Image Processing Reference
In-Depth Information
The use of the squared RDT is more convenient in the case of Euclidean
RDT. The following property holds in which
Q
is a skeleton and each voxel
Q
m
in
Q
has a density value
f
i
m
j
m
k
m
which is equal to the value of the squared
DT.
Property 5.5.
Given an input image (skeleton image)
F
=
{
f
ijk
}
,
f
ijk
=
the value of the squared DT
,
if (
i, j, k
)
∈
Q
(skeleton)
,
(5.17)
0
,
otherwise
,
the set
P
=
{
(
i, j, k
)
}
given in Eq. 5.16 of Def. 5.6 is obtained by calculating
the following image
H
=
{
h
ijk
}
,
p
)
2
−
q
)
2
−
r
)
2
}
h
ijk
=max
p,q,r
{
f
pqr
−
(
i
−
(
j
−
(
k
−
.
(5.18)
The set
P
is obtained by extracting all voxels such that
h
ijk
>
0.
Property 5.6.
The image
given in Property 5.5 is obtained
by the following procedure including intermediate images
H
=
{
h
ijk
}
g
(1)
G
(1)
=
{
ijk
}
and
g
(2)
G
(2)
=
.
(Transformation I)
{
ijk
}
F
→
G
(1)
=
g
(1)
;
g
(1)
ijk
p
)
2
}
{
ijk
}
=max
p
{
f
pjk
−
(
i
−
.
(5.19)
(Transformation II)
g
(2)
;
g
(2)
ijk
g
(1)
G
(1)
→
G
(2)
=
q
)
2
}
{
ijk
}
=max
q
{
iqk
−
(
j
−
.
(5.20)
(Transformation III)
g
(2)
G
(2)
→
H
r
)
2
}
=
{
h
ijk
}
;
h
ijk
=max
r
{
ijr
−
(
k
−
.
(5.21)
This property is easily derived by successively substituting Eqs. (5.19) and
(5.20) into Eq. (5.21).
5.5.9 Example of RDT algorithms - (1) Euclidean squared RDT
Property 5.6 is realized by performing three 1D transformations corresponding
to
i
,
j
,and
k
axis directions which are executed consecutively and indepen-
dently. This fact suggests that an RDT algorithm is constructed by a serial
composition of 1D operations, each of which corresponds to an operation in
each coordinate axis direction. The following algorithm is based on this idea
[Saito94b].
Algorithm 5.11 (Reverse Euclidean squared distance transforma-
tion).
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