Image Processing Reference
In-Depth Information
where
fx
(
x
,
y
) is the first-order differential of the edge magnitude along the
x
axis and
where
c
d
e
0
. .
0
a
b
1
1
1
1
1
b
c
d
e
. .
0
a
2
2
2
2
2
a
b
c
d
e
3
3
3
3
3
A
=
:
:
:
:
:
e
0
. .
0
a
b
c
d
s
-1
s
-1
s
-1
s
-1
s
-1
d
e
. .
a
b
c
s
s
s
s
s
Similarly, by analysis of Equation 6.29 we obtain:
Ay
=
fy
(
x
,
y
) (6.33)
where
fy
(
x
,
y
) is the first-order difference of the edge magnitude along the
y
axis. These
equations can be solved iteratively to provide a new vector
v
〈
i
+1〉
from an initial vector
v
〈
i
〉
where
i
is an evolution index. The iterative solution is
〈〉
i
+1
i
〈〉
(
x
-
x
)
+
Ax
〈〉
i
+1
=
fx
(
x
i
〈〉
,
y
i
)
〈〉
(6.34)
∆
where the control factor ∆ is a scalar chosen to control convergence. The control factor, ∆,
actually controls the rate of evolution of the snake: large values make the snake move
quickly, small values make for slow movement. As usual, fast movement implies that the
snake can pass over features of interest without noticing them, whereas slow movement
can be rather tedious. So the appropriate choice for ∆ is again a compromise, this time
between selectivity and time. The formulation for the vector of
y
co-ordinates is:
〈〉
i
+1
i
〈〉
(
y
-
y
)
+
〈
i
+1
〉
i
〈〉
i
〈〉
Ay
=
fy
(
x
,
y
)
(6.35)
∆
By rearrangement, this gives the final pair of equations that can be used iteratively to
evolve a contour; the complete snake solution is then:
(
)
-1
+
1
1
〈〉
i
+1
i
〈〉
i
〈〉 〈〉
i
(6.36)
x
=
A
I
x
+
fx
(
x
,
y
)
∆
∆
where
I
is the identity matrix. This implies that the new set of
x
co-ordinates is a weighted
sum of the initial set of contour points and the image information. The fraction is calculated
according to specified snake properties, the values chosen for α
and β
. For the
y
co-
ordinates we have
(
)
-1
1
+
1
〈
i
+1
〉
i
〈〉
i
〈〉
i
〈〉
y
=
A
I
y
+
fy
(
x
,
y
)
(6.37)
∆
∆
The new set of contour points then becomes the starting set for the
next
iteration. Note that
this is a continuous formulation, as opposed to the discrete (Greedy) implementation. One
penalty
is the need for matrix inversion, affecting speed. Clearly, the benefits are that co-
ordinates are calculated as
real
functions and the
complete
set of new contour points is
