## 3 Breast Cancer Image Classification
### Image Comparison Distance Functions Using Clusters  
The analysis for image comparison that we used performs a comparison of clusters of features in order to maintain both a contextual and spatial comparison while avoiding an exact registration. We experimented with two different models where the clusters are defined using separators and hyper-volumes. We also experimented with a model that compares small clusters of AFUM features between images. The hyper-volume image comparison can be seen in Figure 6, where the points are assigned to clusters that are defined by large volumes of feature space and have a set spatial relationship between each other. The feature-space hyper-volumes have a pre-set registration with the corresponding volumes in the other image. For simplicity, the volumes are assumed to be non-overlapping and space-filling, but this is not required. Additionally, the volumes are assumed to contain the same hyper-volumes in the images of the left and right breasts out of symmetry. This reduces the number of parameters and increases the ability of the model to be generalized to a larger data set, based on the assumption that there are no important anatomical differences between the left and right breasts and that breast cancer is equally as likely to be in the left or right breast. However, when there is a large natural asymmetry to the breasts this assumption may no longer be valid. In a hyper-volume image comparison, a hyper-volume is assigned all of the suspicious points in the space \(dA\) that the hyper-volume spans. The parameters of the hyper-volumes are learned through parametric learning, and any model can be used to characterize the hyper-volumes in feature space. Exact registration of the suspicious points is avoided by using the volumes for the comparisons as they are registered with the corresponding volume in the other image. The feature space is broken up into hyper-volumes as shown in Figure 6. The agglomerated distance \(D\) shown in Equation 1 is defined for the comparison of the two point feature sets, and the absolute value of the differences compared against an optimized threshold. Since the features are point features, they are represented using the delta function \(\delta\)and there is no weighting function yet introduced:  
\[D=\sum\limits_{dA}\left|\left(\underset{dA}{\overset{\left\|\right\|}{ \left\|\right\|}}df\sum\limits_{i}\delta(f-\overline{a}_{i})-\delta(f- \overline{b}_{i})\right)\right| \tag{1}\]  
The point sets for the images are represented as \(\{a_{i}\}\) and \(\{b_{i}\}\) for images \(a\) and \(b\) respectively. The summation over \(dA\) is done over all of the clusters which are represented by their hyper-volume \(dA\). The integration is done over the actual hyper-volume \(dA\) of the cluster. The summation over \(i\) is done over all of the features. The multi-dimensional integral over feature space provides the agglomeration aspect of the distance metric. The hyper-volumes \(dA\) provide the agglomeration and are learned along with a threshold in order to optimize the performance of the distance measure at classification. This allows the distance metric to be easily adapted to different image types and imaging techniques, as well as providing a method for incorporating feedback into the distance metric. This distance metric compares the distributions of spatially distributed point sets, and is sensitive to variations in the distribution for image comparison. This is useful for applications such as determining the presence of cancer. There are several other variations to this distance metric that have been explored. A variation on this distance metric is shown in Equation 2 that learns a threshold for each cluster \(dA\), which has the advantage of being able to emphasize the importance of some areas in the feature space over others. This can be used to distinguish noisy areas where many spurious suspicious points are found from important areas where even small variations are indicative of a lack of similarity.