Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

Most streamline generation algorithms either provide a particular density of streamlines across the domain or explicitly detect features, such as critical points, and follow customized rules to emphasize those features. However, the former generally includes many redundant streamlines, and the latter requires Boolean decisions on which points are features (and may thus suffer from robustness problems for real-world data). We take a new approach to adaptive streamline placement for steady vector fields in 2D and 3D. We define a metric for local similarity among streamlines and use this metric to grow streamlines from a dense set of candidate seed points. The metric considers not only Euclidean distance, but also a simple statistical measure of shape and directional similarity. Without explicit feature detection, our method produces streamlines that naturally accentuate regions of geometric interest. In conjunction with this method, we also propose a quantitative error metric for evaluating a streamline representation based on how well it preserves the information from the original vector field. This error metric reconstructs a vector field from points on the streamline representation and computes a difference of the reconstruction from the original vector field.

/pdf/similarity-guided-streamline-placement-with-error-evaluation-4rx74td4g9.pdf

Similarity-Guided Streamline Placement with Error Evaluation

We treat streamline selection and viewpoint selection as symmetric problems which are formulated into a unified information-theoretic framework. This is achieved by building two interrelated information channels between a pool of candidate streamlines and a set of sample viewpoints. We define the streamline information to select best streamlines and in a similar manner, define the viewpoint information to select best viewpoints. Furthermore, we propose solutions to streamline clustering and viewpoint partitioning based on the representativeness of streamlines and viewpoints, respectively. Finally, we define a camera path that passes through all selected viewpoints for automatic flow field exploration. We demonstrate the robustness of our approach by showing experimental results with different flow data sets, and conducting rigorous comparisons between our algorithm and other seed placement or streamline selection algorithms based on information theory.

A Unified Approach to Streamline Selection and Viewpoint Selection for 3D Flow Visualization

We present a novel approach for analyzing two-dimensional (2D) flow field data based on the idea of invariant moments. Moment invariants have traditionally been used in computer vision applications, and we have adapted them for the purpose of interactive exploration of flow field data. The new class of moment invariants we have developed allows us to extract and visualize 2D flow patterns, invariant under translation, scaling, and rotation. With our approach one can study arbitrary flow patterns by searching a given 2D flow data set for any type of pattern as specified by a user. Further, our approach supports the computation of moments at multiple scales, facilitating fast pattern extraction and recognition. This can be done for critical point classification, but also for patterns with greater complexity. This multi-scale moment representation is also valuable for the comparative visualization of flow field data. The specific novel contributions of the work presented are the mathematical derivation of the new class of moment invariants, their analysis regarding critical point features, the efficient computation of a novel feature space representation, and based upon this the development of a fast pattern recognition algorithm for complex flow structures.

/pdf/moment-invariants-for-the-analysis-of-2d-flow-fields-5au6c0b8ke.pdf

Moment Invariants for the Analysis of 2D Flow Fields

Effective 3D streamline placement and visualization play an essential role in many science and engineering disciplines The main challenge for effective streamline visualization lies in seed placement, ie, where to drop seeds and how many seeds should be placed Seeding too many or too few streamlines may not reveal flow features and patterns either because it easily leads to visual clutter in rendering or it conveys little information about the flow field Not only does the number of streamlines placed matter, their spatial relationships also play a key role in understanding the flow field Therefore, effective flow visualization requires the streamlines to be placed in the right place and in the right amount This paper introduces hierarchical streamline bundles, a novel approach to simplifying and visualizing 3D flow fields defined on regular grids By placing seeds and generating streamlines according to flow saliency, we produce a set of streamlines that captures important flow features near critical points without enforcing the dense seeding condition We group spatially neighboring and geometrically similar streamlines to construct a hierarchy from which we extract streamline bundles at different levels of detail Streamline bundles highlight multiscale flow features and patterns through clustered yet not cluttered display This selective visualization strategy effectively reduces visual clutter while accentuating visual foci, and therefore is able to convey the desired insight into the flow data

Hierarchical Streamline Bundles

The analysis of time-dependent data is often guided by the question of how dominant structures develop over time. It is important to understand how patterns or structures identified for one time step evolve over time, by changing or moving in the domain. To gain insight into such evolving structural change it is crucial to effectively compare different time steps. This paper proposes a comparison method for twodimensional flow fields. The method is based on a feature description using invariant moments. The specific strength of these moments is their invariance under scaling and rotation, thus facilitating an identification of features even if they occur at other positions, with changed orientation, and variation in size. In addition the moments themselves can be used to define a similarity measure. To evaluate the significance of this concept it has been applied to wind speed data from meteorological simulations.

Comparative Visualization of Two-Dimensional Flow Data Using Moment Invariants

Feature detection and pattern matching play an important role in visualization.Originally developed for images and scalar fields, pattern matching methods become increasingly interesting for other applications, e.g., vector fields. To apply pattern matching to vector fields the basic concepts of convolution and fast Fourier transform (FFT) have to be generalized to vector fields. A formalism supporting an elegant generalization of these concepts is provided by the Clifford Algebra, originally developed for describing geometry and geometric operations. We discuss an application of the Clifford Pattern Matching (CPM). We apply CPM to images for ”Clifford Color Edge Detection” (C2ED), an approach for detecting edges and other features in color images. The basic idea is to treat color value tripels as vectors and apply the pattern matching algorithm to the resulting vector field. We introduce vector-valued filters for edge detection and present results.

/pdf/clifford-pattern-matching-for-color-image-edge-detection-onstlirlot.pdf

Clifford pattern matching for color image edge detection

Interaction between particles in so-called granular media, such as soil and sand, plays an important role in the context of geomechanical phenomena and numerous industrial applications. A two scale homogenization approach based on a micro and a macro scale level is briefly introduced in this paper. Computation of granular material in such a way gives a deeper insight into the context of discontinuous materials and at the same time reduces the computational costs. However, the description and the understanding of the phenomena in granular materials are not yet satisfactory. A sophisticated problem-specific visualization technique would significantly help to illustrate failure phenomena on the microscopic level. As main contribution, we present a novel 2D approach for the visualization of simulation data, based on the above outlined homogenization technique. Our visualization tool supports visualization on micro scale level as well as on macro scale level. The tool shows both aspects closely arranged in form of multiple coordinated views to give users the possibility to analyze the particle behavior effectively. A novel type of interactive rose diagrams was developed to represent the dynamic contact networks on the micro scale level in a condensed and efficient way.

/pdf/visualization-of-particle-interactions-in-granular-media-zeyf9h6xbf.pdf

Visualization of Particle Interactions in Granular Media

Vector fields play an important role in many areas of computational physics and engineering. For effective visualization of vector fields it is necessary to identify and extract important features inherent in the data, defined by filters that characterize certain “patterns”. Our prior approach for vector field analysis used the Clifford Fourier transform for efficient pattern recognition for vector field data defined on regular grids [1,2]. Using the frequency domain, correlation and convolution of vectors can be computed as a Clifford multiplication, enabling us to determine similarity between a vector field and a pre-defined pattern mask (e.g., for critical points). Moreover, compression and spectral analysis of vector fields is possible using this method. Our current approach only applies to rectilinear grids. We combine this approach with a fast Fourier transform to handle unstructured scalar data [6]. Our extension enables us to provide a feature-based visualization of vector field data defined on unstructured grids, or completely scattered data. Besides providing the theory of Clifford Fourier transform for unstructured vector data, we explain how efficient pattern matching and visualization of various selectable features can be performed efficiently. We have tested our method for various vector data sets.

Fast clifford fourier transformation for unstructured vector field data

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