Image Processing Reference
In-Depth Information
By expanding
E
edge
at the perturbed solution by Taylor series, we obtain
E
( ( ) +
v
s
v
( ) =
s
E
x
( ) +
s
εδ
( ),
s
y s
( ) +
εδ
( )
s
edge
edge
x
y
∂
E
∂
E
edge
edge
2
=
Ex
(),
s
y s
() +
()
s
+
ε δ
()
s
+
O
( )
(6.22)
edge
x
y
∂
x
∂
y
xy
ˆˆ
,
xy
ˆˆ
,
This implies that the image information must be twice differentiable which holds for edge
information, but not for some other forms of image energy. Ignoring higher order terms in
(since
is small), by reformulation Equation 6.21 becomes
E
(
v
( ) +
s
v
( )) =
s
E
(
v
( ))
s
snake
snake
s
=1
2
∂
E
dx s
ds
()
ds
ds
()
+ ( )
dxs
ds
()
d
2
()
+
s
()
2
s
edge
x
x
x
+ 2
( )
s
s
ds
2
2
∂
x
ds
s
=0
ˆˆ
xy
,
2
s
=1
ds
ds
()
+ ( )
d
()
+
s
()
2
s
∂
E
dy s
ds
()
dys
ds
2
()
y
y
y
edge
+ 2
( )
s
s
ds
∂
ˆˆ
(6.23)
Since the perturbed solution is at a minimum, the integration terms in Equation 6.23 must
be identically zero:
2
2
y
ds
s
=0
xy
,
s
=1
2
∂
E
dx s
ds
()
ds
ds
()
+ ( )
dxs
ds
()
d
2
()
+
s
()
2
s
edge
x
x
x
()
s
s
ds
= 0
2
ds
2
∂
x
s
=0
ˆˆ
xy
,
(6.24)
s
=1
ds
ds
()
+ ( )
d
2
()
+
s
()
2
s
∂
E
2
dy s
ds
()
dys
ds
()
y
y
y
edge
()
s
s
ds
= 0
2
2
∂
y
ds
s
=0
xy
ˆˆ
,
(6.25)
By integration we obtain
1
s
=1
dx s
ds
()
dx s
ds
()
!
"
d
ds
()
s
()
s
-
()
s
()
sds
+
x
x
s
=0
s
=0
1
1
dxs
ds
2
()
ds
ds
()
dxs
ds
2
()
!
"
d
ds
x
()
s
-
( )
s
()
s
x
2
2
s
=0
s
=0
s
=1
1
∂
E
2
dxs
ds
()
!
"
d
ds
2
2
1
edge
+
( )
s
() +
sds
2
δ
ˆˆ
() = 0
sds
(6.26)
x
x
2
∂
x
s
=0
s
=0
xy
,
Since the first, third and fourth terms are zero (since for a closed contour, δ
x
(1) - δ
x
(0) =
0 and δ
y
(1) - δ
y
(0) = 0), this reduces to
