Biomedical Engineering Reference
In-Depth Information
the presence of noise or in images with occlusions or subjective contours, these
edges can be very irregular or even interrupted. Then the analysis of the scene
and segmentation of objects become a difficult task.
In the so-called active contour models [32], an evolving family of curves con-
verging to an edge is constructed. A simple approach (similar to various discrete
region-growing algorithms) is to put small seed, e.g. small circular curve, inside
the object and then evolve the curve to find automatically the object boundary.
For such moving curves the level set models have been introduced in the last
decade. A basic idea is that moving curve corresponds to a specific level line of
the level set function which solves some reasonable generalization of Eq. (11.1).
The level set methods have several advantages among which independence of di-
mension of the image and topology of objects are probably the most important.
However, a reader can be interested also in the so-called direct (Lagrangian)
approaches to curve and surface evolution (see e.g. [16-18, 39, 40]).
First simple level set model with the speed of segmentation curve modulated
by
g
(
|∇
I
0
(
x
)
|
) (or more precisely by
g
(
|∇
G
σ
∗
I
0
|
)), where
g
is a smooth edge
detector function, e.g.
g
(
s
)
=
1
/
(1
+
Ks
2
), has been given in [6] and [36]. In
such a model, “steady state” of a particular level set (level line in 2D image)
corresponds to boundary of a segmented object. Due to the shape of the Perona-
Malik function
g
, the moving segmentation curve is strongly slowed down in a
neighborhood of an edge, leading to a segmentation result. However, if an edge
is crossed during evolution (which is not a rare event in noisy images), there
is no mechanism to go back. Moreover, if there is a missing part of the object
boundary, the algorithm is completely unuseful (as any other simple region-
growing method).
Later on, the curve evolution and the level set models for segmentation
have been significantly improved by finding a proper driving force in the form
−∇
g
(
|∇
I
0
(
x
)
|
) [7-9, 30, 31]. The vector field
−∇
g
(
|∇
I
0
(
x
)
|
) has an important
geometric property: It points toward regions where the norm of the gradient
∇
I
0
is large (see Figs. 11.4 and 11.5). Thus if an initial curve belongs to a neighborhood
of an edge, then it is driven toward this edge by this proper velocity field. Such
motion can also be interpreted as a flow of the curve on surface
g
(
|∇
I
0
(
x
)
|
)
subject to gravitational-like force driving the curve down to the narrow valley
corresponding to the edge (see Fig. 11.6, [40]).
However, as one can see from Figs. 11.7 and 11.8, the situation is much
more complicated in the case of noisy images. The advection process alone is
