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-
Fig. 6.12. From the left, the maximum and minimum approach and cancel each other to form a degenerate critical point in the middle. This point is perturbed into a regular
point on the right.
Fig. 6.13. The intervals defined by critical point pairs are either disjoint or nested.
sively coarser MS complexes. Each step in the process cancels a pair of critical points, and the sequence of cancellations is determined by the persistence of
the pairs.
Motivation. To simplify the discussion, consider first a generic one- dimensional function h : R
→ R. Its critical points are minima and maxima in an alternating sequence from left to right. In order to eliminate a maximum,
we locally modify h so that the maximum moves toward an adjacent minimum. When the two points meet, they momentarily form a degenerate critical point
and then disappear, as illustrated in Figure 6.12. Clearly, only adjacent critical points can be canceled, but adjacency is not sufficient unless we are willing
to modify f globally. Figure 6.13 shows that the persistence intervals of the critical points are either disjoint or nested. We cancel pairs of critical points in
the order of increasing persistence. The nesting structure is unraveled in this manner from inside out, the innermost pair being removed each time.
Simplification. We now return to function h over M. The critical points of h can be eliminated in a similar manner by locally modifying the height function.
In the generic case, the critical points cancel in pairs of contiguous indices.
a b
c d
e
a Before
e d
c
b After
Fig. 6.14. The cancellation of a and b deletes the arcs ad and ae and contracts the arcs ca
and ab. The contraction effectively extends the remaining arcs of b to c.
More precisely, positive minima cancel with negative saddles and positive sad- dles cancel with negative maxima. We may simulate the cancellation process
combinatorially by removing critical points in pairs from the MS complex. Figure 6.14 illustrates the operation for a minimum b paired with a saddle a.
The operation requires that ab be an arc in the complex. Let c be the other minimum and d
, e the two maxima connected to a. The operation deletes the two ascending paths from a to d and e, and contracts the two descending paths
from a to b and c. In the symmetric case in which b is a maximum, the opera- tion deletes the descending and contracts the ascending paths. The contraction
pulls a and b into the critical point c, which inherits the connections of b.
Definition 6.13 cancellation The combinatorial operation described above and shown in Figure 6.14 for critical points a and b is the cancellation of a and
b .
Cancellation is the only operation needed in the construction of the hierarchy. There are two special cases, namely, when d
= e and when b = c, which cannot occur at the same time. In the latter case, we prohibit the cancellation because
it would change the topology of the 2-manifold. The sequence of cancellations is again in the order of increasing persistence.
In general, paired critical points may not be adjacent in the MS complex. The theorem below shows, however, that they will be adjacent just before they are
canceled, even if the initial QMS complex Q is a poor approximation of the MS complex.
Theorem 6.5 adjacency For every positive i, the i-th pair of critical points
ordered by persistence forms an arc in the complex obtained by canceling the first i
− 1 pairs. Proof
Assume without loss of generality that the i-th pair consists of a negative saddle a
= u
j +1
and a positive minimum z. Consider the component of K
j
that contains z. One of the descending paths originating at a enters this component,
and because it cannot ascend, it eventually ends at some minimum b in the same component. Either b
= z, in which case we are done, or b has already been paired with a saddle c
= a. In the latter case, c has height less than a; it belongs to the same component of K
j
as b and z; and the pair b , c is one of
the first i − 1 pairs of critical points. It follows that when b gets canceled, the
path from a to b gets extended to another minimum d, which again belongs to the same component. Eventually, all minima in the component other than z are
canceled, implying that the initial path from a to b gets extended all the way to z
. The claim follows. We may cancel pairs of critical points combinatorially without the need of
an MS complex, using the simplification algorithms given in Chapter 8. For simplifying terrains, however, we would like to modify the geometry so that
critical points actually disappear. The MS complex provides us with the geo- metric control we need for this modification.
6.3 Linking Number
In the last two sections, we described a measure for topological attributes and showed how it may be applied to simplify a sampled density function. In
this section, we discuss another topological property: linking. Figure 6.15 shows the five linked tetrahedral skeletons we last saw in Chapter 1. Intuitively,
we say an object is linked if components of the object cannot be separated from each other. In this section, we consider the linking number, a topological
invariant that detects linking. As before, we are interested in computing linking in a filtration. To do so, we need to extend the definition of the linking number
to simplicial complexes.
The mathematical background needed for this section is rather brief, so I present it here in the first two sections instead of placing it in a separate chap-
ter. My treatment follows Adams 1994, a highly readable introductory book, as well as Rolfsen 1990, the classic textbook on knots and links. The last
section includes new results. I extend the linking number to graphs and define a canonical basis for the set of homological 1-cycles in a simplicial complex.
