Obtaining a minimal skeleton

In section 6.1.3 we extracted the topological skeleton of the projected 2D vector field u. While these critical points and separation curves segment u into areas of different flow behavior, not all of them are necessarily needed to do the segmentation into areas of different F-classification. A redundancy with respect to the segmentation of w is in- troduced to that skeleton since only the w-inflow/outflow behavior has to be considered in order to do the segmentation. In other words, some neighboring substructures of the 2D skeleton may have the same F-classification and thus, the separation between them does not reflect different sectors of F-classification. Figures 6.9a-b illustrate this.

A minimal skeleton representing the different sectors of the F-classification only has to be found: structural elements of the 2D skeleton with identical behavior as their neighbors have to be either merged with them or completely removed. For this, we convert the 2D skeleton into a graph representation, where critical points reflect nodes, separation curves correspond to edges between two nodes, and inner areas are represented by their associated curves and points. We consider a graph to emphasize that any geometrical information can be discarded for the following. To minimize the skeleton, we have to

• remove edges, if they exhibit the same F-classification as their neighboring areas,

• merge areas, if an edge belonging to all of them has been removed,

• remove unconnected nodes (i.e., not connected to an undeleted edge), if they have the same F-classification as their surrounding area,

(a) Redundant skeleton. (b) Substructures of the upper face.

(c) Minimal skeleton. (d) Upper face after removal and merging.

Figure 6.9: F-classified topological skeleton of u before and after the removal of re- dundant elements.

(a) 184 first order critical points. The box around the molecule represents the chosen area for topological simplification.

(b) Topologically simplified representation with one higher order critical point elucidates the far field behavior of the benzene.

Figure 6.10: Topological representations of the electrostatic field of the benzene molecule.

• remove nodes, if they possess the same F-classification as all their connected edges,

• merge two edges, if both have been connected to the same deleted node. Figures 6.9c-d illustrate the outcome of this process. Note that nodes cannot be merged since they correspond to isolated critical points in the skeleton, and areas can not be removed since their union covers the whole 2D domain on s. The obtained graph represents the minimal skeleton needed to distinguish between sectors of different F- classification. Critical points in this skeleton correspond to directions of straight in- flow/outflow, while the remaining curves give the separation surfaces in between the different sectors (i.e., areas).

Our algorithm needed 4 seconds on our hardware to extract, F-classify and min- imize the skeleton of the test data set. This is much faster than the naive approach depicted in figure 6.5. Furthermore, our algorithm guarantees to capture all features on s. For example, figure 6.9c shows in contrast to figure 6.5 an outflow separation curve between two hyperbolic areas on the upper face.

The resulting icon for the example used above is shown in figure 2.27b on page 42.

6.2

Applications

Figures 6.10 – 6.13 visualize the electrostatic field around a benzene molecule. This data set was calculated on a 1013 regular grid using the fractional charges method described in [SS96]. Its topological richness is shown in figure 6.10a: it consists of 184 first order critical points.

This field describes the force of the electrostatic potential of the benzene molecule upon a positive point charge given in a certain location. If such a point charge is situated very close to the molecule, the closest atom will exert the highest force on it, i.e., attract or repel it. The influence of a single atom decreases the farther the point charge is located from the whole molecule. Instead, all atoms have nearly the same influence. One might say that the molecule as a whole is exerting force on a somewhat far located point charge. Thus, it is possible to distinguish between a near and a far field. Furthermore, sources and sinks of the electrostatic field represent minima and maxima of the potential. See [Bad90] for a further discussion of classifying atoms and molecules based on field topology.

(a) Extracted and F-classified skeleton. (b) Minimal skeleton.

Figure 6.11: Benzene data set: high level of topological abstraction.

These properties give a good setting for our algorithm. By placing a large box around the whole molecule, we yield a high level of abstraction. Figure 6.11a shows the box around the whole molecule together with the extracted and F-classified topological skeleton of the projected 2D vector field u. The minimal skeleton is depicted in figure 6.11b. As it can be seen here, there is a star-shaped inflow area (blue), an outflow line (red), and an elliptic area (green) between them in the visible parts of the box. Figure 6.10b shows the icon for this area together with stream lines of w. It clearly shows the behavior of the far field of the benzene molecule, if one compares it with figure 6.12.

