Biomedical Engineering Reference
In-Depth Information
where
x
y
1
x
y
1
=
T
(
p
)
.
(35)
As a segmentation using shape knowledge, the task is to calculate
w
and
pose parameters
p
. The strategy for this calculation is quite similar to the image
alignment for training. The only difference is the specially defined energy function
for minimization.
Energy minimization is based on Chan and Vese's active model as defined by
Eq. (23), which is equivalent (up to a term that does not depend upon the evolving
curve), to the energy functional below:
S
u
A
u
+
.
S
v
A
v
E
cv
=
−
(
µ
2
A
u
+
ν
2
A
v
)=
−
(36)
Before explaining the special features of the above equation, additional terms are
defined. The regions inside and outside the zero level set are denoted as
R
u
and
R
v
:
=
{
(
x, y
)
∈
R
2
:Φ(
x, y
)
>
0
}
.
(37)
The area in
R
u
is
A
u
, the area in
R
v
is
A
v
, the sum intensity in
R
u
is
S
u
, the
sum intensity in
R
v
is
S
v
, the average intensity in
R
u
is
µ
=
R
u
=
{
(
x, y
)
∈
R
2
:Φ(
x, y
)
<
0
}
,
R
v
S
A
u
, and the average
S
A
v
intensity in
R
v
. The Chan-Vese energy functional can be viewed
as a piecewise constant generalization of the Mumford-Shah functional. Based
on this energy definition and the corresponding solution, the next step is to find
the corresponding
w
and
p
to implicitly determine the segmenting curve. The
gradients of the energy
F
are taken with respect to
w
and
p
. The gradient descent
scheme like Hill climbing or Rprop [63], can be utilized to finally find the value
of
w
and
p
.
In addition to the energy function introduced above, several other similar
region-based energy functions are defined in [37]. The reader may refer to Tsai's
work for more details.
In the following, we applied the hand shape into a segmentation procedure
(see Figure 24).
We now present another example using the shape prior active contour based
on level set methods, which involves mitotic cell segmentation. The first step is
to align all the training images until the overlapping error is minimized. Then
principal components analysis (PCA) is then utilized to extract the eigenvalues
and eigenvectors for the shape variance. Next, the mean shape is computed. With
all this information, a target image is tested. The initial contour is the mean
shape. After using the Rprop method, the program yields the final segmentation
is
ν
=
