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Figure 10.4: The cross-entropy (CE), reversed cross-entropy (RCE), and symmetric divergence (SD) profiles of an MR/Em image pair in the vicinity of a manual registration when the y-rotation angle changes independently.

Figure 10.5: The cross-entropy (CE), reversed cross-entropy (RCE), and symmetric divergence (SD) profiles of an MR/Em image pair in the vicinity of a manual registration when the z-rotation angle changes independently.

Figure 10.7: The cross-entropy (CE), reversed cross-entropy (RCE), and symmetric divergence (SD) profiles of an MR/Em image pair in the vicinity of a manual registration when the y-translation changes independently.

directions, but sizable in the z direction (ca. 4 mm). The cross-entropy curve is the smoothest.

Although we don’t know if the peak appearing in the neighborhood of the manual registration represents the global optimal, the presence of a local maxima does indicate that the three measures are suitable as registration criteria. The difference in the peak positions is expected since the similarity measures are different. Furthermore, the Em image used in our study has a low resolution (7.12 mm in all three directions) and the difference in the peak positions is not significant compared to the voxel size.

10.5.1.4 Numerical Stability for Maximization

As mentioned earlier, it could cause numerical problems when there are zero joint probabilities in the reversed cross-entropy and symmetric divergence

Figure 10.9: The reversed cross-entropy profiles of an MR/Em image pair in the vicinity of a manual registration when the x-rotation angle changes independently. The numbers adjacent to the profiles indicate the α value used to compute a small probability value which is used to replace zero pdfs. The bottom profile represents the case where the contribution from zero joint pdfs was ignored in the profile computation.

there is not even a peak around the manual registration. These two undesirable features render the reversed cross-entropy measure unsuitable as a registration measure if the zero pdfs are ignored. If the zero pdfs are taken into account and as α increases, the profiles become smooth and then rough again. It seems that

α= 100 is a reasonable choice, which gives a relatively smooth profile. When

αis small, the extraneous pdfs replacing the zero pdfs causes the summation of the pdfs to significantly deviate from unity. Therefore, a small α is not ideal. When α is large, the contribution of each extraneous pdf becomes dominant, the total reversed cross-entropy increases, and the contribution from nonzero pdfs is diminished gradually. The latter behavior makes a large α not ideal either.

When α = 100, the summation of all pdfs, including the extraneous ones, is at

Figure 10.10: The reversed cross-entropy profiles of an MR/Em image pair in the vicinity of a manual registration when the x-translation changes independently. The numbers adjacent to the profiles indicate the α value used to compute a small probability value which is used to replace zero pdfs. Ignored means that those pdfs are ignored when computing the profile.

most 1.01. The upper bound can be achieved only when there is 1 nonzero pdf which is 1, and all other pdfs are zero. Based on these observations, we chose

α = 100 in our calculation.

10.5.2Registration by Cross-Entropy, Reversed Cross-Entropy, and Symmetric Divergence Minimization

10.5.2.1 Registration Performance

The standard deviations of four independent manual registration results were, in vector form, (3.48, 0.53, 3.11, 1.68, 2.98, 1.83). The likely priori was calculated

Table 10.5: Average and standard deviation of MR/Em registration parameters obtained from cross-entropy, reversed cross-entropy, and symmetric divergence minimization. The angles are in degrees and translation offsets in mm

using the average manual registration parameters, as mentioned earlier. Refer to section 10.4.3 for details on the experimental setup.

Table 10.5 lists the statistics of the registration results for two sets of experiments. As the table reveals, the registration parameters are very close to those of the manual results and the differences of these registration parameters are well below the detection threshold of a trained technician. This is expected since the priori was calculated based on the manual registration results. Also revealed by Table 10.5 is that both reversed cross-entropy and symmetric divergence outperformed cross-entropy minimization in terms of success rate, and that symmetric divergence has the highest overall success rate.

Note that the implementation of this minimization process is not yet optimized, e.g., one may use Paren density estimation with a Gaussian kernel to add some smoothness and stability to the similarity functions. Nevertheless, these results indicate that cross-entropy, reversed cross-entropy, and symmetric divergence minimization can yield a very good registration if a good prior estimation is available.

10.5.2.2 Behavior Around True Registration

The behavior of these three similarity measures was also checked in the vicinity of the manual registration, with the desirable prior calculated from that same manual registration.

Figure 10.11: The cross-entropy (CE), reversed cross-entropy (RCE), and symmetric divergence (SD) profiles (with a likely priori) of an MR/Em image pair in the vicinity of a manual registration when the x-rotation changes independently.

Figures 10.11–10.13 show the representative profiles of these similarity measures as a function of rotation angles about the x-, y-, and z-axis, respectively. Figures 10.14–10.16 show the representative profiles of these similarity measures as a function of translation along the x-, y-, and z-axis, respectively. As revealed by all these figures, there are valleys in the profiles around the manual registration parameters, which is one of the necessary conditions for a good registration measure. As expected, the minimum values in all cases are zero since the priori pdf and the estimated pdf at this registration are identical.

10.5.2.3 Numerical Stability for Minimization

In the cross-entropy minimization calculation, if the priori pdf is zero, the crossentropy is not stable. To address this case, one can either ignore the zero priori pdf terms or substitute the zero pdfs with small positive numbers and use the