Image Processing Reference
In-Depth Information
by
µ
a processing (image operation) to replace by a 1-voxel a 0-voxel
which is at the unit distance from a figure. Then let us denote by
φ
k
(
)
a figure obtained after
k
times of successive application of
φ
to an image
F
F
, and denote by
µ
(
φ
k
(
)) one time of application of
µ
to
φ
k
(
F
F
). Then
if there exists a voxel which belongs to any figure in
φ
k−
1
(
F
), but does
not belong to
µ
(
φ
k
(
)), such a voxel is called a
skeleton with the distance
k
. The entire skeleton with all values of
k
is simply called a
skeleton
of an
image
F
.
(4) Let us call a set of all voxels within the distance
r
from a voxel P a
digital
ball of the radius
r
centered at P. Next, consider an image
F
F
and its
distance transformation
G
. Suppose that a suitable subset
Q
of 1-voxels
F
in
and distance values on them are given. Let us imagine a set of digital
balls centered at each voxel in the set
Q
and the radius of the same value
as the distance value on each voxel. If a set sum of all such digital balls
exactly coincides with the set of all 1-voxels in
F
,thenaset
Q
is called
a
skeleton
of an image
F
or a
skeleton
of a distance transformation
G
.
(4) which are different
from each other. Skeletons derived by those definitions are not exactly the
same. For a given image
Definition 5.5 includes four kinds of definitions (1)
∼
, its skeleton is not always determined uniquely
even if we adopt one of them.
F
The following are reasons for ambiguity in the definition of the skeleton.
(i) A local maximum point cannot be easily defined. For example, the type of
neighborhood employed in the detection of local maxima and local minima
cannot be fixed beforehand. In fact, a local maximum will be different
according to which of the 6- and the 26-neighborhood is adopted. In the
fixed neighborhood DT, we can select the same neighborhood as was used
in the calculation of DT for the extraction of local maxima. This strategy
will not be applied in the case of Euclidean DT. It is assumed that the
second item of Definition 5.5 will satisfy (1) and (3) of Definition 5.5. This
is not always true, however, when some specific types of the neighborhood
are used in the definition of local maxima.
(ii) A set of voxels stated in Definition 5.5 (1), that is, a voxel set from which
an original figure is restored exactly, is not determined uniquely. Even
the minimum set of voxels needed for an original figure to be restored
exactly from it is not always given uniquely. The algorithm to obtain such
a minimal set of voxels may not always be derived easily [Borgefors97,
Nilsson97].
(iii) This definition of the skeleton is less significant, unless a good algorithm
to extract a skeleton and to restore an original figure from the skeleton is
known. Only such a definition of the skeleton, if those algorithms exist, is
acceptable from the viewpoint of practical applications.
(iv) The importance of each of the above factors varies depending on the kinds
of DT, in particular, on the type of the distance function employed in DT.
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