Reworking geometric morphometrics into a methodology of transformation grids

Today's typical application of geometric morphometrics to a quantitative comparison of organismal anatomies begins by standardizing samples of homologously labelled point configurations for location, orientation, and scale, and then renders the ensuing comparisons graphically by thin-plate spline as applied to group averages, principal components, regression predictions, or canonical variates. The scale-standardization step has recently come under criticism as inappropriate, at least for growth studies. This essay argues for a similar rethinking of the centering and rotation, and then the replacement of the thin-plate spline interpolant of the resulting configurations by a different strategy that leaves unexplained residuals at every landmark individually in order to simplify the interpretation of the displayed grid as a whole, the "transformation grid" that has been highlighted as the true underlying topic ever since D'Arcy Thompson's exposition of 1917. For analyses of comparisons involving gradients at large geometric scale, this paper argues for replacement of all the Procrustes conventions by a version of my two-point registration of 1986 (originally Francis Galton's of 1907). The choice of the two points interacts with another non-Procrustes concern, interpretability of the grid lines of a coordinate system deformed according to a fitted polynomial trend rather than an interpolating thin-plate spline. The paper works two examples using previously published cranial data; there result new findings pertinent to the interpretation of both of these classic data sets. A concluding discussion suggests that the current toolkit of geometric morphometrics, centered on Procrustes shape coordinates and thin-plate splines, is too restricted to suit many of the interpretive purposes of evolutionary and developmental biology.