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11. TOPOLOGICAL PERSISTENCE
boundary at level v ). As the length of persistence intervals represented the lifespan of topological
features, in persistence diagrams the distance of a point from the diagonal WuDv represents
the lifespan of the associated topological feature. In turn, the lifespan of a feature reflects its
importance: points far from the diagonal describe important or global features, i.e., the long-
lived ones, whereas points close to the diagonal describe local information such as smaller details
and noise.
Figure 11.2 .b/ shows the 0th persistence diagram obtained from the 0th persistent Betti
numbers of a surface model filtered by the height function (Fig. 11.2 .a/ ). Note that, in the case
of the 0 -degree homology, persistence diagrams substantially coincide with the representation of
size functions by cornerpoints and cornerlines [ 24 ].
.a/
.b/
Figure 11.2: .a/ A 3D model with the “height” function color-coded (left), and the corresponding
0th persistence diagram. e 0th persistent diagram analyzes the behavior of the connected compo-
nents. e topological events registered by the red points are the birth of connected components and
their death when the components joins together. e vertical line can be seen as a point at infinity,
representing a topological feature that will never die . .b/ .
Persistence diagrams inherit their invariance properties directly from the associated func-
tion f , which can be chosen according to the application. Moreover, this kind of representation
allows one to turn the problem of comparing two shapes into the simpler comparison of two cor-
responding persistence diagrams: this can be done by using, e.g., the bottleneck or the Hausdorff
distances (see section 6.2 ), which are proven to be stable with respect to small perturbations of
the data [ 67 ].
e formal definition of persistence diagrams is based on the notion of multiplicity [ 67 , 83 ].
In what follows, the symbol
C [f.u;1/Wu2Rg .
denotes the set
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