientific stakeholders in a coordinated fashion using a range of visualization, mathematics, statistics, computer and computational science and data management technologies.

The notions of Finite-Time Lyapunov Exponent (FTLE) and Lagrangian Coherent Structures provide a strong framework for the analysis and visualization of complex technical flows. Their definition is simple and intuitive, and they are built on a deep theoretical foundation. We apply these concepts to enable the analysis of flows in the immediate vicinity of the boundaries of flow-embedded objects by limiting the Lagrangian analysis to surfaces closely neighboring these boundaries. To this purpose, we present an approach to approximate FTLE fields over such surfaces. Furthermore, we achieve an effective depiction of boundary-related flow structures such as separation and attachment over object boundaries and specific insight into the surrounding flow using several specifically chosen visualization techniques. We document the applicability of our methods by presenting a number of application examples.

The bioelectric activity that takes place throughout the human body enables the action of muscles and the transmission of information in nerves. A variety of imaging modalities have therefore been developed to assess this activity. In particular, electrocardiography for the heart and electrocardiography and magnetoencephalography for the brain permit to measure non-invasively the resulting electric signal. Beyond their obvious clinical applications these techniques also open the door to a computational reconstruction of the physiological activity at the origin of this signal through the numerical solution of so-called inverse problems. In this case as well as in basic bioengineering research effective postprocessing tools are necessary to facilitate the interpretation of measured and simulated bioelectric data and permit to derive anatomical and functional insight from it. In contrast to scalar quantities for which effective depictions generally exist and are routinely used, the vector-valued nature of this information bears specific challenges that are insufficiently addressed by the visualization tools typically available to biomedical practitioners. This paper reports on the application of advanced vector visualization techniques to the postprocessing analysis of bioelectric fields as they arise in cardiovascular and inverse source reconstruction research. Our work demonstrates the ability of the corresponding visual representations to improve the interpretation of the data and support new insight into the underlying physiology.

The focus of this article is on how one group of researchersthe DOE SciDAC Visualization and Analytics Center for EnablingTechnologies (VACET) is tackling the daunting task of enabling knowledgediscovery through visualization and analytics on some of the world slargest and most complex datasets and on some of the world's largestcomputational platforms. As a Center for Enabling Technology, VACET smission is the creation of usable, production-quality visualization andknowledge discovery software infrastructure that runs on large, parallelcomputer systems at DOE's Open Computing facilities and that providessolutions to challenging visual data exploration and knowledge discoveryneeds of modern science, particularly the DOE sciencecommunity.

We present a method to extract and visualize vortices that originate from bounding walls of three-dimensional time- dependent flows. These vortices can be detected using their footprint on the boundary, which consists of critical points in the wall shear stress vector field. In order to follow these critical points and detect their transformations, affected regions of the surface are parameterized. Thus, an existing singularity tracking algorithm devised for planar settings can be applied. The trajectories of the singularities are used as a basis for seeding particles. This leads to a new type of streak line visualization, in which particles are released from a moving source. These generalized streak lines visualize the particles that are ejected from the wall. We demonstrate the usefulness of our method on several transient fluid flow datasets from computational fluid dynamics simulations.

An optimal combustion process within an engine block is central to the performance of many motorized vehicles. Associated with this process are two important patterns of flow: swirl and tumble motion, which optimize the mixing of fluid within each of an engine's cylinders. The simulation data associated with in-cylinder tumble motion within a gas engine, given on an unstructured, timevarying and adaptive resolution CFD grid, demands robust visualization methods that apply to unsteady flow. Good visualizations are necessary to analyze the simulation data of these in-cylinder flows. We present a range of methods including integral, feature-based, and image-based schemes with the goal of extracting and visualizing these two important patterns of motion. We place a strong emphasis on automatic and semi-automatic methods, including topological analysis, that require little or no user input.We make effective use of animation to visualize the time-dependent simulation data. We also describe the challenges and implementation measures necessary in order to apply the presented methods to time-varying, volumetric grids.

The recently introduced notion of Finite-Time Lyapunov Exponent to characterize Coherent Lagrangian Structures provides a powerful framework for the visualization and analysis of complex technical flows. Its definition is simple and intuitive, and it has a deep theoretical foundation. While the application of this approach seems straightforward in theory, the associated computational cost is essentially prohibitive. Due to the Lagrangian nature of this technique, a huge number of particle paths must be computed to fill the space-time flow domain. In this paper, we propose a novel scheme for the adaptive computation of FTLE fields in two and three dimensions that significantly reduces the number of required particle paths. Furthermore, for three-dimensional flows, we show on several examples that meaningful results can be obtained by restricting the analysis to a well-chosen plane intersecting the flow domain. Finally, we examine some of the visualization aspects of FTLE-based methods and introduce several new variations that help in the analysis of specific aspects of a flow.

The SciDAC2 Visualization and Analytics Center for Enabling Technologies (VACET) began operation on 10/1/2006. This document, dated 11/27/2006, is the first version of the VACET project management plan. It was requested by and delivered to ASCR/DOE. It outlines the Center's accomplishments in the first six weeks of operation along with broad objectives for the upcoming future (12-24 months).

Current methods for extracting models of white matter architecture from diffusion tensor MRI are generally based on fiber tractography. For some purposes a compelling alternative may be found in analyzing the first and second derivatives of diffusion anisotropy. Anisotropy creases are ridges and valleys of locally extremal anisotropy, where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. We describe a crease extraction algorithm that generates high-quality polygonal models of crease surfaces, then demonstrate the method on a measured diffusion tensor dataset, and visualize the result in combination with tractography to confirm its anatomic relevance.

