Image Processing Reference
In-Depth Information
Correctness of this algorithm is shown by enumerating all possible pro-
jection graphs for
Nc
(
)=
1
[Yonekura82a, Yonekura82c]. Examples of a
projection graph are also included in [Yonekura80b, Yonekura80c].
The test as given in Algorithm 4.3 is obtained for the 26-connectivity case
by noting that
x
R
[26]
(X
,
)=
1
,
and
H
[26]
(X
,
)=
Y
[26]
(X
,
x
x
x
)=
0
,
(4.52)
are equivalent to the following:
R
[6]
(X
,
)=
1
,
and
H
[6]
(X
,
)=
Y
[6]
(X
,
x
x
x
)=
0
.
(4.53)
Algorithm 4.3 (Deletability test -
26
-connectivity case).
(1) Calculate the connectivity number
Nc
[26]
(
.Goto(2),if
Nc
[26]
(
x
)at
x
x
)=
is not deletable.
(2) After inverting
1
and
0
of all voxels in the 26-neighborhood, apply the
step (2) and all steps following that of Algorithm 4.2.
1
.Otherwise,
x
For the 18-connectivity case, see [Yonekura80c].
4.9 Path and distance functions
In this section we introduce how to define a distance measure on a digitized
image. We need to understand two concepts, path and distance function. A
path is a sequence of voxels and a distance function is a digital version of the
Euclidean distance in the continuous space.
4.9.1 Path
We will begin with a formal definition of a path.
Definition 4.12 (Path).
A sequence of voxels
x
0
(=
u
)
,
x
1
,...,
x
K
(=
v
)
such that
x
i
+1
∈N
(
m
)
(
1
,m
=
6
,
18
,
18
,
26
x
i
)
,i
=
0
,
1
,...,
K
−
(4.54)
is called a
path
from a voxel
, or more strictly an
m
-connected
path
(
m
=
6
,
18
,
18
,
26
). The number of voxels
K
contained in a path is
called
length
of a path. Given two voxels
u
to a voxel
v
u
and
v
, a path of the specified
connectivity with the minimum length from
is called
minimal path
.
Note here that the length of a path defined above does not always give the
distance between two voxels. Even the length of the minimal path does not
always become a distance measure in the ordinary sense.
u
to
v
Before proceeding to the distance function, we will extend a path to a
more general one called a variable neighborhood path.
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