issue, the topological changes caused by a geometric simplification algorithm are often touted as a feature of the algorithm Garland and Heckbert, 1997;
Hoppe et al., 1993; Popovi´c and Hoppe, 1997; Schroeder et al., 1992. Dey et al. 1999 describe a topology-preserving decimation operation that disal-
lows topological changes all together. In general, however, geometrical con- cerns override topological ones, and there is little control or understanding of
the resulting topological changes.
There has been little work, moreover, in the area of topological simplifica- tion. Rossignac and Borrel 1993 use a global grid and simplify the topology
within grid elements. He et al. 1996 use low-pass filters for volume grid data sets. Their work does not apply, however, to polygonal objects, unless they
are voxelized. El-Sana and Varshney 1998 approach simplification using α-
shape inspired ideas and convolution. Wood and Guskov 2001 eliminate small tunnels by growing regions on a surface. None of the work considers the
problem using a theoretical foundation or a well-defined topological measure.
8.1.2 Approach and Goals
In this book, I advocate the approach of using persistence within the framework of filtrations. The topological complexity of a filtration is reflected in its Betti
numbers. Consequently, I consider topological simplification to be a process that decreases a space’s Betti numbers. If we view a filtration as a history of
a growing complex, simplification is a process that does not allow short-lived cycles to ever exist. Simply put, a cycle cannot be born unless it has a long
life, and persistence controls the prerequisite life-time for existence. There are two goals in the simplification process:
1. elimination of nonpersistent cycles, 2. and maintenance of the filtration.
As stated, it is not clear whether any conflicts exist between achieving the above two goals.
The simplification process reorders the simplices in the filtration to elimi- nate nonpersistent cycles. It is the entire history of a growing complex that is
being simplified, however, and not a single complex. Some may argue, there- fore, that no simplification has taken place: The same simplices exist as before
in the filtration, but in a new order. This argument is based on notions from geometric simplification, where simplices are removed and new ones are in-
troduced in a single complex. The argument is not valid, however, as the two simplification processes are not analogous. The filtrations in this book exist
in a geometric context, and the order of simplices has meaning. For example,
s u
su t
st tu
v w
sw uv
tw tuw
suw stu stw
suv uw
sv
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17
[ [
[ [
[ [
[
persistence index
Fig. 8.1. The k-triangles that intersect the new axis at p = 2 have persistence 2 or larger.
The simplex pairs representing cycles of persistence less than 2 are boxed.
a topologically simplified filtration of a Morse complex specifies a sequence of geometric modifications to the Morse complex. In other words, there is a
level of indirection between topological simplification and the meaning of that simplification.
8.2 Reordering Algorithms
In this section, we present two reordering algorithms for simplification. These algorithms are successful in simplifying a filtration in most cases. Conflicts
occur, however, between the goals of simplifying and maintaining a filtration. We will discuss such conflicts in the next section and provide algorithms for
simplification in the presence of conflicts.
8.2.1 Persistent Betti Number Algorithm
We get inspiration for simplification methods through an algorithm for com- puting persistent Betti numbers. By the k-triangle theorem Theorem 7.2 in
Section 7.2.1, the p-persistent kth Betti number of K
l
is the number β
l ,p
k
of k
-triangles that contain the point l, p in the index-persistence plane. To com-
pute these numbers for a fixed p, we intersect the k-triangles with a horizontal line at p. Figure 8.1 illustrates this operation by modifying Figure 7.5, the
k -triangles of our example filtration. The algorithm for p-persistent Betti num-
bers is similar to the function B
ETTI
-N
UMBERS
given in Figure 7.3. We go through the filtration from left to right and increase
β
p k
whenever we encounter
β
1 2
3 4
5 6
7 8
9
index persistence
7 6
5 4
3 2
1 1
2 3
4
Fig. 8.2. Persistent 0-th Betti numbers of the first ten complexes in the filtration of Figure 7.2 and for persistence up to 7.
the left endpoint of a k-interval longer than p. Similarly, we decrease β
p k
when- ever there is a right endpoint of a k-interval longer than p, p positions ahead
of us. Figure 8.2 shows the results of the algorithm applied to our example filtration for k
= 0.
8.2.2 Migration
The intersection of the k-triangles and the horizontal line at p is a collection of half-open intervals. We interpret these intervals as k-intervals of a simplified
version of the original filtration. Our goal is to reorder the filtration so that this interpretation is valid, that is, we wish to obtain a new filtration whose Betti
numbers are the p-persistent Betti numbers of the original filtration.
Definition 8.1 persistent complexes Let {K
l
} be a filtration. K
l ,p
is the l-th complex in a reordered filtration, where cycles with persistence less than p are
eliminated. We call K
l ,p
a p-persistent complex. The algorithm for reordering is clear from Figure 8.1. For each pair
σ
i
, σ
j
, we move
σ
j
to the left, closer or all the way to σ
i
. The new position of σ
j
is max
{i, j − p}. If j − p ≤ i, then σ
i
and σ
j
no longer form an interval as they both occupy the same index in the new filtration.
There is a complication in the reordering algorithm that occurs whenever a negative simplex attempts to move past one of its faces. To maintain the filtra-
tion ordering, we must move the face along with its coface. For example, if we increase p to 4 in Figure 8.1, then stu will move to index 11 past its face tu at
index 12. Moving a face along with a simplex will not change any Betti num- bers if the face represents a cycle whose persistence is less than p. At the time
[ [
[ [
[ [
[ [
[
s t
1
u
2 3
4 5
6 7
8 9
10 11
12
tu
13 14
15
stu
16 17
v w
suv stw suw
tuw uw
su sv
uv tw
sw st
[
index
persistence
Fig. 8.3. Alternative visualization of the result of the function P
AIR
-S
IMPLICES
in Section 7.2.1. The squares of s and stw are unbounded and not shown. The light
squares represent 0-cycles and the dark squares represent 1-cycles.
we move it, the face is already co-located with its matching negative simplex, and the two cancel each other’s contributions. We may then grab the pair and
move it with the simplex, moving the pair tu,tuw with stu in our example.
For any moving simplex, however, we must also move all the necessary faces and their matching negative simplices recursively. There is trouble if the face
of a moving negative simplex represents a cycle whose persistence is at least p
. For instance, when stu encounters the edge su, the triangle suv that is paired with su has not yet reached su. There is a conflict between our two goals of
maintaining a filtration and reordering so the new Betti numbers are the old p
-persistent Betti numbers. We will postpone discussion on conflicts until the next section.
8.2.3 Lazy Migration
Our motivation for formulating persistent homology in Equation 6.1 was to eliminate cycles with low persistence. As a consequence of the formulation,
the life-time of every cycle is reduced regardless of its persistence, leading to the creation of k-triangles. A possibly more intuitive goal would be to elimi-
nate cycles with low persistence without changing the life-time of cycles with high persistence. In other words, we replace k-triangles by k-squares as illus-
trated in Figure 8.3. We may also define square Betti numbers, analogs to Betti numbers, for a filtration.
Definition 8.2 square Betti numbers The p-persistent kth square Betti num-
