Image Processing Reference
In-Depth Information
Remark 5.14.
Let us add several known results of algorithms to extract
skeletons and to restore an original input figure.
First, the relationship between Def. 5.5 (1) and others in Def. 5.5 is
presently not known exactly in the case of the Euclidean DT. Definition 5.5 (2)
is not applicable to the Euclidean DT as was already stated in (i) concerning
the ambiguity of the definition of the skeleton.
Second, Def. 5.5 (3) and 5.5 (4) are applicable to all types of DT, because
they refer to algorithms to restore an input figure. Concerning extraction of
the skeleton, however, they suggest only a trivial cut and try procedure, that
is, extracting a suitable subset of voxels and testing whether an input figure
is restored or not.
All of Def. 5.5 (1)
∼
(4) are applicable to the fixed neighborhood DT
employing the 6-, 18-, and 26-neighborhood and eventually coincident with
each other, except for the ambiguity pointed out in (i)
(iv) above. The
algorithm to extract skeletons will be obtained by Def. 5.5 (2), (3) and (4)
will give procedures to restore an input figure from its skeleton.
For the variable neighborhood DT, a neighborhood sequence should be
taken into consideration in all of Def. 5.5 (2)
∼
∼
(4). It is conjectured that
Def. 5.5 (2)
(4) are eventually coincident also in the variable neighborhood
DT, although it has not been proven theoretically.
∼
5.5.8 Reverse distance transformation
The process to restore an input figure from its skeleton and distance values
on it is called
reverse distance transformation
. Let us define this more clearly
below.
Definition 5.6 (Reverse distance transformation (RDT)).
Given a set
of voxels
Q
=
{
Q
1
,
Q
2
,...,
Q
m
}
,whereQ
m
=(
i
m
,j
m
,k
m
), and a density
value
{
f
i
m
j
m
k
m
}
on a voxel Q
m
.Then consider a procedure to obtain a set of
all voxels
P
=
{
(
i, j, k
)
}
such that
P=(
i, j, k
);
∃
m, d
(P
,
Q
m
)
<f
i
m
j
m
k
m
,
Q
m
∈
Q
.
(5.16)
This procedure is called
reverse distance transformation
(
RDT
) by the dis-
tance function
d
(
).
The procedure is called
reverse squared distance transformation
(
RSDT
),
if
d
2
(P
,
Q
m
) is used instead of
d
(P
,
Q
m
).
∗∗∗
,
∗∗∗
in Def. 5.6 is the skeleton in the sense of Def. 5.5 (1). If
the density value
f
ijk
The voxel set
Q
is equal to the value of the DT, the result of RDT
P
coincides with an original input image.
The meaning of Eq. 5.16 is as follows. At every voxel P, let us represent
by Q
∗
avoxelof
which is nearest to P. Then, we make a voxel P
belong to
P if the distance
d
(P
,
Q
∗
) is smaller than the distance value of Q
∗
. This rule
is applied to every voxel P.
Q
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