Biomedical Engineering Reference
In-Depth Information
T
voxels
x
in the overlapping regions of the images
, we plot the intensity
A
A
,
B
of this voxel in image
A
,
A
(
x
) against the intensity of the corresponding
A
voxel in image
). The joint histogram can be normalized by dividing
by the total number of voxels
B
,
B
T
(
x
A
T
N
in
, and regarded as a joint probability
A
,
B
T
distribution function (PDF)
p
AB
of images
A
and
B
. We use the superscript
T
T
to emphasize that
. Due to the quantization of image
intensity values, the PDF is discrete, and the values in each element represent
the probability of pairs of image values occurring together. The joint entropy
p
AB
changes with
T
H
(
A
,
B
) is therefore given by:
T
T
p
AB
(
a
,
b
)
log
p
AB
(
a
,
b
)
HA
,
B
(
)
(3.22)
a
b
The number of elements in the PDF can either be determined by the range
of intensity values in the two images or from a reduced number of intensity
“bins.” For example, MR and CT images registered could have up to 4096
(12 bits) intensity values, leading to a very sparse PDF with 4096 by 4096 ele-
ments. The use of 32 to 256 bins is more common. In the above equation,
a
and
b
represent either the original image intensities or the selected intensity
bins.
As seen in Figure 3.1, as misregistration increases the brightest regions
of the histogram get less bright, and the number of dark regions is reduced.
If we interpret the joint histogram as a joint probability distribution, then
misregistration involves reducing the highest values in the PDF and reduc-
ing the number of zeros in the PDF; this will increase the entropy. Con-
versely, when registering images we want to find a transformation that
will produce a small number of PDF elements with very high probabilities
and give us as many zero probability elements in the PDF as possible,
which will minimize the joint entropy.
The simple form of the equation for joint entropy (Equation 3.22) can hide an
important limitation of this measure. As we have emphasized with the
super-
T
script on the joint probabilities, joint entropy is dependent on
. In particular,
is very dependent on which is undesirable, and also on the interpola-
tion algorithm used to transform the image at each iteration. The overlap
dependence can be made clear by the following example. A change in
T
T
p
AB
T
A
,
B
B
T
may
alter the amount of air surrounding the patient overlapping in the images
T
A
and
. Since the air region contains noise that will tend to occupy the lowest
value intensity bins (e.g.,
B
a
0,
b
0), changing this overlap will alter the joint
T
probability
p
AB
T
(0, 0). If the overlap of air increases,
p
AB
(0, 0) will increase,
T
reducing the joint entropy
H
(
A
,
B
). If the overlap of air decreases,
p
AB
(0, 0) will
reduce, increasing
). A registration algorithm that seeks to minimize
joint entropy will tend, therefore, to maximize the amount of air in , which
may result in an incorrect solution. The interpolation dependence of is
clear if we remember that interpolation algorithms will tend to blur images,
which sharpens the corresponding image histogram, changing the joint histo-
gram and consequently joint probability distribution
H
(
A
,
B
T
A
,
B
T
p
AB
p
A
T
.
