Biomedical Engineering Reference
In-Depth Information
Given a partition of the image domain defined by
S
(
t
), the descriptor of the
inner region is defined as
k
in
(
x
;
t
)
=−
log(
P
in
(
x
;
t
))
(5.12)
where
P
in
is the probability for a voxel
x
to belong to
in
. An analogous definition
holds for the outer region.
Following the ideas of the theory of GAC, the energy functional associated
to the region-based model is defined as
E
(
t
)
=
ζ
k
in
(
x
;
t
)
d
x
+
ζ
k
out
(
x
;
t
)
d
x
+
η
g
(
x
)
da
(5.13)
in
(
t
)
out
(
t
)
S
(
t
)
where
ζ
and
η
control the contribution of the regional and boundary information,
respectively.
The evolution of the regions in which the domain is divided can be simplified
by expressing it in terms of the evolution of the boundary
S
(
t
). So the evolution
can be expressed by the partial differential equation
∂
S
(
x
;
t
)
∂
t
=
F
(
x
,
t
)
n
(5.14)
where
F
(
x
;
t
) is the evolution speed.
The gradient descent flow associated to the minimization of the functional
E
(
t
)is
4
∂
S
(
x
;
t
)
∂
t
=
ζ
(
k
out
−
k
in
)
n
−
η
(
g
κ
+∇
g
,
n
)
n
(5.15)
So the associated level set equation will be
φ
t
+
ζ
(
k
out
−
k
in
)
|∇
φ
|−
η
(
g
κ
|∇
φ
|+∇
g
∇
φ
)
=
0
.
(5.16)
As region descriptors, we propose to use the negative logarithm of the prob-
abilities learned from the kNN rule. When using multiple features, like in our
approach, a non-linear PDF estimation technique can better adapt to the distri-
butions of the underlying tissues than the traditionally used Gaussian PDFs. So,
the probability of the inner region is computed as
P
in
=
P
(
I
(
x
)
=
i
|
C
0
)
(5.17)
4
In the case of time-dependent region descriptors, other additive terms extracted from
in
(
t
)
∂
k
in
∂
t
d
x
and
out
(
t
)
∂
k
out
∂
t
d
x
appear in the Euler Lagrange equation. In the case of Gaussian
and kNN descriptors, it can be shown that these terms have zero contribution (see [16] for the
case of Gaussian descriptors).
