Biomedical Engineering Reference
In-Depth Information
and subsampling in each direction) [69, 109]. At each resolution level, the simi-
larity
I
(
A
,
T
(
B
)) is maximized w.r.t. the parameters of the transformation using
a Powell's algorithm [110]. We calculate the joint histogram on the overlapping
part of
A
with
T
(
B
) by partial volume interpolation, the latter being known to
provide a smoother cost function.
8.3.2.3
Intensity Correction
The hypothesis of luminance conservation is strong and cannot stand when
considering a large database. Actually, studies nowadays involve distributed
databases. Since the MR acquisition can come from different systems, the inten-
sity difference of MR images of different subjects needs to be corrected prior to
registration. Let us formulate the problem as:
Given two 3D images
I
1
and
I
2
, and their histograms
h
1
and
h
2
, the problem is
to estimate a correction function
g
such that corresponding anatomical tissues
of
g
(
I
1
) and
I
2
have the same intensity, without registering volumes
I
1
and
I
2
.
Estimation of Mixture Model.
The intensity correction
f
should be
anatomically consistent, i.e., the intensity of gray matter (resp. white mat-
ter) of
g
(
I
1
) should match the intensity of gray matter (resp. white matter)
of
I
2
. To ensure this coherence, we estimate a mixture of
n
Gaussian distri-
butions [3, 83, 86, 122, 149] that models the two histograms
h
1
and
h
2
using
the expectation-maximization (EM) algorithm [44] or a stochastic version, the
stochastic expectation maximization (SEM) algorithm [23].
Basically, the EM algorithm consists of two steps: Step E where conditional
probabilities are computed, and step M where mixtures parameters are esti-
mated so as to maximize the likelihood. Contrary to the EM algorithm, the SEM
algorithm consists in adding a stochastic “perturbation” between the E and M
step. The labels are then randomly chosen from their current conditional distri-
bution. The SEM algorithm is supposed to be less sensitive to initialization but
also to converge more slowly than the EM algorithm.
It is well known that the MR histogram can be roughly modeled as the mix-
ture of five Gaussian laws modeling the main tissues: background, cerebrospinal
fluid (CSF), gray matter (GM), white matter (WM) and a mixture of fat and
muscle. The Gaussian mixture has proved to be relevant for fitting MR-T1 his-
tograms [83]. It has also been shown that mixture tissues (interface gray-CSF and
