The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Applying known techniques to convey this information proves hard and sometimes even impossible. We present several adaptions of popular graph drawing approaches to the problem of contour tree drawing and evaluate them. We identify five esthetic criteria for drawing contour trees and present a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria. Our implementation is fast and effective for contour tree sizes usually used in interactive systems (around 100 branches) and also produces readable pictures for larger trees, as is shown for an 800 branch example.

Abstract We present an algorithm that allows stream surfaces to recognize and adapt to vector field topology. Standard stream surface algorithms either refine the surface uncontrolled near critical points which slows down the computation considerably and may lead to a poor surface approximation. Alternatively, the concerned region is omitted from the stream surface by severing it into two parts thus generating an incomplete stream surface. Our algorithm utilizes topological information to provide a fast, accurate, and complete triangulation of the stream surface near critical points. The required topological information is calculated in a preprocessing step. We compare our algorithm against the standard approach both visually and in performance.

Abstract We introduce a novel technique for the construction of smooth stream surfaces of 4th order precision. While common stream surface techniques use linear interpolation for generating seed points for new streamlines in the refinement phase, we use Hermite interpolation. The derivatives needed for Hermite interpolation are obtained by integration along the streamlines. This yields stream surfaces of4th order precision. Additionally, we analyse the accuracy ofthe well known Hultquist approach and our new algorithm and proof that Hultquist's method is exact for linear vector fields. We compare both methods using the well known distance based and a novel error based refinement strategy. Our resulting surface is C1‐continuous, enabling improved rendering among other benefits.

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.

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.