Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 95

first figure at some point. In the Procrustes superimposition, the first
step is to translate the second figure so that the centroids of the two fig-
ures match. Once this is done, no further translation is allowed and the
relative position is held constant during the rotation step. Mimic the
translation step by overlaying the transparencies to match the cen-
troids and push a pin through the superimposed centroids. Now rotate
Figure 3.6. Two quadrilaterals, each defined by four, two-dimensional landmarks super-
imposed on their centroids. This superimposition demonstrates a basic tendency of the
Procrustes fitting criterion: corresponding landmarks farthest from the centroid are
matched closely at the cost of mismatching those that are closer to the centroid.
Additionally, after rotation, the landmark locations of points 2 and 4 over-estimate the
distance between landmarks 2 and 4. Similarly, the placement of landmarks 1 and 3
after rotation tend to underestimate the distance between landmarks 1 and 3 along the
Y-axis (not readily apparent in this figure; after Lele, 1993). Moreover, estimates of vari-
ability for those landmarks lying farther from the centroid are reduced, while estimates
of variability for landmarks lying close to the centroid are amplified (data not shown).