Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 140

normal child ( Figure 4.5c ). It is clear that the two grids are quite dif-
ferent, leading to different scientific conclusions about the locus,
magnitude and direction of the differences between the two forms.
Looking at the inverse-squared distance weighted grid ( Figure 4.5b ),
we see that the grid has been markedly compressed on the lower left
corner. This represents compression of both the posterior aspect of the
neurocranium (landmarks 10 and 11) and the distance from the back
of the palate (landmark 6) to the anterior aspect of the foramen mag-
num (landmark 10). A less obvious constriction is noted on the upper
right corner of the grid while the lower right corner is stretched. This
indicates constriction of the upper face (landmark 1) and stretching of
the lower face (landmarks 3, 4, and 6). Within the grid, anteroposteri-
or expansion or stretching is shown local to the pituitary fossa region
of the cranial base (landmarks 7, 8, 9). The external shape of the min-
imum bending energy grid shows comparatively less deformation
( Figure 4.5c ). The face (landmarks 1, 2, and 4) shows some superoinfe-
rior constriction, while additional restriction occurs along the

a

b

c
Figure 4.5 The noninvariance of the deformation approach is shown in this figure.
Part 4.5a provides the initial grid with landmarks in place on the estimated mean
form of the normal sample. Panel 4.5b shows deformation of the normal mean form
into the mean form estimated for the Apert sample based on the inverse squared dis-
tance weighted deformation. Panel 4.5c gives the same deformation but this time
based on minimum bending energy. The deformation grids produced from the compari-
son of the same data sets differ according to the choice of the side condition. Scientific
conclusions of the two deformation analyses differ.