Image Processing Reference
In-Depth Information
(Transformation III) Perform the following transformation of the above image
H
in the
k
-direction (1D weighted minimum filter). Denote
an output image by
referring to
H
S
=
{
s
ijk
}
(Fig. 5.12 (c)).
z
)
2
;
1
s
ijk
=min
z
{
h
ijz
+(
k
−
≤
z
≤
N
}
.
(5.4)
Then the following property holds true concerning an image
S
.
Property 5.4.
A value of a nonzero voxel in an image
obtained by
Transformation III is equal to the square of the Euclidean distance from a
nonzero voxel to the nearest 0-voxel. This is ascertained easily as follows
[Saito93, Saito94a].
(Proof) From equation (5.2),
S
x
)
2
;
f
ijk
=
0
,
1
g
ijk
=min
x
{
(
i
−
≤
x
≤
L
}
(5.5)
= the squared distance to the closest 0-voxel in the same row as (
i, j, k
)
.
By substituting Eq. (5.5) with Eq. (5.3), we obtain
x
)
2
;
f
xyk
=
0
,
1
y
)
2
;
1
h
ijk
=min
y
{
min
x
{
(
i
−
≤
x
≤
L
}
+(
j
−
≤
y
≤
M
}
x
)
2
+(
j
y
)
2
;
f
xyk
=
0
,
1
=min
y
{
min
x
{
(
i
−
−
≤
x
≤
L
;
1
≤
y
≤
M
}
x
)
2
+(
j
y
)
2
;
f
xyk
=
0
,
1
=min
(
x,y
)
{
(
i
−
−
≤
x
≤
L
;
1
≤
y
≤
M
}
(5.6)
= the squared distance to the closest 0-voxel in the same plane as (
i, j, k
)
.
By substituting the result to Eq. (5.4),
x
)
2
+(
j
y
)
2
;
s
ijk
=min
z
{
min
(
x,y
)
{
(
i
−
−
z
)
2
;
1
f
xyk
=
0
,
1
≤
x
≤
L,
1
≤
y
≤
M
}
+(
k
−
≤
z
≤
N
}
x
)
2
+(
j
y
)
2
+(
k
z
)
2
;
z
{
(
x,y
)
{
(
i
−
−
−
=min
min
f
xyk
=
0
,
1
≤
x
≤
L,
1
≤
y
≤
M
;
1
≤
z
≤
N
}
x
)
2
+(
j
y
)
2
+(
k
z
)
2
;
=min
(
x,y,z
)
{
(
i
−
−
−
f
xyk
=
0
,
1
≤
x
≤
L,
1
≤
y
≤
M
;
1
≤
z
≤
N
}
.
(5.7)
Thus it is shown that an image
S
is the squared EDT of an image
F
.
If a voxel is not cubic but a parallelepiped, the above transformations
should be modified as follows: assume here that the ratio among the lengths
of three edges of a voxel is
1
(
i
-direction):
α
(
j
-direction):
β
(
k
-direction).
Then, using the same notations as in the Transformations I
III,
(Transformation II') Use the following equation instead of Eq. (5.3).
∼
y
))
2
;
1
h
ijk
=min
y
{
g
iyk
+(
α
(
j
−
≤
y
≤
M
}
.
(5.8)
Search WWH ::
Custom Search