Identifying natural groupings of identical brains containing differing known deformations.

<p><i>Left Column</i>: Brains 1 through 10 contained a protrusion, and brains 11 through 20 contained an indentation, in the DLPFC. <i>Middle Column</i>: Brains 1 through 10 contained a protrusion, and brains 11 through 20 contained an indentation, in the occipital cortex (OC). <i>Right Column</i>: Brains 1 through 10 had a protrusion at the OC, the brains 11 to 20 had an indentation at the OC whereas brains 21 through 30 contained a protrusion at the DLPFC and brains 31 through 40 had an indentation at the DLPFC. The deformed brains were normalized to the template to compute signed Euclidean distances. Those distances were mapped onto a sphere using a conformal mapping. We then applied either the lifted interpolate (<i>top two rows</i>) or the lifted butterfly (<i>bottom row</i>) to compute the scaling coefficients. Hierarchical clustering computed distances between features using either the average linkage or Ward's linkage. Left and middle dendrograms demonstrate that the brains were correctly clustered into two groups, one with indentations only and the other with protrusions only. Right dendrogram shows that the brains were correctly clustered into four groups: one with only protrusions in the OC, another with only protrusions in the DLPFC, another one with only indentations in the OC, and the last with only indentations in the DLPFC. Furthermore, Ward's distances between groups were larger than the average distances, indicating good separation of groups according to the type of synthetic deformation that was introduced into the data. Ward distances for feature vectors generated using lifted interpolate were generally greater than those generated using lifted butterfly, motivating our subsequent use of the lifted interpolate wavelet to compute scaling coefficients and use of Ward's linkage to cluster brains into naturalistic groups. <b>Dp</b>, protrusion in DLPFC; <b>Di</b>, indentation in DLPFC; <b>Op</b>, protrusion in OC; <b>Oi</b>, indentation in OC.</p>