Image Processing Reference
In-Depth Information
useful as the base for deriving the surface or axis thinning algorithms which
will be presented in the following section.
By using procedures of deletability determination given in the previous
chapter, an algorithm for shrinking is obtained as stated below:
Algorithm 5.2 (Shrinking).
Input:
F
=
{
f
ijk
}
. Binary image of the size
I
×
J
×
K
. 1-Voxels represent
figures and 0-voxels represent the background.
(1) Repeat (2)
(4) for
k
=1
,
2
,...,K
.
(2) Repeat (3) for
j
=1
,
2
,...,J
.
(3) Repeat the following for
i
=1
,
2
,...,I
.
If
∼
x
ijk
x
ijk
is a 1-voxel and deletable, then replace
by a 0-voxel (delete
x
ijk
) and proceed to the next voxel.
If otherwise, proceed to the next voxel with doing nothing.
(4) If at least one 1-voxel was deleted in (1)
∼
(3) above, then go to (1). Stop
if no deletion occurred in (1)
∼
(3).
This algorithm can be executed only on the memory area storing an input
image, that is, only one 3D array is needed. Only an input image is referred
to during execution, that is, one 3D array is enough for execution. As soon
as a deletable voxel is found, it is deleted at once. Only a 1-voxel in an input
image is changed, and a 0-voxel is never changed. Thus, this algorithm is a
sequential type according to the classification given in Chapter 2. According to
the classification of algorithms for a 2D image [Yokoi79], this is classified into
the type I below and sequential. An output image (a result) of the shrinking
algorithm applied to an input image
F
is called the
shrunk skeleton
of an
image
F
.
Remark 5.1.
A shrinking algorithm for a 2D image is classified as follows:
(Type I algorithm) Only a 1-voxel can be deleted. A shrunk skeleton of a sim-
ply connected component is a single 1-voxel included in an input connected
component.
(Type II algorithm) Both a 1-voxel and a 0-voxel can be rewritten. A shrunk
skeleton of a simply connected component is a single voxel. This shrunk
skeleton may be either inside or outside of an input connected component.
Details are presented in [Yokoi79].
Many of these properties will be extended to a 3D image, although a
detailed report has not yet been published.
Property 5.1.
A shrunk skeleton of Algorithm 5.2 is always contained in an
input figure. Topological properties of a shrunk skeleton (numbers of cavities,
holes, and components) are the same as those of an input figure.
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