Quasi Morse-Smale Complex Hierarchical Morse-Smale Complexes

6.2.3 Filtration

Having discussed the resolution of PL artifacts, we may now return to our original goal of applying persistence to 2-manifolds. In Section 2.5, we intro- duced two filtrations, constructed by sorting the vertices of K according to the associated function h and taking the first j lower or upper stars, respectively. Without loss of generality, we will focus on the filtration of lower stars, that is, K i = 1 ≤ j ≤ i St u j . Our goal is to show this filtration is meaningful with respect to persistence and the MS complex. To do so, we show a correspondence be- tween the critical points of a triangulated 2-manifold and the persistence pairs discussed in Section 6.1. As in that section, we will assume that such pairs exist and that the underlying space is torsion-free. Let us consider the topological changes that occur at time i in a filtration. As |K| is a closed connected 2-manifold, only β , β 1 , β 2 are nonzero and β 2 is at most 1 during the manifold sweep. When vertex u i enters complex K i , it brings along its lower star St u i . As shown in Figure 6.6, the lower star consists of a number of wedges. It is clear by induction that each wedge has one more edge than it has triangles. Applying the Euler-Poincaré Theorem Theorem 4.2.5 to our 2-manifold, we get: χ = v − e + t = β − β 1 + β 2 , 6.4 where v , e, f are the number of vertices, edges, and triangles in the filtration, respectively. Once we have unfolded the multiple saddles, vertex u i may be one of the following types: minimum: St u i = u i , so a minimum vertex is a new component and χ i = χ i − 1 + 1. We know that β i = β i − 1 + 1 because of the new component and β i 1 = β i − 1 1 and β i 2 = β i − 1 2 , as there are no other simplices to create such cycles. Substituting, we get χ i = β i − 1 +1+β i − 1 1 +β i − 1 2 = χ i − 1 + 1, as expected. So, a minimum creates a new 0-cycle and acts like a positive vertex in the filtration of a complex. The negative simplex that destroys this 0-cycle is added at a time j i. Therefore, the vertex is unpaired at time i. regular: St u i is a single wedge, bringing in one more edge than triangles, giving us χ i = χ i − 1 + 1 − 1 = χ i − 1 . As St u i is nonempty, no new component has been created and β i = β i − 1 . St u i is also nonempty, no 2-cycle is created either, and β i 2 = β i − 1 2 . Substituting into Equa- tion 6.4, we get β i 1 = β i − 1 1 . Therefore, no topological changes occur at regular vertices. All the cycles created at time i are also destroyed at time i. That is, the positive and negative simplices in St u i cancel each other, leaving no unpaired simplices. Table 6.1. Critical points, the unpaired simplex in their lower star, and the induced topological change. The last is specified in C notation, where β k ++ ⇔ β i k = β i − 1 k + 1, and β k – – is defined similarly. critical unpaired action minimum vertex β ++ saddle edge β – – or β 1 ++ maximum triangle β 1 – – or β 2 ++ saddle: St u i has two wedges, bringing in two more edges than triangles. The new vertex and two extra edges give us χ i = χ i − 1 + 1 − 2 = χ i − 1 − 1. A saddle does not create a new component, being connected in two directions to the manifold through its lower star. If this saddle connects two components, it destroys a 0-cycle and β i = β i − 1 − 1. Otherwise, it creates a new 1-cycle and β i 1 = β i − 1 1 +1. This means that all the simplices in a saddle are paired, except for a single edge whose sign corresponds to the action of the saddle. We have χ i = χ i − 1 − 1 in either case. maximum: St u i = St u i and has the same number of edges and triangles. So, χ i = χ i − 1 + 1 for the single vertex. If the maximum is the global max- imum, β i 2 = β i − 1 2 + 1 = 1. Otherwise, the lower star covers a 1-cycle and β i 1 = β i − 1 1 − 1. As no new component is created, the positive ver- tex is paired with a negative edge, leaving a single unpaired triangle that is positive or negative, depending on the action of the maximum. We have χ i = χ i − 1 + 1 in both cases. Table 6.1 displays the association between critical points and simplices that do not arrive at the same time with their persistence counterparts. We call a critical point positive or negative, according to the sign of its associated un- paired simplex. A 0-cycle is created by a positive minimum and destroyed by a negative saddle. A 1-cycle is created by a positive saddle and destroyed by a negative maximum. This association gives us persistence intervals for critical points, as shown in Figure 6.10. There is a natural relationship between these filtrations and the MS complex. If we relax the definition of a filtration to include k-cells, then we may construct a filtration of an MS complex for applying persistence. In this filtration, a minimum is still a vertex, a saddle is represented by an arc a path of edges, and a maximum is represented by a region a set of triangles. Once again, h + + + + − − Fig. 6.10. Each critical point is either positive or negative. We use time-based persis- tence to measure the life-time of critical points. Fig. 6.11. The critical points of a section of data set Iran in Section 12.5. Minima pits, saddles passes, and maxima peaks are in increasingly lighter shades of gray. Dam¯avand , the highest peak in Iran, is visible over the Caspian sea in the northeast corner. The Mesopotamian valley, in the southwest corner, is bordered by the Zagros mountain range. we get the same persistence intervals as above, since the MS complex captures the critical points and their connectivity. The filtration of simplices is a refined version of the filtration of the MS complex. Both filtrations contain geometry in the ordering of their components. Persistence correctly identifies the critical points through the unpaired simplices. In fact, this is precisely how we will identify critical points for terrains in Chapter 9, as shown in Figure 6.11 for the critical points of the data set Iran. Finally, note that we may also use the filtration composed of upper stars for computation. In this filtration, minima and maxima exchange roles, and saddles change signs. The persistence of critical points remains unchanged, however, as the same pairs of critical points define cycles.

6.2.4 Hierarchy

The length of the persistence intervals of critical points gives us a measure of their importance. We use this measure to create a hierarchy of progres-