The 3-dimensional Case

We now extend the notion of separatrix persistence to three dimensions, which assigns an importance measure to each point of a separation line and surface.

We need to consider two differences compared to the 2-dimensional case: Firstly, an optimal simplification based on the Morse inequalities of a Morse-Smale complex cannot be expected anymore, see Section 3.5.5. Hence, the coarsest level of detail in V probably contains insignificant and spurious topological structures. Secondly, the hierarchy is no longer based on persistent homology. As discussed in Section 4.4, a pair of critical points which is removed in the simplification process is not necessarily a persistence pair. The definition of separatrix persistence in three dimensions will address these facts.

Similar to the 2-simensional case, we first assess the local strength of separation by considering the evolution of isocontours. Consider a single 1-saddle s and its com- binatorial separation surface S as shown in Figure 5.5. The separation surface is the boundary between the two volumes governed by the minima m1and m2. Our goal is

to define the strength of this separation for each point on S. To do so, we observe how the evolution of an isocontour affects the separation between the volumes. Let f(m1) > f (m2). Also note that f (s) > f (m1) by construction. For an increasing iso-

S s

m1 m2

Figure 5.5: Illustration of the local feature strength of separation surfaces. The evolution of isocontours is de- picted as yellow surfaces.

Sa Sb

s

sa sb

`2

Figure 5.6: Illustration of the local fea- ture strength of separation lines. The evolution of isolines is depicted as black lines.

zero components r < f (m2) < f (m1) < f (s)

one component around m2 f(m2) ≤ r < f (m1) < f (s)

two components around m1and m2 f(m2) < f (m1) ≤ r < f (s)

the two components merge at the saddle s f(m2) < f (m1) < r = f (s)

one component intersecting the separation surface f(m2) < f (m1) < f (s) < r

The separation surface is pierced by the isocontour for the first time when the two components merge at the saddle. This infinitesimal small hole constitutes a breach of the separation between the two volumes. In other words, the saddle is the weakest point of separation between the two volumes. With further increasing r, the hole becomes larger, and we find that the outer parts of S provide the strongest separation between the volumes. Mathematically speaking, we define the local strength of separation I3DS

for all points x ∈ S as

I3DS(x) = P(s) + h(x, s), (5.5)

which has its smallest value at the saddle: I3DS(s) = P(s) denotes the persistence of s

and thereby the “life time” of the weakest point on the separation surface. Note that P(s) = P(m1) = h(s, m1), if (s, m1) is a persistence pair, which is the case in a simple

scalar field as described above. Later on, we will consider scalar fields with more topological structures, where(s, m1) may not be a persistence pair. A statement similar

to (5.5) holds for combinatorial separation surfaces emanating at 2-saddles.

The definition for the two separation lines `1, `2 of a saddle s follows the same

ideas, except that these lines do not separate volumes, but areas on two neighboring separation surfaces Sa, Sbcoming from two saddles sa, sbwith f(sa) > f (sb) (Figure

5.6). Therefore, we observe the evolution of isocontours of f restricted to these sur- faces, i.e., we consider isolines. They emanate at saand sb, merge at s, and create an

(a) Local Importance (b) Separatrix Persistence

Figure 5.7: Extremal lines scaled by local feature strength (left) and separatrix persis- tence (right). Small fluctuations in the data cause an improper representation of the dominant extremal structures using the local feature strength. Separatrix persistence, in contrast, reveals the global structure.

increasingly larger hole in`1and`2. It turns out, we can define the local strength of

separationI3D`for all points x ∈`1∪ `2very similar to (5.5):

I3D`(x) = P(s) + h(x, s). (5.6)

The use of the local strengths of separation I3D`and I3DSdoes not accommodate the

global gestalt of the function; as shown in Figure 5.7a for a synthetic data set. Local perturbations cause an erratic and unintuitive behavior of I3D` and I3DS – if applied

directly to the unsimplified Morse-Smale complex.

We use the hierarchy of combinatorial gradient fieldsV = (Vi) from Section 3.5.3

to successively remove small perturbations and gain an increasingly global view of the topological features. Note that the connectivity of critical points changes withinV . Additionally, combinatorial separation lines and surfaces may merge during the sim- plification process. A point x on a separatrix can therefore separate multiple critical points. Hence, we need to determine the maximal strength of separation for x by con- sidering (5.5) and (5.6) over all elements of(Vi). We define for separation surfaces:

Definition 2 (Separatrix Persistence 3D for Surfaces) Given is a hierarchy of com- binatorial gradient fieldsV = (Vi)i=0,...,m. Let S be a separation surface. At a hierar-

chy level i, the surface S emanates at a saddle s. At most two extrema are connected to the saddle s at level i: let e denote the extremum with the smallest persistence. The Separatrix Persistence S3Dis defined for each point x∈ S as

S3D(x) = max

i=0,...,m(Pmax(s, e) + h(x, s)) , (5.7)

wherePmax(., .) denotes the maximal persistence of two critical points.

= max

i (P(e) + h(x, s)) (5.10)

but only if the saddle-extremum pairs (s, e) obtained by the hierarchy are actually persistence pairs. As discussed in Section 4.4, this is not necessarily the case in 3- dimensional scalar fields. This may yield the situation that a saddle point with a low persistence is connected to an extremum with a high persistence. Taking only the per- sistence of the saddle into account would result in an underestimation of the emerging separatrices. We therefore use the maximum of persistence Pmax(s, e) in (5.7) of two

neighboring critical points(s, e), since it estimates the largest strength of separation. Note that a separatrix exists only up to a level Vjin the hierarchy(Vi). Hence, (5.7)

is effectively computed for the levels V0, . . . ,Vj, and not further examined for levels

k> j.

Separatrix persistence for separation lines follows a similar scheme:

Definition 3 (Separatrix Persistence 3D for Lines) Given is a hierarchy of combina- torial gradient fieldsV = (Vi)i=0,...,m. Let` be a separation line. At a hierarchy level i,

the separation line` emanates from a saddle s . From the saddle s a separation surface emanates which has several other saddles in its boundary: let t denote the one with the smallest persistence. TheSeparatrix Persistence S3Dis defined for each point x∈ ` as

S3D(x) = max

i=0,...,m(Pmax(s,t) + h(x, s)) . (5.11)

Figure 5.7b shows the minimal lines of a synthetic data set that have been scaled by separatrix persistence. In contrast to the local importance I3D, separatrix persistence

reflects the global gestalt of the function well.

A straightforward approach in computing S3D(x) is to iterate over each saddle s in

each level of the hierarchy(Vi)i=0...mand compute (5.7) and (5.11) for each point x on

the separatrices of s. However, a more efficient approach is possible by exploiting that a simplification step Vi→ Vi+1creates only local changes in the Morse-Smale complex

(Section 3.5.5): only one pair of critical points gets removed with every simplification step. Hence, we compute (5.7) and (5.11) only for the separatrices that are affected in this step; and continue to the next level in the hierarchy. At Vm, we evaluate (5.7) and

(5.11) for the separatrices of the (few) remaining saddles. Since separatrices are given as discrete set of links in the cell graph G, we assign the importance values only to the nodes incident to these links. An explicit sampling of separatrices is not necessary. This makes computing separatrix persistence very efficient, and can actually be done while building the hierarchy.