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Substituting the terms in the above equation, we obtain a pair of independent
Euler-Lagrange equations,
αx ss + βx ssss + ∂E ext
∂x
=0
and
αy ss + βy ssss + ∂E ext
∂y
=0 .
This is best to solve numerically. The Euler-Lagrange equations with
f x ( i )= ∂E ext /∂x i
and
f y ( i )= ∂E ext /∂y i
are discretized, yielding
α i ( ν i
ν i− 1 )
α i +1 ( ν i +1
ν i )+ β i− 1 ( ν i− 2
2 ν i− 1 + ν i )
2 β i ( ν i− 1
2 ν i + ν i +1 )+ β i− 1 ( ν i
2 ν i +1 + ν i +2 )
+( f x ( i ) ,f y ( i )) = 0 .
Writing the equation in matrix forms, one for x and another for y , yields
Ax + f x ( x, y )=0
and
Ay + f y ( x, y )=0 .
We can now solve for position vectors iteratively by,
x t =( A + γI ) 1 ( γx t− 1
f x ( x t− 1 ,y t− 1 ))
and
y t =( A + γI ) 1 ( γy t− 1
f u ( x t− 1 ,y t− 1 )) .
Amini, Weymouth, and Jain identified several problems with the above cal-
culus of variations approach as originally proposed by Kass, Witten, and Ter-
zopoulos in 1987. In particular, they raised the following objections:
1. There is a significant risk that the above procedure does not converge.
2. Optimality cannot be guaranteed as the Euler-Lagrange equations are a
necessary but not a sucient condition for optimality in a local sense.
3. Constraints are required to be differentiable, which cannot be guaranteed
in general.
4. The requirement for differentiability of the images will lead to instability
unless the image is smoothed leading to poor localization of features.
5. If a snake is not subject to appropriate external forces, it will contract to
a line or a point.
6. If a snake is not placed close to image features, it will not get attracted.
For these reasons, they proposed the dynamic programming approach to
minimizing the energy [3]. This method has now been adopted as the standard
algorithm by most researchers and will be used henceforth in this chapter.
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