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the filtration and becomes a boundary at level
j
. More recently,
-intervals have been described
as sets of points in the extended planes, called
persistence diagrams
, as detailed in the next section.
P
K
0
K
1
K
2
K
3
K
4
K
5
K
6
+∞
0
1
1
2
β
0
3
2
4
3
5
β
1
4
6
Figure 11.1:
e persistent homology of a filtered complex can be represented by
P
-intervals [
24
].
11.2 PERSISTENCE DIAGRAMS
In this section we focus on the filtration of a space
X
by the increasing values of a real function
f
defined on it. As we have seen in chapters
8
-
10
, this is a common scenario for shape analysis
applications. e function
f
can be used to define a filtration, made of the subspaces
X
u
D
f
1
.1;u/
. Each subspace
X
u
includes the points of
X
where the function takes values less
than
u
. ese subspaces can be nested by inclusion: whenever
u < v2R
, there is an inclusion
X
u
!X
v
. According to the theory seen in the previous section, the inclusion of
X
u
into
X
v
induces a homomorphism of homology groups
H
k
.X
u
/!H
k
.X
v
/
for every
k2Z
, whose image
is the
k
th persistent homology group
of
.X;f /
at
.u;v/
. e group consists of the
k
-homology
classes that live at least from
H
k
.X
u
/
to
H
k
.X
v
/
. Assuming that this group is finitely generated,
we call its rank the
k
thpersistentBettinumberofthepair
.X;f /
, and denote it by
f
.u;v/
. Roughly
speaking, the
k
-th persistent Betti number
f
.u;v/
counts the number of
k
-homology classes
which survive while passing from
X
u
to
X
v
.
e success of persistent homology in shape analysis applications is due to the fact that a
simple and compact description of the
k
th persistent Betti numbers of
.X;f /
exists, provided
by the corresponding persistence diagram. Persistence diagrams are multi-sets of points in the
half-plane
C
Df.u;v/2RRWu < vg
. In analogy with persistence intervals, a point
.u;v/
in the persistence diagram indicates that there exists a topological event that starts at level
u
of
the filtration and ends at level
v
of the filtration (a cycle that is completed at level
u
and becomes a
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