Image Processing Reference
In-Depth Information
between two voxels is required so that we may use the concept of distance in
the same way as the Euclidean distance in the continuous space. Pay attention
to the distance measure that is only applicable to a pair of voxels ordered
on a cubic array. Noting that each voxel is defined by an integer triad that
represents the number of a row, a column, and that of a plane, we can obtain
the definition below:
Definition 4.14 (Distance function).
If a mapping
A
×
A
→
R
d
(
x
,
y
):
(4.61)
R
is the set of non-
negative real numbers, satisfies all of the following relations, the mapping
(function)
d
(
where
A
is the set of all of integer triads (
i, j, k
)and
x
,
y
) is called a
distance function
(or
distance measure
)ona3D
image.
reflective law:
d
(
x
,
y
)=
0
⇔
x
=
y,
∀
x
∈
A
,
∀
y
∈
A
.
(4.62)
symmetric law:
d
(
x
,
y
)=
d
(
y
,
x
)
,
∀
x
∈
A
,
∀
y
∈
A
.
(4.63)
triangle law:
d
(
x
,
y
)
<d
(
x
,
z
)+
d
(
z
,
y
)
,
∀
x
∈
A
,
∀
y
∈
A
,
∀
z
∈
A
.
(4.64)
This is the digital version of the axiom of the distance metric. A number
of distance functions have been defined in past literatures. Important ones
among them are shown in Table 4.6. Several examples are given in Fig. 4.15
[Kuwabara82]. Note here the shapes of equidistance (contour) surfaces. Ta-
ble 4.6 also shows clear expressions of the distance values between two arbi-
trary voxels.
Remark 4.16 (Euclidean distance).
The well known Euclidean distance
d
E
(
u
,
v
) between
u
=(
u
x
,u
y
,u
z
)and
v
=(
v
x
,v
y
,v
z
), is given as follows and
also utilized anytime if necessary:
v
x
)
2
+(
u
y
−
v
y
)
2
+(
u
z
−
v
z
)
2
]
1 /2
.
d
E
(
u
,
v
)=[(
u
x
−
(4.65)
Remark 4.17.
Distance functions in Table 4.6 have their own advantages
and disadvantages. The 6-neighbor distance and the 26-neighbor distance
have been used most frequently in practical applications, because they are
calculated most easily. As is found by Fig. 4.15, however, the bias from the
Euclidean distance is rather large, and their contours are quite different from
those of the Euclidean distance, that is, a group of concentric circles. The use
of the neighborhood sequence
somewhat compensates for
this defect. In this case, contours become closer to circles. The major reasons
for such errors are that the ratio of the distance to a 6-adjacent voxel to that of
{
6
,
18
}
or
{
6
,
26
}
a diagonal-adjacent one is assumed to be
1
:
1
instead of
1
:
√
2
or ,
1
:
√
3
,
only integers are accepted as the distance values, and only local operations
are employed in the calculation of distance values to save time. Therefore,
improvement is achieved in several different ways.
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