In the context of vector field data visualization, it is often desirable to construct a hierarchical data representation. One possibility to construct a hierarchy is based on clustering vectors using cerain similarity criteria. We combine two fundemental approaches to cluster vectors and construct hierarchical vector field representations. For clustering, a locally constructed linear least-squares approxiamation is incorporated into a similarity measure that considers both Euclidean distance between point pairs (for which dependent vector data are given) and difference in vector values. A modified normalized cut (NC) method is used to obtain a near-optimal clustering of a given discrete vector field data set. To obtain a hierarchical representation, the NC method is applied recursively after the construction of coarse-level clusters. We have applied our NC-based segmentation method to simple, analytically defined vector fields as well as discrete vector field data generated by turbulent flow simulation. Our test results indicate that our proposed adaptation of the original NC method is promising method as it leads to segmentation results that capture the qualitative and topological nature of vector field data.

In order to obtain insight into a complex vector field, it is often necessary to construct a hierarchical representation of the field. One way to construct such a hierarchy is based on grouping vectors together using certain similarity criteria. In this paper, we present a study of a 2D vector field clustering technique that is based on piecewise linear vector field approximations and an extension of a data clustering method called Normalized Cut (NC). Specifically, two steps are taken to implement the extended NC method. First, a similarity measurement for vector data is defined. Second, an eigenproblem solver is used to find the eigenvector used for partitioning. After the construction of first-level clusters, we can obtain a finer-level clustering by recursively applying the same procedure to intermediate clusters. The resulting clusters capture the features around the critical points.

We describe a system supporting the interactive exploration of threedimensional scientific data sets in a virtual reality (VR) environment. This system aids a scientist in understanding a data set by interactively placing and manipulating visualization primitives,e. g., isosurfaces or streamlines, and thereby finding features in the data and understanding its overall structure. We discuss how the requirement of interactivity influences the architecture of the visualization system, and how to adapt standard visualization techniques to work under real-time interaction constraints. Though we have implemented our visualization system to work with multiple types of data sets structures -cartesian, tetrahedral, curvilinear-hexahedral and adaptive mesh refinement (AMR)- we will focus on AMR grids and show how their inherent multiresolution structure is useful for interactive visualization.

Summary. We describe a system supporting the interactive exploration of threedimensional scientific data sets in a virtual reality (VR) environment. This system aids a scientist in understanding a data set by interactively placing and manipulating visualization primitives, e. g., isosurfaces or streamlines, and thereby finding features in the data and understanding its overall structure. We discuss how the requirement of interactivity influences the architecture of the visualization system, and how to adapt standard visualization techniques to work under real-time interaction constraints. Though we have implemented our visualization system to work with multiple types of data sets structures - cartesian, tetrahedral, curvilinear-hexahedral and adaptive mesh refinement (AMR) - we will focus on AMR grids and show how their inherent multiresolution structure is useful for interactive visualization.

Researchers doing scientific computations are attempting to accurately model physical phenomena. When these physical phenomena take place at a variety of different spatial and temporal scales it can be more efficient and accurate to model them at different levels of detail in an adaptive, hierarchical manner. We present a framework for visualizing adaptive, hierarchical computations- a conceptual framework and an implementation framework. Given that researchers have already defined a hierarchical structure for their data and are performing their computations using this structure, it has become important to provide a visualization tool which accurately represents this data and visualizes it directly. The tool we have designed for this purpose was built using the Visualization Toolkit, VTK, and one of its interpretive interfaces, Tcl/Tk. In addition to creating a visualization tool, we are developing extensions of visualiztion techniques and algorithms to hierarchiacal data (i.e., seamless isosurface generation).

Adaptive mesh refinement (AMR) is a numerical simulation technique used in computational fluid dynamics (CFD). By using a set of nested grids of different resolutions, AMR combines the simplicity of structu red rectilinear grids with the ability to adapt to local changes in complexity in th e domain. Without proper interpolation on the boundaries of grids of different lev els of a hierarchy, discontinuities can arise. Treating locations of data values giv en at cell centers of AMR grids as vertices of a dual grid creates gaps between hierarchy levels. Using an index-based tessellation approach that fills these gaps with ``stitch cells'' we define an interpolation scheme that avoids discontinuities at level boundaries. We use the resulting interpolation scheme to generate volume-rendered images. Modifying transfer functions on a per-level basis allows us to emphasize (or de-emphasize) a specific level and gain a better understanding of the underlying hierarchical structure.

The complexity of physical phenomena often varies substantially over space and time. There can be regions where a physical phenomenon/quantity varies very little over a large extent. At the same time, there can be small regions where the same quantity exhibits highly complex variations. Adaptive mesh refinement (AMR) is a technique used in computational fluid dynamics (CFD) to simulate phenomena with drastically varying scales concerning the complexity of the simulated variables. Using multiple nested grids of different resolutions, AMR combines the topological simplicity of structured-rectilinear grids, permitting efficient computation and storage, with the possibility to adapt grid resolutions in regions of complex behavior. We present methods for direct volume rendering of AMR data. Our methods utilize AMR grids directly for efficiency of the visualization process. We apply a hardware-accelerated rendering method to AMR data supporting interactive manipulation of color-transfer functions and viewing parameters. We also present a cell-projection-based rendering technique for AMR data.