. . . . . . . .
bc x
bx cx
. . . . . . . .
abc ax
acx abx
σ
g
σ
g . . . .
. . . .
σ
h
σ
h
before: after:
b c
a b
c a
starring x
Fig. 8.6. The conflict exists between moving abc toward σ
g
and keeping bc ahead of abc
. We subdivide edge bc and order the new simplices to resolve the conflict.
The persistence algorithm produces new pairs x, bx and ax, acx that have no
effect on Betti numbers. After acx enters, the complex is homotopy equivalent to the old complex just before abc enters. The edge cx replaces bc and the
triangle abx replaces abc in the filter. Consequently, the algorithm produces pairs
σ
g
, abx and cx, σ
h
. As cx is not a face of abx, we have removed the conflict and preserved the Betti numbers of a refined filtration.
8.3.3 Diminution
Often times, simplices have structural meaning in a filtration, and conflicts signal properties of the structure the simplices describe. We may not wish to
tamper with this structure through subdivision, as such action may not have any meaning within our filtration. For example, in
α complex filtrations, sim- plices are ordered according to a particular growth model. The ordering of the
new simplices specified by subdivision in Figure 8.6 might not have a corre- sponding set of balls that would generate the filtration under the growth model.
We may attempt to reduce the effect of conflicts on Betti numbers without eliminating the conflicts. Recall that a simplex pair
σ
i
, σ
j
defines a k-cycle that may be visualized by a k-triangle, as in Figure 8.1. Whenever
σ
i
occurs in a conflict, we allow it to be dragged to a new location. This clearly changes
the Betti numbers of the reordered filtration, so they no longer match the p- persistent Betti numbers of the original filtration. If we just follow the reorder-
ing algorithms from the last section, however, we may never destroy a cycle, as in Figure 8.7a. On the other hand, we may modify the reordering algorithms
to allow
σ
j
to reach σ
i
through the various schemes displayed graphically in Figure 8.7b–e. For example, we also allow
σ
j
to move faster during reorder- ing, whenever
σ
i
is moved. This method creates a pseudo-triangle with the
+ −
p l
a Naive
+ −
p l
b Shift
+ −
p l
c Wormhole
+ −
p l
d Pseudo-triangle
+ −
p l
e Sudden Death
Fig. 8.7. Reordering algorithms and regions of influence. We show the k-triangle in each case for comparison. The regions are transparent filled polygons, and darker re-
gions correspond to areas of overlap.
same area as the cycle’s k-triangle, as shown in Figure 8.7d. Therefore, this algorithm allows each k-cycle to have the same effect on Betti numbers as it
would in the absence of conflicts, but at different times. As such, it seems to be the ideal algorithm for reordering in the presence of conflicts.
8.4 Topology Maps
Before presenting the experiments, we introduce a powerful tool for visualiz- ing the topology of a space. We have already seen that persistence is correctly
visualized as k-triangles in the index-persistence plane, as in Figure 8.1. In general, we may only view the triangles in each dimension separately. For ex-
ample, we may look at the persistent Betti numbers of data set FAU as surfaces in three dimensions, as shown in Figure 8.8. If the only nonzero Betti numbers
are
β ,
β
1
, and β
2
, we may use color to assemble a single image presenting all the values at once. The space of all colors is three-dimensional and may
be parametrized by a three-dimensional coordinate system Foley et al., 1996. There are many such coordinate systems called color models. We use the CMY
color model, as described in Figure 8.9. This color model is appropriate as it is
