Streamline-based Analysis

Streamlines are a fundamental construct that can represent the complete behavior of steady vector fields. As a result, some of the most common techniques to visualize vector fields, for example, glyph-based visualization, line integral convolution [20], and image-based flow visualization [208], highlight the streamline behavior of the flow. Using streamline- based approaches, it is possible to perform higher-order [122, 180, 181, 182, 205, 214, 215] and combinatorial [26, 27, 173, 174, 197, 198] analysis, design [52, 172, 199, 225], compari­ son [44, 201], compression [200], and simplification [206] of vector fields, etc.

2 .5 .1 .1 V e c to r field to p o lo g y . The foremost example of streamline-based analysis is the vector field topology. Topological approaches are attractive as they produce an abstract yet complete description of the global flow behavior, as illustrated in Figure 2.6. Such tehcniques provide a simpler visualization of the flow, and, therefore, form an ideal starting point for analysis as well as visualization.

Topology-based approaches for vector fields describe the global flow structure through the behavior o f its streamlines. The limit sets of streamlines— critical points and closed orbits— are of particular interest, as is the topological skeleton of the flow. The topo­ logical skeleton of 2D vector fields was introduced to the visualization community by Helman and Hesselink [89], who compute it by segmenting the domain using the separa-

trices— streamlines traced from saddles o f the field along the directions of its eigenvectors.

The critical points of the field form the nodes o f the skeleton, which are connected by the separatrices. These ideas were extended to surface flows [90] and 3D flows [91] by the same authors. Globus et al. [69] proposed a similar technique for 3D vector fields by using a combination of glyphs and streamlines to depict the global structure.

At the same time, streamsurface was introduced for visualizing 3D vector fields by Hultquist [96] to represent the surface spanned by an infinite set of streamlines seeded at an arbitrary curve. The streamsurfaces were shown to correspond to separation and attachment lines. Since then the streamsurface computation has been extended to more

F ig u re 2.6. Streamline-based visualizations of steady vector fields highlight the global be­ havior of streamlines. As compared to glyph-based (top-left), streamline-based (top-right), and texture-based (bottom -left) visualizations of vector fields, the topology-based (b ot­ tom-right) visualization provides an abstract summary o f the field by displaying only the important features. The topological skeleton of a 2D vector field is a graph with critical points as nodes connected by saddle separatrices as edges.

efficient and generalized techniques, for example, by Scheuermann et al. [179] and Garth et al. [66]. Some of the other techniques for visualizing 3D vector field topology include those proposed by Thiesel et al. [202] and Mahrous et al. [132].

The following sections discuss the three important components of topological skeleton: streamlines, critical points, and vortices. For a detailed discussion on topological methods in vector field visualization, the reader may refer to the surveys by Garth and Tricoche [65], Laramee et al. [116], Post et al. [168], and Scheuermann and Tricoche [183].

2 .5 .1 .2 T ra cin g strea m lin es. Streamlines are typically computed using numerical integration of varying accuracy, such as the Euler integration or the Runge-Kutta (RK) techniques [170]. These techniques take a small step in the current direction of the vector, and recompute the new direction using an interpolation scheme to repeat the process. The higher-order RK methods are known to be more accurate than the Euler integration. Higher- order means greater accuracy, but more processing time. Another factor that influences the correctness and time behavior is the step-size. A shorter step-size in particle trajectory calculation means the need for more steps and a longer process. Each integration step

introduces some error, which is typically known as an upper bound with respect to the step-size. During the integration, however, these small errors get compounded, and it becomes nontrivial to determine the net error incurred during the streamline tracing.

One way to avoid numerical integration errors is to use an analytical solution to stream­ lines. The local exact method (LEM) of Kipfer et al. [108], which follows the lead of Nielson and Jung [149], solves an ordinary differential equation for piecewise linear vector fields defined on simplicial meshes, making it possible to represent the position of the particle as a function of time. Although the solution is more expensive than numerical integration, LEM can compute the exact path o f a particle in a triangle when its entry point to the triangle is known. Consequently, it removes the need to perform step-wise numerical integration, and, hence, is free from the cumulative integration error. However, the exit point is calculated numerically as an intersection with the triangle edges, which is still prone to numeric errors. Despite this, it is the most accurate technique available since it does not incur integration error. As a result, the LEM will be used for streamline integration in Chapters 6 and 7.

2 .5 .1 .3 D e t e c t io n o f cr itic a l p o in ts . Critical points are singularities (zeros) of the field, and can be classified into various kinds depending upon their behavior. Figure 2.7 shows the classification o f nondegenerate singularities based on the eigenvalues of the Jacobian of the field, as explained by Helman and Hesselink [89].

(a) (b) (c)

F ig u re 2.7. First-order critical points in a vector field can be classified based upon the eigenvalues of the Jacobian of the field computed at the critical point. (a) Sink and attracting focus; (b) source and repelling focus; and (c) saddle and center.

Most of the early attempts at identification of critical points were based on numerical analysis. For example, isolated nondegenerate singularities were identified using numerically integrated tangent curves (streamlines) [89]. Lavin et al. [118] extract singularities in a linear vector field V = A x + o by numerically solving the system A x = 0 for each cell in a triangulated domain. This technique was extended to 3D by Batra and Hesselink [10].

The detection of singularities has also been extended to higher-order singularities. Scheuer­ mann et al. [180, 181, 182] extract higher-order critical points by identifying regions whose winding number is greater than 1 or less than - 1 , and using a polynomial approximation to represent the field. Tricoche et al. [205] analyze higher-order singularities in 2D by partitioning the neighborhood of the singularity into sectors of different flow behavior. The topological analysis o f higher-order singularities provides a foundation for the design and simplification of vector fields. Tricoche et al. [206] simplify the topology o f vector fields by merging clustered singularities within a convex polygon into higher-order singularities. These ideas have been extended to 3D by Weinkauf et al. [214, 215]. It is more challenging to identify singularities in nonlinear vector fields. Li et al. [122] subdivide the simplicial mesh and compute the vector field by side-vertex interpolation in polar coordinates. Singularities are then ensured to be located at the vertices.

In general, detection of singularities can be reformulated as solving nonlinear systems of equations. The Newton-Raphson method and Broyden’s method can be used to solve such systems. However, techniques aimed at solving generic nonlinear systems are sensitive to perturbation and not guaranteed to find all the solutions. For multivariate rational splines, Elber and Kim [50] apply the bisection method to localize the potential regions containing roots. However, computational complexity is a major concern with their method. In contrast, the proposed work aims at computing critical points consistently in both 2D and 3D in a significantly more efficient manner.