Image Processing Reference
In-Depth Information
2
2
2
ds
ds
v
()
2
ds
ds
v
()
E
=
( )
s
- (
L
+ )
+ ( )
s
(6.39)
int
2
where the length adjustment ε when positive, ε > 0, and added to the contour length
L
causes the contour to expand. When negative, ε < 0, this causes the length to reduce and
so the contour contracts. To avoid imbalance due to the contraction force, the technique can
be modified to remove it (by changing the continuity and curvature constraints) without
losing the controlling properties of the internal forces (Xu, 1994) (and which, incidentally,
allowed corners to form in the snake). This gives a contour no prejudice to expansion or
contraction as required. The technique allowed for integration of prior shape knowledge;
methods have also been developed to allow
local shape
to influence contour evolution
(Williams, 1992; Berger, 1991).
Some snake approaches have included factors that attract contours to regions using
statistical models (Ronfard, 1994) or texture (Ivins, 1995), to complement operators that
combine edge detection with region-growing. Also, the snake model can be generalised to
higher dimensions and there are 3D snake
surfaces
(Wang, 1992; Cohen, 1992). Finally, an
approach has introduced snakes for moving objects, by including velocity (Peterfreund, 1999).
6.3.5
Further snake developments
Snakes have been formulated not only to include local shape, but also phrased in terms of
regularisation
(Lai, 1995) where a single parameter controls snake evolution, emphasising
a snake's natural compromise between its own forces and the image forces. Regularisation
involves using a single parameter to control the balance between the external and the
internal forces. Given a regularisation parameter λ, the snake energy of Equation 6.7 can
be given as
1
E
(
v
( )) =
s
{
λ
E
(
v
( )) + (1 -
s
λ
)
E
(
v
( ))}
s
ds
(6.40)
snake
int
image
s
=0
Clearly, if
= 0, then the
snake will be attracted to the selected image function only. Usually, regularisation concerns
selecting a value in between zero and one guided, say, by knowledge of the likely confidence
in the edge information. In fact, Lai's approach calculates the regularisation parameter at
contour points as
λ
= 1 then the snake will use the internal energy only whereas if
λ
2
σ
η
λ
=
(6.41)
i
2
2
σ
+
σ
η
i
2
is the variance of the noise at the
point (even digging into Lai's PhD thesis provided no explicit clues here, save that 'these
parameters may be learned from training samples' - if this is impossible a procedure can
be invoked). As before, λ
i
lies between zero and one, and where the variances are bounded
as
i
2
appears to be the variance of the point
i
and σ
η
where
σ
1
1
+
= 1
(6.42)
2
2
σ
σ
η
i
This does actually link these
generalised
active contour models to an approach we shall
