Image Processing Reference
In-Depth Information
Fig. 4.6.
Illustration of requirements to the simplicial decomposition.
(3) Two edge elements are connected by one common vertex only.
The simplicious decomposition for a given 3D image is not always uniquely
determined for a given figure.
4.3 Euler number
We will now introduce a widely used topological property of a figure called
the Euler number or the genus.
Definition 4.8 (Euler number, genus).
Consider a 3D digitized figure
C
and its simplicious decomposition. Then the
Euler number
(
genus
)
E
of a
figure (a connected component)
C
is defined by
E
(
C
)=
n
0
−
n
1
+
n
2
−
n
3
,
(4.5)
where
n
k
s denote the numbers of
k
-dimensional simplexes (
k
-simplexes). The
value of the Euler number varies according to the type of the connectivity.
The following equation is known as
Euler-Poincare's formula
in the field
of topology.
Property 4.1.
E
=
b
0
−
b
1
+
b
2
,
(4.6)
where
b
0
= number of 1-components (0-dimensional Betti number),
b
1
=num-
ber of holes in all of 1-components (1-dimensional Betti number),
b
2
=number
of cavities in all of 1-components (2-dimensional Betti number). For a given
binary image
F
, the sum of the Euler numbers of all connected components
in
F
is called the Euler number of an image
F
.
E
(
C
) of a 3D figure
C
defined above is
Remark 4.8.
The Euler number
intuitively described as follows.
E
(
C
)=
1
−
the number of handle in
C
+ the number of cavities in
C.
(4.7)
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