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Fig. 9.5.
Balloon applied to the cell image segmentation problem (from [11]).
9.4 Minimizing the Snake Energy Using the Calculus of
Variations
As minimizing the snake energy is an optimization problem we can use tech-
niques from calculus of variations. In particular, we will use Lagrangian mul-
tipliers.
Following the development of Amini, Weymouth, and Jain [3], we let
E
ext
=
E
image
+
E
forces
where
E
ext
is the external energy. Substituting (9.4)
into (9.3), we have
α
(
s
)
∂s
ν
(
s
)
+
β
(
s
)
∂s
2
ν
(
s
)
E
snake
=
1
0
2
2
∂
2
∂
+
E
ext
(
ν
(
s
))
ds.
(9.9)
For simplicity, we represent the integrand by
F
(
s, ν
s
,ν
ss
), then the Euler-
Lagrange necessary condition for minimization is derived by
∂
2
∂s
2
F
ν
ss
=0
.
∂
∂s
F
ν
s
+
F
ν
=
(9.10)
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