Image Processing Reference
In-Depth Information
4.10 Border surface
In a 2D continuous image, a border line (outline) of a region (or a figure) will
be intuitively obvious if a figure is well formed. In a continuous 3D figure, the
counterpart of a borderline is a border surface of a 3D object (3D figure). For
a 3D digitized figure, the definition (or the concept) of a border surface is not
as clear. There are several ways of defining a border surface of a 3D object in
the 3D digitized space as follows.
(a)
A set of border voxels
: For the definition of a border voxel, see Section 4.5.3
(a). In this case, a border surface of an object (a 3D connected compo-
nent)
P
always belongs to
P
, that is, the border surface of a 3D figure
(connected component)
P
is a set of the outermost layer of voxels in
P
.
(b)
Continuous surface
: The surface of a 3D figure (a 3D connected com-
ponent)
P
is the surface of a 3D continuous figure corresponding to a
digitized figure
P
(see Definition 4.5). Here the border surface of
P
is also
a continuous figure consisting of parts of faces of cubes corresponding to
border voxels.
(c)
A set of border voxels
(
outside
): A set of 0-voxels such that there exists at
least one 1-voxel of a
k
-connected component
P
in their
k
-neighborhood.
In this case a border surface of a 3D object
P
consists of 0-voxels and
belongs to the background.
(d)
Border voxels
(
outside and inside of a figure
): This is the set sum of (a)
and (c) above.
In the discussion concerning border surfaces and their processing, we should
express explicitly which one of the above (a)
(d) or others is treated.
In this topic we define a border surface as follows.
∼
Definition 4.15 (Border surface).
Let us consider an
m
-connected com-
ponent of 1-voxels
P
and an
m
-connected component of 0-voxels
Q
.Thena
border surface
of
P
to
Q
,
B
(
P, Q
) is a set of all voxels in
P
that have at
least one voxel of
Q
in their
m
-neighborhood. Here the
m
-connectivity means
the admissible type of the connectivity of 0-voxels when the
m
-connectivity
is employed for 1-voxels. We call the remainder of
P
after removing
B
(
P, Q
)
from
P
the
inside
of the component
P
.
Let us give again the definition of a border voxel for the convenience of
explanation.
Definition 4.16 (Border voxel).
A border voxel of an
m
-connected com-
ponent
P
of the value
f
(=
0
or
1
)isavoxelin
P
that has at least one voxel
of the value
1
−
f
in the
m
-neighborhood.
Remark 4.22.
For a connected component with a cavity, more than one bor-
der surface exists; one of them is the outside surface and the rest are on cav-
ities. In other words, a border surface consists of more than one component.
Therefore, in order to specify one of them, we need to designate a 1-component
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