Biomedical Engineering Reference
In-Depth Information
the sum of the entropies of the individual images. The more similar (i.e., less
independent) the images are, the lower the joint entropy compared to the
sum of the individual entropies.
H
(
A
,
B
)
H
(
A
)
H
(
B
)
(3.21)
The concept of joint entropy can be visualized using a joint histogram calcu-
lated from images
A
and
B
, examples of which are shown in Figure 3.1. For all
FIGURE 3.1
Example 2D histograms from Hill et al.
43
for (a) identical MR images of the head, (b) MR
and CT images of the head, and (c) MR and PET images of the head. For all modality
combinations, the left panel is generated from the images when aligned, the middle panel
when translated by 2 mm, and the right panel when translated by 5 mm. Note that, while
the histograms are quite different for the different modality combinations, misregistration
results in a dispersion or blurring of the signal. Although these histograms are generated
by lateral translational misregistration, misregistration in other translation or rotation di-
rections has a similar effect.
