Fiber Surfaces: Generalizing Isosurfaces to Bivariate Data

Scientific visualization has many effective methods for examining and exploring scalar and vector fields, but rather fewer for bivariate fields. We report the first general purpose approach for the interactive extraction of geometric separating surfaces in bivariate fields. This method is based on fiber surfaces: surfaces constructed from sets of fibers, the multivariate analogues of isolines. We show simple methods for fiber surface definition and extraction. In particular, we show a simple and efficient fiber surface extraction algorithm based on Marching Cubes. We also show how to construct fiber surfaces interactively with geometric primitives in the range of the function. We then extend this to build user interfaces that generate parameterized families of fiber surfaces with respect to arbitrary polygons. In the special case of isovalue‐gradient plots, fiber surfaces capture features geometrically for quantitative analysis that have previously only been analysed visually and qualitatively using multi‐dimensional transfer functions in volume rendering. We also demonstrate fiber surface extraction on a variety of bivariate data.     

Non‐iterative Second‐order Approximation of Signed Distance Functions for Any Isosurface Representation

Signed distance functions (SDF) to explicit or implicit surface representations are intensively used in various computer graphics and visualization algorithms. Among others, they are applied to optimize collision detection, are used to reconstruct data fields or surfaces, and, in particular, are an obligatory ingredient for most level set methods. Level set methods are common in scientific visualization to extract surfaces from scalar or vector fields. Usual approaches for the construction of an SDF to a surface are either based on iterative solutions of a special partial differential equation or on marching algorithms involving a polygonization of the surface. We propose a novel method for a non‐iterative approximation of an SDF and its derivatives in a vicinity of a manifold. We use a second‐order algebraic fitting scheme to ensure high accuracy of the approximation. The manifold is defined (explicitly or implicitly) as an isosurface of a given volumetric scalar field. The field may be given at a set of irregular and unstructured samples. Stability and reliability of the SDF generation is achieved by a proper scaling of weights for the Moving Least Squares approximation, accurate choice of neighbors, and appropriate handling of degenerate cases. We obtain the solution in an explicit form, such that no iterative solving is necessary, which makes our approach fast.     

Multifield-graphs: an approach to visualizing correlations in multifield scalar data

We present an approach to visualizing correlations in 3D multifield scalar data. The core of our approach is the computation of correlation fields, which are scalar fields containing the local correlations of subsets of the multiple fields. While the visualization of the correlation fields can be done using standard 3D volume visualization techniques, their huge number makes selection and handling a challenge. We introduce the Multifield-Graph to give an overview of which multiple fields correlate and to show the strength of their correlation. This information guides the selection of informative correlation fields for visualization. We use our approach to visually analyze a number of real and synthetic multifield datasets.     

Invariant crease lines for topological and structural analysis of tensor fields

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.     

Generation of accurate integral surfaces in time-dependent vector fields

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.     

Local and singularity-free G 1 triangular spline surfaces using a minimum degree scheme

We develop a scheme for constructing G ¹ triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G ¹ triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples.     

ISA and IBFVS: image space-based visualization of flow on surfaces

We present a side-by-side analysis of two recent image space approaches for the visualization of vector fields on surfaces. The two methods, image space advection (ISA) and image-based flow visualization for curved surfaces (IBFVS) generate dense representations of time-dependent vector fields with high spatio-temporal correlation. While the 3D vector fields are associated with arbitrary surfaces represented by triangular meshes, the generation and advection of texture properties is confined to image space. Fast frame rates are achieved by exploiting frame-to-frame coherency and graphics hardware. In our comparison of ISA and IBFVS, we point out the strengths and weaknesses of each approach and give recommendations as to when and where they are best applied.     

Ray Tracing Generalized Tube Primitives: Method and Applications

We present a general high‐performance technique for ray tracing generalized tube primitives. Our technique efficiently supports tube primitives with fixed and varying radii, general acyclic graph structures with bifurcations, and correct transparency with interior surface removal. Such tube primitives are widely used in scientific visualization to represent diffusion tensor imaging tractographies, neuron morphologies, and scalar or vector fields of 3D flow. We implement our approach within the OSPRay ray tracing framework, and evaluate it on a range of interactive visualization use cases of fixed‐ and varying‐radius streamlines, pathlines, complex neuron morphologies, and brain tractographies. Our proposed approach provides interactive, high‐quality rendering, with low memory overhead.     

Interactive comparison of scalar fields based on largest contours with applications to flow visualization

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.     

