Graphics Reference
In-Depth Information
C
Definition 11.3
Multiplicity.
e
multiplicity
f
.u;v/
of
.u;v/2
n
is the finite, non-
negative number defined by:
e
f
.u;v/:
f
.u;v/D
min
e0
uCeve
(11.2)
e
multiplicity
f
.u;1/
of
.u;1/
is the finite, non-negative number given by:
f
.u;1/D
min
e0;uCev
f
.uCe;v/
f
.ue;v/:
(11.3)
Having extended the notion of multiplicity to a multidimensional setting, the definition
of persistence space is completely analogous to the one of persistence diagram for a real-valued
continuous function.
Set
n
C
n
[f.u;1/Wu2R
n
g
and
n
D@
C
D
n
.
Definition 11.4
Persistence Space.
e persistence space Spc
.f /
is the multiset of all points
p2
n
such that
f
.p/ > 0
, counted with their multiplicity, union the points of
n
, counted
with infinite multiplicity.
Similarly to persistence diagrams, persistence spaces share invariance properties with the
associated functions. Also, they can be stably compared by using, e.g., the Hausdorff distance or
a generalization of the bottleneck distance.
Note that, in practice, only an approximation of Spc
.f /
can be computed. Indeed, one
of the main issues in dealing with persistence spaces is that, in contrast to persistence diagrams,
they are not discrete collections of points. In [
39
], the authors show that, in contrast with what
happens for 1-dimensional persistence, it is not possible to have a
discrete and complete
descriptor
of the topological information within a multi-parameter filtration. erefore, in practical appli-
cations we can only consider approximations of persistence spaces, that is, a sample of the points
with positive multiplicity. [
21
,
22
] present solutions to the approximation and the comparison of
persistence spaces.
11.4 CONCEPTS IN ACTION
Persistent homology and statistical shape analysis applied to human jaws
We have seen how the
complex problem of comparing two shapes can be cast as the comparison of persistence diagrams,
which is easy and stable. is has been done in [
85
] to compare human jaws, which is an impor-
tant task in orthodontics to monitor the effects of ongoing treatments. e authors propose to
incorporate the comparison of persistence diagrams in the pipeline of the standardized landmark
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