Image Processing Reference
In-Depth Information
and a 0-component that are located on both sides of the border surface. This
is the reason that two components
P
and
Q
are included in Definition 4.15.
We need to describe explicitly a border surface of a 1-component
P
that faces
on a 0-component
Q
or a border surface between a 1-component
P
and a
0-component
Q
.
Property 4.7.
A border surface
B
(
P, Q
) in Definition 4.15 (more strictly, a
border surface of a 1-component
P
to a 0-component
Q
) has the following
properties.
(i)
B
(
P, Q
) is a set of border voxels of a connected component
P
.Aborder
voxel here is defined as Definition 4.16.
(ii) A set of voxels in the inside of
P
contains no voxel of a connected compo-
nent
Q
in the
m
-neighborhood. In this sense, two components
P
and
Q
are
separated by the border surface
B
(
P, Q
).
(iii) A border surface
B
(
P, Q
) is the minimum figure (in the sense that the
number of voxels is minimum) satisfying (i) and (ii) above.
Proof of these properties are given in [Matsumoto84].
In this section we explained only the outline of the approach to study
properties of a border surface of a 3D figure and parts of basic properties of
a border surface. Further study will be needed in order to obtain a method
to distinguish individual surfaces of each connected component. For instance,
we need a method to give different marks to different components. Details of
such algorithms will be presented in the next chapter.
Remark 4.23.
Image processing algorithms and the digital geometry of 3D
images were first reported in [Park71]. They discussed the labeling of con-
nected components and gave a definition of the Euler number of a 3D figure.
Only the 6- and the 26-connectivity was treated there. Three-dimensional im-
age processing began to be studied around 1980. For example, detection and
following of border surfaces by Herman et al. [Artzy85, Herman78, Liu77], con-
nectivity index and topology preservation by T.Yonekura et al. [Yonekura80a,
Yonekura80b, Yonekura80c, Yonekura82a, Yonekura82b, Yonekura82c, Yone-
kura82d], and other areas of study was researched. Research reports are col-
lected in [Kong85, Kong89, Toriwaki85a, Toriwaki85b]. [Kong85, Kong89]
mainly pointing out many problems requiring study, but references are lim-
ited and research from Japan is not included. [Toriwaki02a, Toriwaki02b,
Toriwaki85a, Toriwaki85b] will complement this. More recent research is
included in [Toriwaki02a, Toriwaki02b, Toriwaki04, Klette98, Nikolaridis01,
Rogalla01, Bertland01]. A border surface and related topics including bor-
der following algorithms were presented in detail in [Artzy85, Udupa94,
Herman98].
Search WWH ::
Custom Search