Biomedical Engineering Reference
In-Depth Information
arise even when the images are preprocessed with more robust segmentation
approaches, such as image foresting transformation [23] or other fuzzy tech-
niques [70, 76]. These problems make threshold-based methods not very ade-
quate for deformable models initialization.
In the following section, we discuss an approach to improve the automatic
detection of an initial curve.
7.3.2 Mathematical Morphology for Initialization
The use of mathematical morphology to initialize deformable models is a subject
with few references in the literature [59, 76].
For the particular case of medical images, the general idea is to isolate objects
of interest (such as lungs, arteries, heart, bones, etc.) in the scene and to work
with them individually, avoiding neighboring interference of other objects, noise,
spurious artifacts, or background.
Mathematical morphology is a known set of mathematical tools used in dig-
ital image processing area to perform linear transformations on the shapes of
images's regions. There are two basic morphological operations:
erosion
and
dilation
. They will be defined next to make this text self-contained.
Let us take the image
X
and a template
B
, the
structuring element
. They will
be represented as sets in two-dimensional Euclidean space. Let
B
x
denote the
translation of
B
so that its origin is located at
x
. Then the erosion of
X
by
B
is
defined as the set of all points
x
such that
B
x
is included in
X
, that is,
erosion
:
X
B
={
x
:
B
x
⊂
X
}
.
(7.23)
Similarly, the dilation of
X
by
B
is defined as the set of all points
x
such that
B
x
hits
X
, that is, they have a nonempty intersection:
dilation
:
X
⊕
B
={
x
:
B
x
∩
X
=
φ
}
.
(7.24)
These two operations are the base of all more complex transformations in
mathematical morphology. For example, we can use an
opening
which consists
of an erosion followed by a dilation of the result. This operation allows one to
disconnect two different regions for treating them separately. The dual of open-
ing is the close operation, which consists of an erosion over the dilation's result.
The effect of closing an image is rightly the opposite of opening: It connects
weak separated regions (see [40] for a review of other useful operations).
