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FIGURE 2 Example unregistered CT (left) and PET (right) images. The top panels are the original axial images, and the lower panel the reformatted coronal images. As in Fig. 1, the intensities in these images are very different, and also the modalities have different fields of view. In this case, the CT has a smaller field of view than the PET both in the slice plane and in the through-plane direction.

FIGURE 2 Example unregistered CT (left) and PET (right) images. The top panels are the original axial images, and the lower panel the reformatted coronal images. As in Fig. 1, the intensities in these images are very different, and also the modalities have different fields of view. In this case, the CT has a smaller field of view than the PET both in the slice plane and in the through-plane direction.

exclude from the histogram all places where the two image volumes do not overlap. An interesting observation was that the appearance of joint histograms changes in a visually striking way as registered images are misaligned [13,14]. This visual change is especially striking if the images are blurred to increase the partial volume effect prior to the calculation of the 2D histograms. Example joint histograms for MR and CT images and MR and PET images at various stages of misregistration are shown in Fig. 3. These joint histograms are functions of the two images and of the registration transformation T Although the change in the appearance of the histograms for these two modality combinations is different, there are some important similarities: In particular, structure that is visible in the histogram at registration appears to disperse with misalignment, with an increasing number of low-intensity values appearing in the histogram.

It is possible to describe many different voxel similarity measures with reference to joint histograms. For example, cross-correlation makes the assumption that at registration, the values in the joint histogram lie on a straight line. The correlation coefficient CC can be calculated directly from the joint histogram as

N {(Ea,b h(a, b)(b - b)2)(£a,b h(a, b)(a - a)2)}1/2"

The variance of intensity ratios measure tries to reduce the spread of this joint histogram about the mean position for each vertical (or horizontal) line in the histogram. As in Eqs. (4) and (5), there are two versions of this measure: minimizing the spread of modality A with respect to modality B,

b and minimizing the spread of modality B with respect to modality A,

FIGURE 3 Example 2D histograms from Hill etal. [13] 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, although the histograms are quite different for the different modality combinations, misregistration results in a dispersion of the signal. Although these histograms are generated by lateral translational misregistration, misregistration in other translation or rotation directions has a similar effect.

FIGURE 3 Example 2D histograms from Hill etal. [13] 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, although the histograms are quite different for the different modality combinations, misregistration results in a dispersion of the signal. Although these histograms are generated by lateral translational misregistration, misregistration in other translation or rotation directions has a similar effect.

where Ga (b) is the standard deviations of intensity values a in modality A corresponding to a given intensity value b in modality B (similarly for Gb(a)) and h(a) is the projection of the joint histogram h(a, b) onto the b = 0 axis. As well as describing existing similarity measures in terms of this joint histogram, it is also possible to use the histograms as a basis for new measures. The most important of these are the measures based on information theory.

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