Biomedical Engineering Reference
In-Depth Information
4.2.7. Estimation of intensity probability density functions
A segmented image
I
consists of homogeneous regions characterized by sta-
tistical properties related to a visual consistency. The inter-region transitions are
assumed to be smooth. Let Ω
⊆
R
p
be an open and bounded
p
-dimensional do-
main. Let
I
:Ω
→
R
be the observed
p
-dimensional image data. We assume that
the number of classes
K
is known. Let
p
i
(
I
) be the intensity probability density
function of class
i
. Each density function must represent the region information to
discriminate between two different regions. In our experience, Gaussian models
show satisfactory results in medical image segmentation. In this work, we use such
density functions and associate the mean
µ
i
, variance
σ
i
, and prior probability
π
i
with each class
i
. The priors satisfy the obvious condition:
K
π
i
=1
.
(11)
i
=1
In accordancewith the estimationmethod in [30], themodel parameters are updated
at each iteration as follows:
Ω
Ω
I
(
x
))
2
dx
H
α
(
φ
i
)
I
(
x
)
dx
Ω
H
α
(
φ
i
)(
µ
i
−
,
and
σ
i
Ω
µ
i
=
=
.
(12)
H
α
(
φ
i
)
dx
H
α
(
φ
i
)
dx
We propose the following equation to estimate the prior probability by counting
the number of pixels in each region and dividing it by the total number of pixels:
H
α
(
φ
i
)
dx
Ω
π
i
=
H
α
(
φ
i
)
dx
.
(13)
i
=1
Ω
Here,
H
α
(
·
) is the Heaviside step function defined in [43] as a smoothed differen-
tiable version of the unit step function. The function
H
α
(
·
) changes smoothly at
the boundary of each region. By the above equations, the model parameters are
estimated based on the region information and the main equation.
4.2.8. Evolutionary curve/surface model
The term
ν
=
±
1 in Eq. (5) specifies the direction of front propagation. Sev-
eral approaches were developed to make all fronts either contracting or expanding
(see, e.g., [8]) in order to evolve in both directions and avoid overlaps between
the regions. The problem can be reformulated as classification of each point at
the evolving front. If the point belongs to the associated class, the front expands;
otherwise, it contracts.
•
PDE System
: The classification decision is based on the Bayes decision [44] at
point
x
as follows:
i
∗
(
x
) = arg
i
=1
,··· ,K
(
π
i
p
i
(
I
(
x
)))
.
max
(14)
