Biomedical Engineering Reference
In-Depth Information
following definition of the zero level set:
Φ(
x
(
t
)
,t
)=0
.
(3)
Taking the partial differential of the above equation, we have the following
equations system:
Φ
t
+
∇
Φ(
x
(
t
)
,t
)
.x
(
t
)=0
,
(4)
∇
Φ
|∇
Φ
|
=
x
(
t
)
·
F
n,
where
n
=
,
(5)
Φ
t
+
F
|∇
Φ
|
=0
,
given
Φ(
x, t
=0)
.
(6)
From these equations, the time evolution of the level set function Φ can be de-
scribed, and the zero level set of this evolving function is always identified with
the propagating interface.
There is also an efficient way to solve the initial value problem as just intro-
duced: the narrow band method. For readers who are interested in the details of
this method, please refer to [42].
3.
BASIC APPLICATION OF LEVEL SET METHODS
The level set method includes numerous advantages: it is implicit, parameter
free, allows easy estimation of the geometric properties of the evolving front or
surface, can change the topology, and is intrinsic. Therefore, one can conclude
that it is a very convenient framework for addressing numerous applications of
computer vision and image processing.
In this section we provide some typical examples of applications in image
processing, without complicated controlling or regularizers.
3.1. Curvature-Based Applications
In differential geometry, any simple closed curve moving under its curvature
collapses nicely to a circle and then disappears. The wider applications include the
study of the surface tension of an interface, the evolution of boundaries between
fluids, and image noise removal. In order to prove this theorem and apply it to
further applications, we chose the level set method to track the propagation of the
front. The only factor affecting front propagation is the curvature. The speed
F
is
defined as below. We only need to update the level set value in the whole domain
and choose the positions of the zero crossing pixels as the locations of the new
zero level set:
F
=
f
(
F
curv
)
,
(7)
where
f
() is a function defined on the curvature. In the following example, two
different definitions are used. One is the signed curvature itself, and the other is
