Biomedical Engineering Reference
In-Depth Information
Figure 1. Left
: A graph of the image intensity function
I
0
(
x
)
.
Right
: Image given by the
intensity
I
0
(
x
)
plotted together with arrows representing the vector field
−∇g
(
|∇I
0
(
x
)
|
)
.
See attached CD for color version.
there is no mechanism to reverse the motion since
F
is always positive. Moreover,
if there is a missing part of the object boundary, such an algorithm, as with any
other simple region-growing method, is completely useless.
Not only the segmentationmodels but also image smoothing (filtering) models
and methods have been suggested either using the original Perona-Malik idea of
nonlinear diffusion depending on an edge indicator (cf. [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34]) or using geometrical PDEs in the level set formulations
(cf. [35, 36, 37, 38, 16, 39, 40, 41, 42]).
Later on, the level set models for image segmentation were significantly im-
proved by introducing a driving force in the form
I
0
(
x
)
|
) ([43, 44, 45,
−∇
g
(
|∇
−∇
g
(
|∇
I
0
(
x
)
|
) has an important geometric property:
it points toward regions where the norm of the gradient
46, 47]). The vector field
I
0
is large (see Figure 1,
illustrating the 2D situation). If an initial segmentation curve or surface belongs to
a neighborhood of an edge, it is driven automatically to this edge by the velocity
field.
However, the situation is more complicated in the case of noisy images (see
Figure 2). The advection is not sufficient, the evolving curve can be attracted
to spurious edges, and no reasonably convergent process is observed. Adding a
curvature dependence (regularization) to the normal velocity
F
, the sharp curve
irregularities are smoothed, as presented in the right-hand part of Figure 2. It turns
out that an appropriate regularization term is given by
g
0
k
, where the amount of
curve intrinsic diffusion is small in the vicinity of an un-spurious edge. Following
this 2D example, we can write the geometrical equation for the normal velocity
v
∇
