Biomedical Engineering Reference
In-Depth Information
4.2.1. Likelihood map estimation
Likelihood maps require estimation of the conditional probability density
function. In this section we provide a brief introduction to one of the multiple
approaches available for density estimation, and we will use it, for illustration, in
our experiments with gaussian mixture models.
A gaussian mixture model is a semi-parametric technique that allows estima-
tion of an approximation to a density function by combining gaussian functions.
One of the main advantages of this technique is that is able to smooth over gaps
given a sparse data set:
1
exp
−
2
(
x
−
µ
)Σ
−
1
(
x
−
µ
)
.
G
(
x
,µ,
Σ) =
|
Σ
|
1
/
2
2
π
The model is formulated as a linear model in the following way:
K
MG
(
x
,
Θ) =
α
r
G
(
x
,µ
r
,
Σ
r
)
,
r
=1
where
G
(
x
,µ
r
,
Σ
r
) is a multidimensional gaussian function,
k
is the number of
gaussians involved, Θ=(
µ
1
,...,µ
K
,
Σ
1
,...,
Σ
K
) are the gaussian's mean val-
ues and standard deviation matrices.
In order to adjust the parameters of the model Θ, we will use the
expecta-
tion maximization
process. This process assumes that we know the number
K
of
gaussian functions that will approximate the probability density function. Given
the number of gaussian functions desired, a pre-initialization is performed using
a k-means algorithm. This algorithm looks for a certain number of cluster rep-
resentatives in an unsupervised manner. Once the initial parameters are set
θ
0
,
the expectation maximization algorithm optimizes the model parameters given a
set of features points. The basic idea is to iteratively estimate the likelihood of
the data. This is done in two steps: the first,
expectation
, is concerned with the
probability model building assuming that the current parameters are the optimal
ones; the second,
maximization
, looks for the optimal parameters assuming that
the model is the one obtained in the expectation step.
As a result of this process an accurate estimate of an approximation to the
likelihood is computed. This model is used to find the likelihood values of the
whole working image, resulting in the likelihood map. The likelihood map for a
pixel located at (
x, y
) is computed using the following expression:
L
(
x, y,
Θ) =
MG
(
x
(
x, y
)
,
Θ)
However, the likelihood maps suffer form several drawbacks: First, there is a
lack of accuracy at the real boundaries of the regions. It is easy to see that due to the
