Two different concepts are commonly used for extracting extremal structures: the local analysis due to ridges/valleys and the global point of view by means of topology. In this work, we concentrate on the topological view. Nevertheless, we provide a discussion of the similarities and differences between the local and the global approach in the following.
A minimal/maximal point is canonically defined in arbitrary dimensions. How- ever, its higher dimensional generalizations cannot be defined in a canonical way, i.e., several equated definitions exist for extremal lines or surfaces. This is documented throughout the literature [KvD93, Dam99, LLSV99, SWTH07, PS08, SPFT12]. Be- sides their global definition as topological separatrices [Max70], another frequently used concept are Ridges/Valleys, which goes back to De Saint-Venant [dSV52]. The
(see, e.g., Koenderink and van Doorn [KvD93]). Nowadays, there is a consensus that both approaches have their merits, and López et al. [LLSV99] provide an exhaustive evaluation of the equated local and global definitions.
A recent variant of ridges/valleys is the Height Ridge definition [Ebe96]. This definition is local and builds on the first and second derivatives of f , i.e., the gradient ∇ f and the Hessian H f . The local ridge definition investigates the convexity/concavity of f in a local neighborhood of a given point. As elegantly formulated by Peikert and Sadlo [PS08], ridge lines in a scalar field are found at locations where the vectors ∇ f and H f · ∇ f are parallel. They can be extracted using the Parallel Vectors operator [PR99, POS+11]. In case of 3-dimensional scalar fields, ridge surfaces can be found as parts of the zero level set ∇ f · e1with λ1< 0, where e1is the eigenvector to the smallest
eigenvalue λ1 of H f . A consistent orientation of the eigenvectors at the vertices of
each cell is necessary to extract this level set. This can be achieved using a principal component analysis [FP01].
In contrast, separatrices represent the border of adjacent compartments where f behaves monotonically [Sma61b]. The separatrices literally separate the gradient flow within these compartments. Let us consider the following example: the analytic func- tion f :[−1, 1]2→ R consisting of an anisotropic Gauss function g, a local distortion
d, and a rotation r with angle θ =π/4:
g(x, y) = 0.05 e−(200 x2+300 (y−0.1768)2) d(x, y) = x + g(x, y) (cos(5x) sin(20y + 10.6066))) r(x, y) = cos(θ ) − sin(θ ) sin(θ ) cos(θ ) d(x, y) y
f(x, y) = sin(2π r1(x, y)) cos(2π r2(x, y))
(5.1)
Figure 5.1 provides an illustration of this example. Consider f as a height field and assume that a separatrix is given connecting a saddle with a maximum, see Figure 5.1a. Clearly, the central separatrix separates the flow (indicated by the arrows) of the left and right side. Imagine that rain pours onto this terrain and water assembles around the two minima. The water is rising and the shape of the water level is defined by the integral lines of the flow. If the water level continues to rise, it will reach the saddle at some time. This is the lowest point of separation. Continuing the rainfall, the two water basins meet themselves along the separatrix. The water on the left side is separated from the water on the right side. The coloring of the arrows in Figure 5.1a indicates this. This separation represents a (weak) monotony break of f .
The monotony break that can be observed in Figure 5.1a is not locally given. The saddle point is connected to two maxima and to two minima. These two minima are essential in order to understand the extremal characteristic of the central separatrix. Each of the two minima gives rise to compartments that meet each other along the separatrix. In order to check for a monotony break, one needs to investigate if a set of particles inserted in the gradient flow assembles in the same minimum or not. In contrast to the local ridge definition, this kind of characteristic is global and cannot be determined locally.
In fact, this global nature of a separatrix makes it very useful for feature extraction. Real-world data sets are usually affected by noise. The small fluctuations caused by the noise create local distortions similar to the example shown in Figure 5.1b. Within
(a) Ridge lines. (b) Separatrices.
Figure 5.2: Extremal lines of the scalar field (3.1) following two different definitions.
small regions, the local attracting/repelling behavior of a separatrix can change. This happens when the local landscape described by f changes from convex to concave, or vice versa. However, its global nature prevents interruptions along the separatrix in contrast to the local ridge/valley definitions. In order to interrupt the course of a separatrix, the distortion must be so strong that new critical points are introduced.
It has been pointed out by Sahner et al. [SWTH07] that every separatrix can be assigned a ridge counterpart: each saddle point of f gives rise to ridges as well as separatrices (they do not need to coincide at any other places). However, not every ridge can be assigned a separatrix counterpart [SWTH07]. Intuitively, this happens when a ridge-creating fluctuation of f – as shown in Figure 5.1 – does not break its monotony. This is also nicely shown by the “Ridges without Critical Points” example of Peikert and Sadlo [PS08].
By definition (Section 2.2), separation lines are tangential to the gradient ∇ f . Ridges lines, on the other hand, are defined as features where ∇ f is parallel to H f · ∇ f . In fact, additionally requiring that they are also tangential to ∇ f yields an overdeter- mined system as discussed by Schindler et al. [SPFT12].
There are several differences between ridges and separatrices from an algorithmic point of view: ridges are local features based on the first and second derivatives, which eases their extraction using parallel algorithms. This might be difficult for separatrices due to their global nature. On the other hand, separatrices can be extracted combinato- rially without any derivatives.
Figure 5.2 shows the ridge lines (left) and separation lines (right) of the three di- mensional scalar field defined by Equation (3.1) on page 53. It can be seen that they largely coincide and that both are in the center of the shown iso-surfaces which con- firms their extremal characteristic. Interestingly, this example indicates that the local and the global approach besides their different definitions can give very similar results. Therefore, we will compare our topological approach in the subsequent sections to the local analysis of extremal features.