Interactive Visualization of Function Fields by Range-Space Segmentation

We present a dimension reduction and feature extraction method for the visualization and analysis of function field data. Function fields are a class of high-dimensional, multi-variate data in which data samples are one-dimensional scalar functions. Our approach focuses upon the creation of high-dimensional range-space segmentations, from which we can generate meaningful visualizations and extract separating surfaces between features. We demonstrate our approach on high-dimensional spectral imagery, and particulate pollution data from air quality simulations.     

Crease Surfaces: From Theory to Extraction and Application to Diffusion Tensor MRI

Crease surfaces are two-dimensional manifolds along which a scalar field assumes a local maximum (ridge) or a local minimum (valley) in a constrained space. Unlike isosurfaces, they are able to capture extremal structures in the data. Creases have a long tradition in image processing and computer vision, and have recently become a popular tool for visualization. When extracting crease surfaces, degeneracies of the Hessian (i.e., lines along which two eigenvalues are equal) have so far been ignored. We show that these loci, however, have two important consequences for the topology of crease surfaces: First, creases are bounded not only by a side constraint on eigenvalue sign, but also by Hessian degeneracies. Second, crease surfaces are not, in general, orientable. We describe an efficient algorithm for the extraction of crease surfaces which takes these insights into account and demonstrate that it produces more accurate results than previous approaches. Finally, we show that diffusion tensor magnetic resonance imaging (DT-MRI) stream surfaces, which were previously used for the analysis of planar regions in diffusion tensor MRI data, are mathematically ill-defined. As an example application of our method, creases in a measure of planarity are presented as a viable substitute.     

Mesh-driven vector field clustering and visualization: an image-based approach

Vector field visualization techniques have evolved very rapidly over the last two decades, however, visualizing vector fields on complex boundary surfaces from computational flow dynamics (CFD) still remains a challenging task. In part, this is due to the large, unstructured, adaptive resolution characteristics of the meshes used in the modeling and simulation process. Out of the wide variety of existing flow field visualization techniques, vector field clustering algorithms offer the advantage of capturing a detailed picture of important areas of the domain while presenting a simplified view of areas of less importance. This paper presents a novel, robust, automatic vector field clustering algorithm that produces intuitive and insightful images of vector fields on large, unstructured, adaptive resolution boundary meshes from CFD. Our bottom-up, hierarchical approach is the first to combine the properties of the underlying vector field and mesh into a unified error-driven representation. The motivation behind the approach is the fact that CFD engineers may increase the resolution of model meshes according to importance. The algorithm has several advantages. Clusters are generated automatically, no surface parameterization is required, and large meshes are processed efficiently. The most suggestive and important information contained in the meshes and vector fields is preserved while less important areas are simplified in the visualization. Users can interactively control the level of detail by adjusting a range of clustering distance measure parameters. We describe two data structures to accelerate the clustering process. We also introduce novel visualizations of clusters inspired by statistical methods. We apply our method to a series of synthetic and complex, real-world CFD meshes to demonstrate the clustering algorithm results.     

Direct Visualization of Deformation in Volumes

Deformation is a topic of interest in many disciplines. In particular in medical research, deformations of surfaces and even entire volumetric structures are of interest. Clear visualization of such deformations can lead to important insight into growth processes and progression of disease.
We present new techniques for direct focus+context visualization of deformation fields representing transformations between pairs of volumetric datasets. Typically, such fields are computed by performing a non-rigid registration between two data volumes. Our visualization is based on direct volume rendering and uses the GPU to compute and interactively visualize features of these deformation fields in real-time. We integrate visualization of the deformation field with visualization of the scalar volume affected by the deformations. Furthermore, we present a novel use of texturing in volume rendered visualizations to show additional properties of the vector field on surfaces in the volume.     

Sparse Representation and Visualization for Direct Numerical Simulation of Premixed Combustion

Direct Numerical Simulations of premixed combustion produce terabytes of raw data, which are prohibitively large to be stored, and have to be analyzed and visualized. A simultaneous and integrated treatment of data storage, data analysis and data visualization is required. For this, we introduce a sparse representation tailored to DNS data which can directly be used for both analysis and visualization. The method is based on the observation that most information is located in narrow‐band regions where the chemical reactions take place, but these regions are not well defined. An approach for the visual investigation of feature surfaces of the scalar fields involved in the simulation is shown as a possible application. We demonstrate our approach on multiple real datasets.     

Metric-Driven RoSy Field Design and Remeshing

Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user-defined field topologies. The user has full control of the number, positions, and indexes of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility such that it can design N-RoSy fields with single singularity and mixed-RoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.     

