Biomedical Engineering Reference
In-Depth Information
n
. With the Gaussian distribution assumption,
D
i
can be
It satisfies
n
≤
s
≤
2
simplified as follows:
σ
w
+(
µ
w
−
µ
i
)
2
σ
i
µ
w
)
2
+(
µ
i
−
1
2
D
i
=
+
−
,
4
σ
i
4
σ
w
where
σ
i
is the variance of region
R
i
, and
σ
w
is the variance of a neighborhood
region.
G
i
is the characteristic function used to represent region
R
i
with
n
level set
functions. The evolution equation can be obtained directly from an Euler-Lagrange
formulation. Let
s
n
F
(
φ
1
,φ
2
,...,φ
n
)=
λ
i
·
D
i
·
G
i
(
φ
1
,φ
2
,...,φ
n
)+
ν
j
i
=1
j
=1
··
g
(
|∇
I
(Γ
j
)
|
)
·|∇
H
(
φ
j
)
|
.
(23)
Equation (11) can be written as
E
(
φ
1
,φ
2
,...,φ
n
)=
F
(
φ
1
,φ
2
,...,φ
n
)
d
x
.
Ω
For any
φ
j
, keeping
µ
j
and
σ
j
fixed, the Euler-Lagrange equation of Eq. (11) is
∂F
∂φ
jk
m
∂F
∂φ
j
−
d
dx
k
k
=1
s
m
∂G
i
∂φ
j
−
d
dx
k
(
g
(
|∇
∂
|∇
φ
j
|
∂φ
jk
=
λ
i
·
D
i
·
ν
j
·
δ
(
φ
j
)
·
I
|
)
·
)
i
=1
k
=1
s
m
∂G
i
∂φ
j
−
d
dx
k
(
g
(
|∇
φ
jk
|∇
φ
j
|
)
λ
i
·
D
i
·
ν
j
·
δ
(
φ
j
)
·
|
)
·
=
I
i
=1
k
=1
div
g
(
|∇
s
∂G
i
∂φ
j
−
∇
φ
j
=
λ
i
·
D
i
·
ν
j
·
δ
(
φ
j
)
·
I
|
)
·
|∇
φ
j
|
i
=1
g
(
|∇
div
∇
s
∂G
i
∂φ
j
−
φ
j
=
λ
i
·
D
i
·
ν
j
·
δ
(
φ
j
)
·
I
|
)
·
|∇
φ
j
|
i
=1
∇
,
φ
j
+
∇
g
(
|∇
I
|
)
·
(24)
|∇
φ
j
|