Biomedical Engineering Reference
In-Depth Information
encourages transformations that project
R
into complex parts of
F
. The third
term, the (negative) joint entropy of
R
and
F
, contributes when
R
and
F
are
functionally related. Maximizing MI tends to find as much of the complexity that
is in the separate volumes (maximizing the first two terms) as possible so that at
the same time they explain each other well (minimizing the third term) [25, 31].
1.5.2.1
Computation of MI Metric
Based on the definition of relative entropy, also known as Kullbak Leibler dis-
tance, between two probability mass functions, Eq. (1.19) can be written in terms
of probability distribution functions as follows
I
(
R
(
x
)
,
F
(
T
(
x
)))
p
(
r
(
x
)
,
F
(
T
(
x
)))
p
R
(
x
)
(
r
(
x
)
·
p
F
(
T
(
x
))
(
F
(
T
(
x
)))
.
(1.20)
p
(
r
(
x
)
,
F
(
T
(
x
)))
·
log
2
≡
R
(
x
)
,
F
(
T
(
x
))
where
p
(
x
,
y
) is the joint distribution function and
p
x
(
x
) is the marginal prob-
ability mass functions. The marginals can be obtained directly from the joint
probability function. The joint probability mass function
p
(
r
(
x
)
,
F
(
T
(
x
))) will
be approximated by the normalized joint histogram
H
(
r
(
x
)
,
F
(
T
(
x
))). Here,
normalization refers to scaling of the histogram, such that the sum of approxi-
mated probabilities equals 1.0. The marginals are then approximated from
H
()
by summation over the rows, and then the columns. Computation of
H
( ) in-
volves a complete iteration over each sample in the floating volume. For each
sample, the transformation
T
is applied, to arrive at a coordinate set in the im-
age coordinate system of the reference volume. If the transformed coordinate
is outside the measured reference volume, then the remaining operations are
not executed, and the process starts again with the next sample in the floating
volume. Otherwise, a sample in the reference volume at the transformed coor-
dinates is approximated using trilinear interpolation, and discretized. The two
samples, one from the floating volume, and one from the reference volume, are
then binned in the joint histogram.
Computation of the joint histogram involves the processing of each sample
in the floating volume, application of a transformation to the coordinate of the
sample in the floating volume to obtain a coordinate in the reference volume,
interpolation in the reference volume, and binning in the joint histogram. For a
typical 256 by 256 by 20 MRI volume, there are thus 256 by 256 by 20
=
1,310,720