To achieve a deeper understanding of the brain, scientists, and clinicians use electroencephalography (EEG) and magnetoencephalography (MEG) inverse methods to reconstruct sources in the cortical sheet of the human brain. The influence of structural and electrical anisotropy in both the skull and the white matter on the EEG and MEG source reconstruction is not well understood. In this paper, we report on a study of the sensitivity to tissue anisotropy of the EEG/MEG forward problem for deep and superficial neocortical sources with differing orientation components in an anatomically accurate model of the human head. The goal of the study was to gain insight into the effect of anisotropy of skull and white matter conductivity through the visualization of field distributions, isopotential surfaces, and return current flow and through statistical error measures. One implicit premise of the study is that factors that affect the accuracy of the forward solution will have at least as strong an influence over solutions to the associated inverse problem. Major findings of the study include (1) anisotropic white matter conductivity causes return currents to flow in directions parallel to the white matter fiber tracts; (2) skull anisotropy has a smearing effect on the forward potential computation; and (3) the deeper a source lies and the more it is surrounded by anisotropic tissue, the larger the influence of this anisotropy on the resulting electric and magnetic fields. Therefore, for the EEG, the presence of tissue anisotropy both for the skull and white matter compartment substantially compromises the forward potential computation and as a consequence, the inverse source reconstruction. In contrast, for the MEG, only the anisotropy of the white matter compartment has a significant effect. Finally, return currents with high amplitudes were found in the highly conducting cerebrospinal fluid compartment, underscoring the need for accurate modeling of this space.

We propose a novel point-based approach to view dependent iso-surface extraction. We introduce a fast visibility query system for the view dependent traversal, which exhibits moderate memory requirements. This technique allows for an interactive interrogation of the full visible woman dataset (1GB) at four to fifteen frames per second on a desktop computer. The point-based approach is built on an extraction scheme that classifies different sections of the iso-surface into four categories, depending on the size of the geometry when projected onto the screen. In particular, we use points to represent small and sub-pixel triangles, as well as larger sections of the isosurface whose projection has sub-pixel size. To assign consistent and robust normals to individual points representing such regions, we propose to compute them during post processing of the extracted isosurface and provide the corresponding hardware implementation.

The paper is concerned with the extraction and visualization of structural features in delta wing dataset resulting from a large-scale Reynolds Averaged Navier-Stokes simulation. Its main contributions are two novel schemes designed for this purpose. First, we introduce a new method for fast and precise extraction of separation and attachment lines on the body of the delta wing. Second, we improve significantly standard stream surface computation schemes to achieve accurate integration even in regions exhibiting complex structures. Additional state-of-the-art flow visualization techniques are used to complete the study of the dataset. The main focus is on the interaction between open separation and attachment lines of the wall shear stress on one hand, and vortices that form in the vicinity of the delta wing on the other hand. These features are key factors in flight stability and their study is mandatory in aircraft design. The paper demonstrates the power of a combined use of line-type feature extraction and stream surface computation to deeply improve understanding of three-dimensional structures and ease the analysis of flow separation and vortex breakdown.

Tensor fields can be found in most areas of physics and engineering sciences. Topology-based methods provide efficient means for visualizing symmetric, second-order, planar tensor fields. Yet, for tensor fields with complicated structure typically encountered in turbulent flows, topology-based techniques lead to cluttered pictures that confuse the interpretation and are of little help for analysts. In this paper, we present a hierarchical method that merges close degenerate points into one. This results in a simplified topology and a clarified depiction, though globally maintaining the qualitative properties of the original data. The whole process can be seen as a scaling of the topological information that neglects details of small scale and only retains their structural aspect in the large. We extend here previous work dealing with planar vector fields. Results are demonstrated on a computational fluid dynamics(CFD) dataset.

Piecewise linear interpolation of 2D scattered vector data is a classical, simple and fast scheme to process the discrete information provided by experiments or numerical simulations. Nevertheless, its major drawback is its low order that prevents satisfying approximation of non linear behaviors. For topology-based methods in particular, commonly applied in vector field visualization, it often restricts the structural features found to very few possible configurations which may be insufficient for interpretation. In this paper, on the contrary, we consider piecewise linear vector fields from the modeling viewpoint, showing that they can exhibit arbitrary complex topological features.

International audience; Interdisciplinary efforts in modeling and simulating phenomena have led to complex multi-physics models involving different physical properties and materials in the same system. Within a 3d domain, substructures of lower dimensions appear at the interface between different materials. Correspondingly, an unstructured tetrahedral mesh used for such a simulation includes 2d and 1d substructures embedded in the vertices, edges and faces of the mesh. The simplification of such tetrahedral meshes must preserve (1) the geometry and the topology of the 3d domain, (2) the simulated data and (3) the geometry and topology of the embedded substructures. Although intensive research has been conducted on the first two goals, the third objective has received little attention. This paper focuses on the preservation of the topology of 1d and 2d substructures embedded in an unstructured tetrahedral mesh, during edge collapse simplification. We define these substructures as simplicial sub-complexes of the mesh, which is modeled as an extended simplicial complex. We derive a robust algorithm, based on combinatorial topology results, in order to determine if an edge can be collapsed without changing the topology of both the mesh and all embedded substructures. Based on this algorithm we have developed a system for simplifying scientific datasets defined on irregular tetrahedral meshes with substructures. The implementation of our system is discussed in detail. We demonstrate the power of our system with real world scientific datasets from electromagnetism simulations.