Biomedical Engineering Reference
In-Depth Information
quantitative results comparing the performance of RAGS against the standard
geometric snake.
10.2 The Geometric Snake
Geometric active contours were introduced by Caselles
et al.
[1] and Malladi
et al.
[2] and are based on the theory of curve evolution. Using a reaction-
diffusion model from mathematical physics, a planar contour is evolved with a
velocity vector in the direction normal to the curve. The velocity contains two
terms: a constant (hyperbolic) motion term that leads to the formation of shocks
3
from which more varied and precise representations of shapes can be derived,
and a (parabolic) curvature term that smooths the front, showing up significant
features and shortening the curve. The geodesic active contour, hereafter also
referred to as the
standard geometric snake
, is now introduced. Let
C
(
x
,
t
)be
a 2D active contour. The Euclidean curve shortening flow is given by
C
t
=
κ N,
(10.1)
where
t
denotes the time,
κ
is the Euclidean curvature, and
N
is the unit in-
ward normal of the contour. This formulation has many useful properties. For
example, it provides the fastest way to reduce the Euclidean curve length in the
normal direction of the gradient of the curve. Another property is that it smooths
the evolving curve (see Fig. 10.1).
In [3,4], the authors unified curve evolution approaches with classical energy
minimization methods. The key insight was to multiply the Euclidean arc length
by a function tailored to the feature of interest in the image.
Let
I
:[0
,
a
]
×
[0
,
b
]
→
+
be an input image in which the task of extracting
an object contour is considered. The Euclidean length of a curve
C
is given by
L
:
=
|
C
(
q
)
|
dq
=
ds
,
(10.2)
where
ds
is the Euclidean arc length. The standard Euclidean metric
ds
2
=
dx
2
+
dy
2
of the underlying space over which the evolution takes place is modified to
3
A discontinuity in orientation of the boundary of a shape; it can also be thought of as a
zero-order continuity.
