Biomedical Engineering Reference
In-Depth Information
regions by simply evaluating their sample means and variances:
R
1
I
(
x
)
d
x
I
(
x
)(1
−
H
ε
(
φ
(
x
)))
d
x
Ω
µ
1
=
=
,
(16)
V
1
Ω
(1
−
H
ε
(
φ
(
x
)))
d
x
R
2
I
(
x
)
d
x
I
(
x
)
H
ε
(
φ
(
x
))
d
x
Ω
µ
2
=
=
,
(17)
V
2
H
ε
(
φ
(
x
))
d
x
Ω
R
1
(
I
(
x
)
−
µ
1
)
2
d
x
I
2
(
x
)(1
−
H
ε
(
φ
(
x
)))
d
x
σ
1
Ω
Ω
(1
−
µ
1
=
=
−
,
(18)
V
1
H
ε
(
φ
(
x
)))
d
x
R
2
(
I
(
x
)
−
µ
2
)
2
d
x
I
2
(
x
)
H
ε
(
φ
(
x
))
d
x
σ
2
Ω
Ω
µ
2
=
=
−
.
(19)
V
2
H
ε
(
φ
(
x
))
d
x
Finally, the curve evolution equation can be rewritten as follows:
ν
g
(
|∇
div
∇
∇
∂φ
∂t
=
δ
ε
(
φ
)
φ
φ
·
I
|
)
·
+
∇
g
(
|∇
I
|
)
·
|∇
φ
|
|∇
φ
|
σ
1
+(
µ
1
−
µ
w
)
2
σ
w
+(
µ
1
−
µ
w
)
2
+
+
4
σ
1
4
σ
w
,
σ
2
+(
µ
2
−
µ
w
)
2
σ
w
+(
µ
2
−
µ
w
)
2
−
−
(20)
4
σ
2
4
σ
w
where (
µ
w
,σ
w
) are the local region statistical parameters. When KL divergence
is used, Eq. (20) becomes Paragios's [11, 12, 13] geodesic active regions evolution
equation. Moreover, setting
σ
1
=
σ
2
, the geodesic active regions reduce to Tony
Chan's [9] level set evolution equation. For the proposed energy functional (14),
the two level set evolution equations Eq. (27)-(28) can be found in Appendix A.
2.4. Numerical Implementation
The numerical scheme we used is the finite-difference implicit scheme in-
troduced in [9, 27]. It is unconditionally stable for any time step. Considering
numerical accuracy, let the time step ∆
t
= max(∆
x
1
,
∆
x
2
,...,
∆
x
m
), that is,
the time step is equivalent to the largest space step. When image intensity values
vary drastically, due to the fact that the speed of active contours is dependent on the
voxel intensity value, the velocity discrepancy is large, and thus the convergence
rate is slow. We use normalized velocity to obtain a smooth and slowly varying
velocity as follows:
g
(
z
)=
π
(arctan(
z
θ
))
z
∈
R
,
(21)
