Image Processing Reference
In-Depth Information
Active contours are actually expressed as an
energy minimisation
process. The target
feature is a minimum of a suitably formulated energy functional. This energy functional
includes more than just edge information: it includes properties that control the way the
contour can stretch and curve. In this way, a snake represents a
compromise
between its
own
properties (like its ability to bend and stretch) and
image
properties (like the edge
magnitude). Accordingly, the energy functional is the addition of a function of the contour's
internal energy, its constraint energy, and the image energy: these are denoted
E
int
,
E
image
,
and
E
con
, respectively. These are functions of the set of points which make up a snake,
v
(
s
),
which is the set of
x
and
y
co-ordinates of the points in the snake. The energy functional
is the integral of these functions of the snake, given
s
∈ [0, 1] is the normalised length
around the snake. The energy functional
E
snake
is then:
1
E
=
E s
(
v
( )) +
E
( ( )) +
v
s
E s
(v
( ))
s
(6.7)
snake
int
image
con
s
=0
In this equation: the internal energy,
E
int
, controls
the natural behaviour of the snake and
hence the arrangement of the snake points; the image energy,
E
image
, attracts the snake to
chosen low-level features (such as
edge
points); and the constraint energy,
E
con
, allows
higher level information to control the snake's evolution. The aim of the snake is to evolve
by
minimising
Equation 6.7. New snake contours are those with lower energy and are a
better match to the target feature (according to the values of
E
int
,
E
image
, and
E
con
) than the
original set of points from which the active contour has evolved. In this manner, we seek
to choose a set of points
v
(
s
) such that
dE
d
snake
= 0
v
(6.8)
This can of course select a maximum rather than a minimum, and a second-order derivative
can be used to discriminate between a maximum and a minimum. However, this is not
usually necessary as a minimum is usually the only stable solution (on reaching a maximum,
it would then be likely to pass over the top to then minimise the energy). Prior to investigating
how we can minimise Equation 6.7, let us first consider the parameters which can control
a snake's behaviour.
The energy functionals are expressed in terms of functions of the snake, and of the
image. These functions contribute to the snake energy according to values chosen for
respective weighting coefficients. In this manner, the internal image energy is defined to be
a weighted summation of first- and second-order derivatives around the contour
2
2
2
ds
ds
v
()
+ ( )
ds
ds
v
()
E
=
( )
s
s
(6.9)
int
2
The first-order differential,
d
v
(
s
)/
ds
,
measures the energy due to
stretching
which is the
elastic
energy since high values of this differential imply a high rate of change in that
region of the contour. The second-order differential,
d
2
v
(
s
)/
ds
2
, measures the energy due to
bending
, the
curvature
energy. The first-order differential is weighted by α (
s
) which controls
the contribution of the elastic energy due to point spacing; the second-order differential is
weighted by β (
s
) which controls the contribution of the curvature energy due to point
variation. Choice of the values of α
and β
controls the shape the snake aims to attain. Low
values for α
imply the points can change in spacing greatly, whereas higher values imply