In figure 6.13 we lowered the abstraction level by subdividing the domain in 3 (figure 6.13a) or 9 (figure 6.13b-c) subareas. This clearly shows the presence of a more complex topological behavior if we zoom into regions of interest. This is due to the fact that these detailed regions are governed by the near field because the influence of the individual atoms increases. Figure 6.13 shows that topologically rather complex structures are present which consist of complex areas of different F-classification.

The application of our technique to this topologically complex data set shows its usefulness at various levels of simplification: if a large area of interest is chosen, a rough global topological impression about the global behavior of the vector field can be obtained. Focusing the area of interest to particular smaller areas inside, topolog- ically more complex structures become visible and provide a deeper insight into the topological behavior of the vector field.

Figure 6.12: Benzene data set: High level of topological abstraction. Shown is the resulting higher order icon. Additionally, stream lines of the original vector field have been seeded inside the yellow boxes at the bottom and top.

(a) Medium level of abstraction.

(b) Low level of abstraction. View from side.

(c) Low level of abstraction. Front view.

Further Applications of

Topological Concepts and

Methods

In this chapter we show that topology has strong applications besides the structural depiction of the examined vector field.

The theoretical tools from chapter 2 will be utilized in section 7.1 to construct 3D vector fields from a given topological description consisting of points and lines. The result is a piecewise linear 3D vector field having the same topological entities as given by the design. This first approach to modeling 3D vector fields of arbitrary topology has been developed in the course of this thesis, and can serve as a foundation of a number of further applications.

The extraction methods developed in chapters 3 - 5 can also be used to extract vortex core lines as shown in section 7.2. Three different definitions of vortex core lines will be discussed and we show that all three types can be extracted using the Unified Feature Extraction Architecture. Furthermore, we give a unified notation of cores of swirling motion, present a novel way of describing vortex core lines as extremum lines, and link the fields of topology and vortex analysis by showing how vortex core lines can be described as topological separatrices of a derived vector field.

In section 7.3 we show how the results of a topological analysis can be used to steer other visualization techniques. Among other things, extracted features will be used to automate the seeding of stream lines and particles.

7.1

Construction of Higher Order 3D Vector Fields

While most of the vector fields to be visualized are obtained by simulation or measure- ment processes, Theisel presents an approach to modeling 2D vector fields of higher order topology, i.e., consisting of higher order critical points [The02]. This approach is based on two steps. First, the topological skeleton is interactively modeled by defining critical points and separation curves. Then a piecewise linear vector field is constructed which has the topological skeleton modeled before. The approach is also applied for a topology based compression technique.

In this section we show how to extend this approach to 3D. The methods in [The02] are strictly limited to 2D vector fields because of the following reasons:

• [The02] uses a complete segmentation of the areas around a 2D critical point into sectors of different flow behavior. Such a segmentation of 3D critical points did not exist prior to this thesis in the Visualization and Computer Graphics community. In fact, only first order critical points and the index of higher order critical points [MR02] had been considered before.

• Contrary to the 2D case, separatrices of 3D vector fields are particular stream surfaces. They tend to have a complex behavior even for rather simple vector fields (see section 2.2.4), making it a cumbersome (or even impossible) task to model them for instance as a parametric surface.

• There exists no approach to create a vector field which has a stream surface coinciding with a modeled parametric surface.

In this section we present the (to the best of our knowledge) first approach to modeling 3D vector fields of arbitrary topology. We extend the main ideas of [The02] to 3D and give solutions for the three problems mentioned above. This is based on the complete classification of 3D critical points into an arbitrary number of sectors of different flow behavior as given in section 2.4. To overcome the second and third problem, we adapt the concept of connectors (see section 3.2) by modeling not the separation surfaces themselves but only their intersection curves. The resulting algorithm is a two-step approach. First the user models a 3D topological skeleton, then a vector field is auto- matically constructed from this. For the user, the problem of modeling a vector field is reduced to the problem of modeling a topological skeleton by a number of control polygons.

We start with giving some motivation to create 3D vector fields by a modeling approach in the next section. In section 7.1.2 we introduce an approach to model the topological skeleton of a 3D vector field based on the already discussed theory of higher order critical points (section 2.4). Section 7.1.3 shows how to construct a vector field from a modeled skeleton. Finally, we demonstrate examples in section 7.1.4.