Biomedical Engineering Reference
In-Depth Information
Then the energy functional of Eq. (11) can be written as
E
(
φ
1
,φ
2
)=
λ
1
·
D
1
·
H
(
φ
1
)(1
−
H
(
φ
1
))
d
x
+
λ
2
Ω
·
D
2
·
(1
−
H
(
φ
1
))(1
−
H
(
φ
2
))
d
x
Ω
+
λ
3
·
D
3
·
H
(
φ
2
)
d
x
+
ν
1
·
g
(
|∇
I
|
)
|∇
H
(
φ
1
)
|
d
x
Ω
Ω
+
ν
2
·
g
(
|∇
I
|
)
|∇
H
(
φ
2
)
|
d
x
,
(14)
Ω
where
D
i
is the J-divergence between a voxel's neighborhood probability distribu-
tion and the PDF of the global region
H
(
φ
1
)(1
−
H
(
φ
2
)),
and
H
(
φ
2
), respectively.
ν
1
and
ν
2
are coefficients of the smoothing constraint of
two level set functions.
H
(
φ
2
)), (1
−
H
(
φ
1
))(1
−
2.3. Gradient Flow
To minimize the energy functional, Eq. (10), a two-step algorithm is per-
formed. First, the statistical parameters of the regions are fixed, and one calculates
the Euler-Lagrange equation of the proposed energy functional with respect to the
level set function. For of differentiation of Eq. (11) and numerical implementation,
regularized versions of the Heaviside and Dirac functions are used. We choose the
approximations introduced in [9]:
1+
π
arctan(
ε
)
,
H
ε
(
z
)=
1
2
z
ε
π
(
ε
2
+
z
2
)
δ
ε
(
z
)=
.
Using the gradient descent flow, we obtain the following curve evolution equation:
g
(
|∇
div
∇
∇
∂φ
∂t
=
δ
ε
(
φ
)
φ
φ
ν
·
I
|
)
·
+
∇
g
(
|∇
I
|
)
·
|∇
|
|∇
|
φ
φ
.
+
λ
1
·
D
1
−
λ
2
·
D
2
(15)
The details of the calculations leading to Eq. (15) can be found in Appendix A.
Second, we update the regions' statistical parameters with fixed level set functions.
This is done to find the optimal parameters that minimize the energy functional
with a constant
φ
. As in [29], we can update the parameters of these corresponding
