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More and more analyses of biological shapes are using the techniques of geometric morphometrics based on configurations of landmarks in two or three dimensions. A fundamental concept at the core of these analyses is Kendall’s shape space and local approximations to it by shape tangent spaces. Kendall’s shape space is complex because it is a curved surface and, for configurations with more than three landmarks, multidimensional. This paper uses the shape space for triangles, which is the surface of a sphere, to explore and visualize some properties of shape spaces and the respective tangent spaces. Considerations about the dimensionality of shape spaces are an important step in understanding them, and can offer a coordinate system that can translate between positions in the shape space and the corresponding landmark configurations and vice versa. By simulation studies “walking” along that are great circles around the shape space, each of them corresponding to the repeated application of a particular shape change, it is possible to grasp intuitively why shape spaces are curved and closed surfaces. From these considerations and the available information on shape spaces for configurations with more than three landmarks, the conclusion emerges that the approach using a tangent space approximation in general is valid for biological datasets. The quality of approximation depends on the scale of variation in the data, but existing analyses suggest this should be satisfactory to excellent in most empirical datasets.

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 unnecessary and indeed 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 celebrated exposition of 1917. For analyses of comparisons involving gradients at large geometric scale, this paper argues for replacement of all three of the Procrustes conventions by a version of my two-point registration of 1986 [originally Galton’s of 1907 (Nature 76:617–618, 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 midsagittal 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.

The geometric morphometric (GMM) construction of Procrustes shape coordinates from a data set of homologous landmark configurations puts exact algebraic constraints on position, orientation, and geometric scale. While position as digitized is not ordinarily a biologically meaningful quantity, and orientation is relevant mainly when some organismal function interacts with a Cartesian positional gradient such as horizontality, size per se is a crucially important biometric concept, especially in contexts like growth, biomechanics, or bioenergetics. “Normalizing” or “standardizing” size (usually by dividing the square root of the summed squared distances from the centroid out of all the Cartesian coordinates specimen by specimen), while associated with the elegant symmetries of the Mardia–Dryden distribution in shape space, nevertheless can substantially impeach the validity of any organismal inferences that ensue. This paper adapts two variants of standard morphometric least-squares, principal components and uniform strains, to circumvent size standardization while still accommodating an analytic toolkit for studies of differential growth that supports landmark-by-landmark graphics and thin-plate splines. Standardization of position and orientation but not size yields the coordinates Franz Boas first discussed in 1905. In studies of growth, a first principal component of these coordinates often appears to involve most landmarks shifting almost directly away from their centroid, hence the proposed model’s name, “centric allometry.” There is also a joint standardization of shear and dilation resulting in a variant of standard GMM’s “nonaffine shape coordinates” where scale information is subsumed in the affine term. Studies of growth allometry should go better in the Boas system than in the Procrustes shape space that is the current conventional workbench for GMM analyses. I demonstrate two examples of this revised approach (one developmental, one phylogenetic) that retrieve all the findings of a conventional shape-space-based approach while focusing much more closely on the phenomenon of allometric growth per se. A three-part Appendix provides an overview of the algebra, highlighting both similarities to the Procrustes approach and contrasts with it.

