Biomedical Engineering Reference
In-Depth Information
2.1.4. Forces
The net force
f
i
is critical to operation of the DDC. It consists of a weighted
vector sum of an image force (
f
img
i
f
int
i
), an internal force (
), and a damping force
f
i
):
(
f
i
=
w
int
f
int
i
f
img
i
f
i
,
+
w
img
i
+
(2)
i
where
w
img
i
and
w
in
i
are relative weights for the image and internal forces, re-
spectively. Image forces drive vertices toward the closest features. Their design
is application dependent and depends on the particular feature that defines the
object's boundary. For TRUS images of the prostate, image forces act to drive
vertices toward edges. Image forces are defined in terms of an “energy” field,
E
,
associated with each pixel in an image [33]:
E
(
x, y
)=
∇
(
G
σ
∗
I
(
x, y
))
,
(3a)
where (
x, y
) are coordinates of a pixel in the image
I
and
G
σ
is a Gaussian
smoothing kernel with a characteristic width of
σ
. The operator denoted by an
asterisk represents convolution,
denotes the
magnitude of a vector. An image force field can then be computed from
E
at each
pixel in the image using [33]:
∇
is the gradient operator, and
2
∇
E
(
x, y
)
f
img
i
(
x, y
)=
.
(3b)
max
∇
E
(
x, y
)
The energy field as defined by Eq. (3a) consists of local maxima at the edges,
and the force field computed at a pixel near an edge will point toward the edge.
The energy and force fields defined by Eqs. (3a) and (3b) are defined at pixel
locations within the image. The image force acting on vertex
i
of the DDC with
coordinates (
x
i
,y
i
) can be obtained from the force field represented by Eq. (3b)
using bilinear interpolation. As noted by Lobregt andViergever [32], the tangential
component of the image force can cause vertices on the DDC to cluster together,
thus resulting in a poor representation of the prostate boundary. In order to prevent
this, only the radial component of the field, i.e., the component in the direction of
r
i
in Figure 1a, is applied to the vertex.
The internal force minimizes local curvature at each vertex, and attempts to
keep the DDC smooth in the presence of image noise. The internal force at vertex
i
is defined by the following equations:
c
i
·
r
i
−
1
2
(
c
i−
1
·
r
i−
1
+
c
i
+1
·
r
i
+1
)
f
int
i
=
r
i
,
(4a)
v
i
(
t
+∆
t
)=
v
i
(
t
)+
a
i
(
t
)∆
t,
(4b)
