Biomedical Engineering Reference
In-Depth Information
3.2
Three-Dimensional Rigid Body
Registration Algorithm with
Special Features
3.2.1
Similarity Measurements
Two similarity measures, mutual information and correlation coefficient (CC),
are used in the registration. Suppose one volume
R
is the
reference
, and
the other
F
is
floating
. Their mutual information
MI(R,F)
is given below
[19]:
p
RF
(
r
,
f
)
p
R
(
r
)
·
p
F
(
f
)
MI
(
R
,
F
)
=
p
RF
(
r
,
f
) log
r
,
f
The joint probability
p
RF
(
r
,
f
) and the marginal probabilities
p
R
(
r
) of the ref-
erence image and
p
F
(
f
) of the floating image, can be estimated from the nor-
malized joint and marginal intensity histogram, respectively. The correlation
coefficient
CC
(
R
,
F
) is given below [26]:
(
R
(
r
)
−
R
(
r
))(
F
(
f
)
−
F
(
f
))
(
R
(
r
)
−
R
(
r
))
2
(
F
(
f
)
−
F
(
f
))
2
CC
(
R
,
F
)
=
Here
R
(
r
)
,
F
(
f
) denote the average intensities of the reference and floating
volumes and the summation includes all voxels within the overlap of both
volumes.
In Fig. 3.1, we compare the two similarity measures at different resolutions.
Plotted are MI and CC values as a function of translation along the transverse
axis where the origin is the optimal transformation. For images at a resolution
of 1/4 voxels along a linear dimension, the CC curves are much smoother than
MI, which is noisy and contains many local maximums as shown in Fig. 3.1a.
In addition, there is a false global maximum in Fig. 3.1a at 18 voxels. At full
resolution, Fig. 3.1c shows that MI is much more peaked than CC, but there is
high frequency noise in the MI curves far from the optimum that give rise to
local maximums that must be avoided. From these figures, we infer that CC is
better at low resolution and that MI is better at full resolution when one is close
