Image Processing Reference
In-Depth Information
where the perturbation is spatial, affecting the x and y co-ordinates of a snake point:
δ
v ( s ) = (δ
x ( s ), δ
y ( s ))
(6.16)
This gives the perturbed snake solution as
v
( ) +
s
εδ
v
( ) = (
s
x s
( ) +
( ), ( ) +
s
y s
εδ
( ))
s
(6.17)
x
y
∧ ∧
( ) and ( ) are the x and y co-ordinates, respectively, of the snake points at the
solution (
where xs
ys
∧ ∧ ∧
s x s y s By setting the constraint energy E con to zero, the snake
energy, Equation 6.7, becomes:
( ) = (
( ), ( )).
1
E
(
v
( )) =
s
{
E
(
v
( )) +
s
E
(
v
( ))}
s
ds
(6.18)
snake
int
image
s
=0
Edge magnitude information is often used (so that snakes are attracted to edges found by
an edge detection operator) so we shall replace E image by E edge . By substitution for the
perturbed snake points, we obtain
1
E
(
v
( ) +
s
ε δ
v
( )) =
s
{
E
(
v
( ) +
s
εδ
v
( )) +
s
E
(
v
( ) +
s
εδ
v
( ))}
s
ds
snake
int
edge
s
=0
(6.19)
By substitution from Equation 6.9, we obtain
E
snake (
v
( ) +
s
v
( ))
s
2
2
s
=1
2
ds
(() +
v
v
() +()
s
d
(()+
v
s
ε δ
v
() +(
s
!
"
=
( )
s
s
E
v
( )+( ))
s
v
s
ds
edge
ds
ds
2
s
=0
(6.20)
By substitution from Equation 6.17,
E
snake (
v
( ) +
s
v
( ))
s
2
2
dx s
ds
()
dx s
ds
()
ds
ds
() +
ds
ds
()
x
x
+ 2
()
s
!
2
2
ds
ds
() +
d
()
s
dy s
ds
()
dy s
ds
()
y
y
+
+ 2
ds
"
s
=1
2
2
2
2
=
dxs
ds
()
dxs
ds
()
d
2
() +
s
d
2
()
s
ds (6.21)
!
x
x
+ 2
s
=0
2
2
2
2
ds
ds
+ ( )
s
!
2
2
d
2
() +
s
d
2
()
s
2
2
dy s
ds
()
dys
ds
()
y
y
+
+ 2
2
2
ds
2
ds
2
"
+
E
(
v
( ) +
s
ε δ
v
( ))
s
"
edge
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