Biomedical Engineering Reference
In-Depth Information
FIGURE 2.11
Diagrammatic explanation in 2D that the superimposition of an aligned image pair contains
less information than a misaligned pair.
Unfortunately, joint entropy on its own does not provide a robust measure
of image alignment, as it is often possible to find alternative (mis)alignments
that result in much lower joint entropy. As an example, an alignment which
results in just the overlap of air surrounding the patient, with the image data
of the patient completely separated, will often produce a global minimum of
entropy. It seems plausible that an appropriate measure might be the differ-
ence between the information in the overlapping volume of the combined or
overlaid images and the information in the corresponding volumes of the
two original images. Such a measure is provided by mutual information,
which was proposed independently by Collignon et al.
27
and the MIT
28
Mutual information is given by the difference between the sum of
the entropies of the individual images at overlap and the joint entropy of the
combined images. As an illustrative example, consider two images of the
same individual, each containing two eyes. Misaligned, the combined images
will contain four eyes, while at alignment there will only be two. There is,
therefore, less “information” in the conventional sense of the word in the
combined images at registration. The extra information at misalignment is
purely artifactual. This concept is explained diagrammatically in Figure 2.11.
At alignment we postulate that the joint entropy is minimized with respect
to the entropy of the overlapping part of the individual images, so that the
mutual information is maximized. Mutual information is a measure of how
one image “explains” the other. It makes no assumption of the functional
form or relationship between image intensities in the two images. Shannon
first presented the functional form of mutual information in 1948.
group.
25
He
defined it as the “rate of transmission of information” in a noisy communica-
tion channel between source and receiver.
The mutual information measure with modifications associated with norma-
lization
29
30,31
and has resulted in fully automated 3D-to-
3D rigid-body registration algorithms that are now in widespread use. The
mathematical description of these measures is provided in the next chapter.
has proved very robust
2.4.4
2D-3D Registration
Registration of x-ray or video images to a 3D-volume image involves estab-
lishing the pose of the x-ray or video image in relation to a previously
acquired CT or MR volume. This has potential applications in image-guided