Visualizing the Positional and Geometrical Variability of Isosurfaces in Uncertain Scalar Fields

We present a novel approach for visualizing the positional and geometrical variability of isosurfaces in uncertain 3D scalar fields. Our approach extends recent work by Pöthkow and Hege [ [PH10] ] in that it accounts for correlations in the data to determine more reliable isosurface crossing probabilities. We introduce an incremental update‐scheme that allows integrating the probability computation into front‐to‐back volume ray‐casting efficiently. Our method accounts for homogeneous and anisotropic correlations, and it determines for each sampling interval along a ray the probability of crossing an isosurface for the first time. To visualize the positional and geometrical uncertainty even under viewing directions parallel to the surface normal, we propose a new color mapping scheme based on the approximate spatial deviation of possible surface points from the mean surface. The additional use of saturation enables to distinguish between areas of high and low statistical dependence. Experimental results confirm the effectiveness of our approach for the visualization of uncertainty related to position and shape of convex and concave isosurface structures.     

Temporally Dense Exploration of Moving and Deforming Shapes

We present our approach for the dense visualization and temporal exploration of moving and deforming shapes from scientific experiments and simulations. Our image space representation is created by convolving a noise texture along shape contours (akin to LIC). Beyond indicating spatial structure via luminosity, we additionally use colour to depict time or classes of shapes via automatically customized maps. This representation summarizes temporal evolution, and provides the basis for interactive user navigation in the spatial and temporal domain in combination with traditional renderings. Our efficient implementation supports the quick and progressive generation of our representation in parallel as well as adaptive temporal splits to reduce overlap. We discuss and demonstrate the utility of our approach using 2D and 3D scalar fields from experiments and simulations.     

Generalized B-spline subdivision-surface wavelets for geometry compression

We present a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation. Our method combines three approaches: subdivision surfaces of arbitrary topology, B-spline wavelets, and the lifting scheme for biorthogonal wavelet construction. The simple building blocks of our wavelet transform are local lifting operations performed on polygonal meshes with subdivision hierarchy. Starting with a coarse, irregular polyhedral base mesh, our transform creates a subdivision hierarchy of meshes converging to a smooth limit surface. At every subdivision level, geometric detail is expanded from wavelet coefficients and added to the surface. We present wavelet constructions for bilinear, bicubic, and biquintic B-spline subdivision. While the bilinear and bicubic constructions perform well in numerical experiments, the biquintic construction turns out to be unstable. For lossless compression, our transform is computed in integer arithmetic, mapping integer coordinates of control points to integer wavelet coefficients. Our approach provides a highly efficient and progressive representation for complex geometries of arbitrary topology.     

Fuzzy Contour Trees: Alignment and Joint Layout of Multiple Contour Trees

We describe a novel technique for the simultaneous visualization of multiple scalar fields, e.g. representing the members of an ensemble, based on their contour trees. Using tree alignments, a graph-theoretic concept similar to edit distance mappings, we identify commonalities across multiple contour trees and leverage these to obtain a layout that can represent all trees simultaneously in an easy-to-interpret, minimally-cluttered manner. We describe a heuristic algorithm to compute tree alignments for a given similarity metric, and give an algorithm to compute a joint layout of the resulting aligned contour trees. We apply our approach to the visualization of scalar field ensembles, discuss basic visualization and interaction possibilities, and demonstrate results on several analytic and real-world examples.     

Hierarchical Correlation Clustering in Multiple 2D Scalar Fields

Sets of multiple scalar fields can be used to model many types of variation in data, such as uncertainty in measurements and simulations or time‐dependent behavior of scalar quantities. Many structural properties of such fields can be explained by dependencies between different points in the scalar field. Although these dependencies can be of arbitrary complexity, correlation, i.e., the linear dependency, already provides significant structural information. Existing methods for correlation analysis are usually limited to positive correlation, handle only local dependencies, or use combinatorial approximations to this continuous problem. We present a new approach for computing and visualizing correlated regions in sets of 2‐dimensional scalar fields. This paper describes the following three main contributions: (i) An algorithm for hierarchical correlation clustering resulting in a dendrogram, (ii) a generalization of topological landscapes for dendrogram visualization, and (iii) a new method for incorporating negative correlation values in the clustering and visualization. All steps are designed to preserve the special properties of correlation coefficients. The results are visualized in two linked views, one showing the cluster hierarchy as 2D landscape and the other providing a spatial context in the scalar field's domain. Different coloring and texturing schemes coupled with interactive selection support an exploratory data analysis.