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and check how much the properties of the original shape have been preserved/distorted; this
amounts to measuring how much the values of the real function representing those properties have
been altered. e natural pseudo-distance offers a framework in which we can plug in different
properties, in the form of different real functions, so as to measure shape (dis)similarity up to
different notions of invariance.
Let us assume a shape is conceptualized as a pair
.S;f /
with
S
a topological space and
f WS!R
a real function that measures some properties. Let
.X;f
X
/
and
.Y;f
Y
/
be two shapes
to be compared. Formally,
d
np
is defined by setting
d
np
.X;f
X
/;.Y;f
Y
//D
inf
h2H
X;Y
sup
P
jf
X
.P/f
Y
.h.P//j;
2
X
where
h
varies in a subset
H
X;Y
of the set
H
of all homeomorphisms between
X
and
Y
. e
subset
H
X;Y
must satisfy the following axioms: the identity map
id
X
2H
X;X
; if
h2H
X;Y
then
the inverse
h
1
2H
Y;X
; if
h
1
2H
X;Y
and
h
2
2H
Y;Z
then the composition
h
2
h
1
2H
X;Z
[
82
].
If
X
and
Y
are not homeomorphic, the natural pseudo-distance is set equal to
1
. In this way,
two objects are considered as having the same shape if and only if they share the same shape
properties, i.e., the natural pseudo-distance between the associated size pairs vanishes.
e
natural pseudo-distance
d
np
is defined as the minimization of the change in measuring
functions due to the application of homeomorphisms between topological spaces and it is related
to the comparison of two shapes through persistence theory (see chapter
11
).
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