A high-level approach to describe the characteristics of a surface is to segment it into regions of uniform curvature behavior and construct an abstract representation given by a (topology)graph. We propose a surface segmentation method based on discrete mean and Gaussian curvature estimates. The surfaces are obtained from three-dimensional imaging data sets by isosurface extraction after data presmoothing and postprocessing the isosurfaces by a surface-growing algorithm. We generate a hierarchical multiresolution representation of the isosurface. Segmentation and graph generation algorithms can be performed at various levels of detail. At a coarse level of detail, the algorithm detects the main feature of the surface. This low-resolution description is used to determine constraints for the segmentation and graph generation at the higher resolutions. We have applied our methods to MRI data sets of human brains. The hierarchical segmentation framework can be used for brain-mapping purposes.

We present a new construction approach for symmetric lifted B-spline wavelets on irregular polygonal control meshes defining two-manifold topolgies. Polygonal control meshes are recursively refined by stationary subdivion rules and converge to piecewise polynomaial limit surfaces. At every subdivision level, our wavelet transforms provide an efficient way to add geometric details that are expanded from wavelet coefficients. Both wavelet decomposition and reconstruction operations are based on local lifting steps and have linear-time complexity.

Trivariate data is commonly visualized using isosurfaces or direct volume rendering. When exploring scalar fields by isosurface extraction is often difficult to choose isovalues that convey ''useful'' information. The significance of visualizations using direct volume rendering depends on the choice of good transfer functions. Understanding and using isosurface topology can help in identifying ''relevent'' isovalues for visualization via isosurfaces and can be used to automatically generate transfer functions. Critical isovalues indicate changes in topology of an isosurface: the creation of new surface components, merginig of surface components or the formation of holes in a surface component. Interseting isosurfaces are usually limited to isolated critical points. Data sets often contain regiions of constant value (i.e., mesh edges, mesh faces or entire mesh cells). We present a method that detects critical points, critical regions and corresponding critical isovalues for a scalar field defined by piecewise trilinear interpolation over a uniform rectilinear grid. We describe how to use the resulting list of critical regions/points and associated values to examine trivariate data.

We review the cell projection method of volume rendering, discussing back-to-front cell sorting, and approximations involved in hardware color computation and interpolation. We describe how the method accommodates cells with non-planar faces using view dependent subdivision into tetrahedra.

The topology of vector fields offers a well known way to show a ''condensed'' view of the stream line behavior or a vector field. The global structure of a field can be shown without time-consuming user interaction. With regard to large data visualization, one encounters a major drawback: the necessity to analyze a whole data set, even when interested in only a small region. We show that one can localize the topology concept by including the boundary in the topology analysis. The idea is demonstrated for a turbulent swirling jet simulation example. Our concept works for all planar, piecewise analytic vector fields on bounded domains.

We present three enchancements to accelerate the extraction of separatrices of three-dimensional vector fields, using intelligently selected ''sample'' streamlines. These enhancements reduce the number of needed sample streamlines and their propagation length. Inflow/outflow matching supports the simultaneous extraction of topologically significant inflow and outflow separatrices in a single pass. An adaptive sampling approach is introduced and used to seed streamlines in a more meaningful and efficient manner. Cell-locking is a new concept that isolates regions of a data set that do not contain separatrices. This concept makes steamline propagation more efficient as streamlines are not propagated through cells that do not influence or contain separatrices. These enhancements enable us to perform separatrix construction for three-dimensional vector field data requiring less overall computation.

We show how to extract a contour line (or isosurface) from quadratic elements--specifically from quadratic triangles and tetrahedra. We also devise how to transform the resulting contour line (or surface) into a quartic curve (or surface) based on a curved-triangle (curved-tetrahedron) mapping. A contour in a bivariate quadratic function defined over a triangle in parameter space is a conic section and can be represented by a rational-quadratic function, while in physical space it is a rational quartic. An isosurface in the trivariate case is represented as a rational-quadratic patch in parameter space and a rational-quartic patch in physical space. The resulting contour surfaces can be rendered efficiently in hardware.

We present a method for efficient computation and storage of approximations of error tables used for error estimation between different time steps in time-varying datasets. The error between time steps is defined as the distance between the data of these time steps. Error tables are used to look up the error between different time steps of a time-varying dataset, their generation itself can be expensive. For n time steps, the exact error look-up table has a memory complexity and preprocessing time complexity of O(n^2). Our approximate error table approach uses trees, where leaf nodes represent original time steps, and interior nodes contain an approximation of the children nodes. The error computed on an edge of a tree describes the distance between the two nodes on that edge. Evaluating the error between two different time steps requires traversing a path between the two leaf nodes, accumulating the errors on the traversed edges. For n time steps, this scheme has a memory complexity and preprocessing time complexity of O(n log(n)), a significant improvement over the exact scheme. As we do not need to calculate all possible n^2 error terms, our approach is a fast way to generate the approximation.

Betti numbers can be used as a means for feature detections to aid in the exploration of complex large-scale data sets. We present a fast algorithm for the calculation of Betti numbers for triangulated isosurfaces, along with examples of their use. Once any isosurface is extracted from a data set, calculating Betti numbers only requires time and space proportional to the isosurfaces, not the data set. Because the overhead of obtaining Betti numbers is small, our algorithm can be used with large data.

We present an algorithm for adaptively extracting and rendering iso- surfaces of scalar-valued volume datasets represented by quadratic tetrahedra. Hier- archical tetrahedral meshes created by longest-edge bisection are used to construct a multiresolution C0-continuous representation using quadratic basis functions. A new algorithm allows us to contour higher-order volume elements efficiently.