Biomedical Engineering Reference
In-Depth Information
FIGURE 2.10
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The joint probability distributions of intensities for aligned MR and
FDG PET volumes
(left), misaligned with a 2 mm translation (middle) and misaligned with a 5 mm translation
(right).
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Also indicated are the regions of the plot approximately corresponging to the scalp,
skull, gray matter, and white matter. The intensity values of the scalp will lie below the
dotted line.
happen if we now repeat these plots at different alignments. The distinctive
pattern in these two images starts to diffuse and disperse, as seen in Figure 2.10
(middle and right). Examples for two MR volumes and an MR and CT vol-
ume of the head are provided in Chapter 3, Figure 3.1. We can see from the
plots in Figure 2.10 why the PIU algorithm works for MR and PET registra-
tion. If we remove the scalp, i.e., everything below the dotted line on Figure 2.10,
then at registration there is only a narrow band of PET intensities for each MR
intensity. This band broadens with misregistration, thus increasing the PIU
measure. Likewise, we can see from the MR and CT plots in the next chapter
how the remapping of the CT intensity will produce a strong linear relation-
ship between MR and the remapped CT image intensities that will reduce
with misalignment. This linear relationship was exploited in the method pro-
posed by Van den Elsen.
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Of more importance, however, these plots provide
insight into an entirely new concept of image registration that is based on
image entropy and information theory.
Information theory dates back to the pioneering work of Shannon in the
1940s.
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Working at Bell Laboratories on how information is transmitted
along a noisy telephone line or radio link, he devised a theory around a new
measure of information. Its mathematical form was the same as the entropy
defined in statistical mechanics, so he called this measure entropy. Entropy is a
measure of disorder; a value for Shannon's entropy can be calculated directly
from the joint probability distribution. Disorder (and entropy) increases with
increasing misregistration in both the joint probability distribution (the plots in
Figure 2.10 become more diffuse) and the visual appearance of the images
when overlaid with one another. This suggests entropy as a possible measure
of image alignment. Minimizing the joint entropy, calculated from the joint
intensity histogram, was proposed by Studholme et al.
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and Collignon
as a
basis for a registration method.
