Behaviour Introduction

One of the important goals of a shape study is to visualize the differences between the classes of shapes. Figure 4-5 and Figure 4-6 show those differences for the right and left complexes. Those differences were found from derivatives of the RBF based classifier

evaluated at examples as described in Section 4.1.3. There are several interesting observations about the figures. First, for a non-linear classifier it is not guaranteed that the gradients computed for the two groups are equal and opposite. However, Figure 4-5 and Figure 4-6 shows that for both classifiers, the changes detected to make one group look more like the other are similar in nature but opposite in sign.

This finding holds for both complexes, and both representations.

Ideally, the shape differences found between the classes would be indpendent of representations. For right complexes, (Figure 4-5), there are strong similarities in the shape differences found by the classifiers based on each displacement fields and distance maps. In particular, at the posterior portion of the tail of the hippocampus (the top of the shapes shown) and the base of the hippocampus (the bottom of the shapes shown), the shape differences found by the two classifiers appear very similar.

For left complexes (Figure 4-6), there are disparities between shape differences found by the classifiers based on each representation. These disparities are particularly evident at the base of the amygdala (the bottom of the shapes in the figure) and the tail of the hippocampus (near the top of the shapes in the figure).

Given that the two representations yielded different morphological differences, we seek evidence as to which answer is the correct one. There are two factors which suggest the answer. Firstly, examining the gradients found at every left complex (not shown), the displacement field based classifier yielded visually similar gradients at each left complex. However, for many of the left complexes, the gradients found using the distance map based classifier were visually very different from each other.

Thus, because the shape differences found were consistent across all left complexes for displacement field based methods, but not for distance map based methods, the shape differences found using displacement fields are more likely. Note that for the right complex, the classifiers based on both representation yielded visually similar gradients at each right complex.

For a second indication of which of the two classifiers yielded the correct answer, we can consider a third classifier. Figure 4-7 shows the morphological differences found between the two groups as determined by a linear classifier using displacement

Diseased Control Dist.

Map

Disp.

Field

inward change outward change

Figure 4-5: The top four plots show the “deformation” of the surfaces relative to the surface normal for the right amygdala-hippocampus complex. For diseased subjects,

“deformation” indicates changes to make the complexes more like a normal control.

For normal subjects, “deformation” indicates changes to make the complexes more diseased. The four pairs of surface plots show deformations of diseased/control sub-jects using distance maps/displacement fields as representations. In each pair of surface plots, the two examples with the largest deformations evaluated at the sup-port vectors of the SVM classifier are shown; the larger one is on the left. The color coding is used to indicate the direction and magnitude of the deformation, changing from blue (inward) to green (no motion) to red (outward). The bottom two plots are the deformations fields used to generate the plots directly above them. Note that motion along the surface does not affect the colors in the surfaces.

Diseased Control

Dist.

Map

Disp.

Field

inward change outward change

Figure 4-6: The top four plots show the deformation of the surfaces relative to the surface normal for the left amygdala-hippocampus complex. The figure is otherwise identical to Figure 4-5.

Linear Classifier Diseased

Right Left Disp.

Field

inward change outward change

Figure 4-7: The “deformation” of the right and left complex determined using a linear classifier based on displacement fields. The results shown were determined by two separate classifiers, one for each side. Both example shapes are from the same diseased subject. The remainder of the figure is described in Figure 4-5.

fields. We consider only a linear classifer based on displacement fields and not a linear classifier based on distance maps due to the poor cross validation accuracy of the linear classifier based on distance maps. Comparing the shape differences in Figure 4-7 with those found in Figure 4-5 and Figure 4-6, there are several similarities in the shape differences found by the linear classifier and the non-linear classifier based on displacement fields, particularly at interface of the hippocampus and amygdala. Thus, the shape differences found using the linear classifier based on displacement fields supports the shape differences found by the non-linear classifier using displacement fields, not those of the non-linear classifier based on distance maps.

Displacements fields have the potential to yield information not present using dis-tance maps. In the bottom of Figures 4-5, 4-6, and 4-7, the derivatives of the classifier based on displacement fields are shown in the form of a vector field. Because distance maps do not use correspondences, it is not possible to show motions tangential to the

surface using distance maps. The vector fields show that there is motion along the surface in several places. Most notably, in Figure 4-6, there is a clear rotation of the tail of the hippocampus. This effect is not clearly visible in the surface changes; it is only visible using the vector field.