Biomedical Engineering Reference
In-Depth Information
where we have defined the gradient vector and the Hessian matrix,
∂
V
g
∂
x
,
∂
V
g
∂
y
,
∂
V
g
∇
V
g
=
(8.14a)
,
∂
z
⎛
⎝
⎞
⎠
.
2
V
g
∂
x
2
2
V
g
∂
y
∂
x
2
V
g
∂
z
∂
x
∂
∂
∂
HV
g
=
2
V
g
∂
x
∂
y
2
V
g
∂
y
2
2
V
g
∂
z
∂
y
∂
∂
∂
(8.14b)
2
V
g
∂
x
∂
z
2
V
g
∂
y
∂
z
2
V
g
∂
z
2
∂
∂
∂
Thus, in closing we note that the level set propagation needed for segmentation
only needs information about the first- and second-order partial derivatives of
the input volumes, not the interpolated intensity values themselves.
8.4.1.2 Computing Partial Derivatives
As outlined above, the speed function
F
in the level set equation, Eq. (8.4), is
based on edge information derived from the input volumes. This requires esti-
mating first- and second-order partial derivatives from the multiple nonuniform
input volumes. We do this by means of moving least-squares (MLS), which is
an effective and well-established numerical technique for computing deriva-
tives of functions whose values are known only on irregularly spaced points
[44-46].
Let us assume we are given the input volumes
V
d
,
d
=
1
,
2
,...,
D
, which
are volumetric samplings of an object on the nonuniform grids
{
x
d
}
. We shall
also assume that the local coordinate frames of
{
x
d
}
are scaled, rotated, and
translated with respect to each other. Hence, we define a world coordinate frame
(typically one of the local frames) in which we solve the level set equation. Now,
let us define the world sample points
{
x
d
}
as
x
d
≡
T
(
d
)
[
x
d
]
,
(8.15)
where
T
(
d
)
is the coordinate transformation from a local frame
d
to the world
frame. Next we locally approximate the intensity values from the input vol-
umes
V
d
with a 3D polynomial expansion. Thus, we define the
N
-order poly-
nomials
N
V
(
d
N
(
x
)
=
C
(
d
)
C
(0)
ijk
x
i
y
j
z
k
000
+
,
d
=
1
,
2
,...,
D
,
(8.16)
i
+
j
+
k
=
1
where the coefficients
C
are unknown. Note that these local approximations
