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 2 ,...,φ n )=
λ i ·
D i ·
G i ( φ 1 2 ,...,φ n )+
ν j
i =1
j =1
··
g ( |∇
I j ) | ) ·|∇
H ( φ j ) |
.
(23)
Equation (11) can be written as
E ( φ 1 2 ,...,φ n )=
F ( φ 1 2 ,...,φ n ) d x .
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 |
 
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