The original exposition of the method of “Cartesian transformations” in D’Arcy Thompson’s On Growth and Form (1917) is still its most cited. But generations of theoretical biologists have struggled ever since to invent a biometric method aligning that approach with the comparative anatomist’s ultimate goal of inferring biologically meaningful hypotheses from empirical geometric patterns. Thirty years ago our community converged on a common data resource, samples of landmark configurations, and a currently popular biometric toolkit for this purpose, the “morphometric synthesis,” that combines Procrustes shape coordinates with thin-plate spline renderings of their various multivariate statistical comparisons. But because both tools algebraically disarticulate the landmarks in the course of a linear multivariate analysis, they have no access to the actual anatomical information conveyed by the arrangements and adjacencies of the landmark locations and the distinct anatomical components they span. This paper explores a new geometric approach circumventing these fundamental difficulties: an explicit statistical methodology for the simplest nonlinear patterning of these comparisons at their largest scale, their fits by what Sneath (1967) called quadratic trend surfaces. After an initial quadratic regression of target configurations on a template, the proposed method ignores individual shape coordinates completely. Those have been replaced by a close reading of the regression coefficients, accompanied by several new diagrams, of which the most striking is a novel biometric ellipse, the circuit of the trend’s second-order directional derivatives around the data plane. These new trend coordinates, directly visualizable in their own coordinate plane, do not conduce to any of the usual Procrustes or thin-plate summaries. The geometry and algebra of the second-derivative ellipses seem a serviceable first approximation for applications in evo-devo studies and elsewhere. Two examples are offered, one the classic growth data set of Vilmann neurocranial octagons and the other the Marcus group’s data set of midsagittal cranial landmarks over most of the orders of the mammals. Each analysis yields intriguing new findings inaccessible to the current GMM toolkit. A closing discussion suggests a variety of ways by which innovations in this spirit might burst the current straitjacket of Procrustes coordinates and thin-plate splines that together so severely constrain the conversion of landmark locations into biological understanding. This restoration of a quantitative diagrammatic style for reporting effects across regions and gradient directions has the potential to enrich landmark-driven comparisons over either developmental or phylogenetic time. Extension of the paper’s quadratic methods to the next polynomial degree, cubics, probably won’t prove generally useful; but close attention to local deviations from globally fitted quadratic trends, however, might. Ultimately there will have to emerge a methodology of landmark configurations, not merely landmark locations.

The textbook literature of principal components analysis (PCA) dates from a period when statistical computing was much less powerful than it is today and the dimensionality of data sets typically processed by PCA correspondingly much lower. When the formulas in those textbooks involve limiting properties of PCA descriptors, the limit involved is usually the indefinite increase of sample size for a fixed roster of variables. But contemporary applications of PCA in organismal systems biology, particularly in geometric morphometrics (GMM), generally involve much greater counts of variables. The way one might expect pure noise to degrade the biometric signal in this more contemporary context is described by a different mathematical literature concerned with the situation where the count of variables itself increases while remaining proportional to the count of specimens. The founders of this literature established a result of startling simplicity. Consider steadily larger and larger data sets consisting of completely uncorrelated standardized Gaussians (mean zero, variance 1) such that the ratio of variables to cases (the so-called “ p / n ratio”) is fixed at a value y . Then the largest eigenvalue of their covariance matrix tends to ( 1 + y ) 2 , the smallest tends to ( 1 - y ) 2 , and their ratio tends to the limiting value ( ( 1 + y ) / ( 1 - y ) ) 2 , whereas in the uncorrelated model both of these eigenvalues and also their ratio should be just 1.0. For y = 1 / 4 , not an atypical value for GMM data sets, this ratio is 9; for y = 1 / 2 , which is still not atypical, it is 34. These extrema and ratios, easily confirmed in simulations of realistic size and consistent with real GMM findings in typical applied settings, bear severe negative implications for any technique that involves inverting a covariance structure on shape coordinates, including multiple regression on shape, discriminant analysis by shape, canonical variates analysis of shape, covariance distance analysis from shape, and maximum-likelihood estimation of shape distributions that are not constrained by strong prior models. The theorem also suggests that we should use extreme caution whenever considering a biological interpretation of any Partial Least Squares analysis involving large numbers of landmarks or semilandmarks. I illuminate these concerns with the aid of one simulation, two explicit reanalyses of previously published data, and several little sermons.

Persons with brain damage consequent to prenatal alcohol exposure have typically been diagnosed with either fetal alcohol syndrome (FAS) or fetal alcohol effects (FAE), depending on facial features. There is great variability of behavioral deficits within these groups. We sought to combine neuroanatomical measures with neurocognitive and neuromotor measures in criteria of greater sensitivity over the variety of consequences of alcohol exposure. To this end, midline curves of the corpus callosum were carefully digitized in three dimensions from T1-weighted MR scans of 15 adult males diagnosed with FAS, 15 with FAE, and 15 who were unexposed and clinically normal. From 5 h of neuropsychological testing we extracted 260 scores and ratings pertaining to attention, memory, executive function, fine and gross motor performance, and intelligence. Callosal midline shape was analyzed by new morphometric methods, and the relation of shape to behavior by partial least squares. The FAS and FAE subgroups have strikingly more variability of callosal shape than our normal subjects. With the excess shape variation are associated two different profiles of behavioral deficit unrelated to full-scale IQ or to the FAS/FAE distinction within the exposed subgroup. A relatively thick callosum is associated with a pattern of deficit in executive function; one that is relatively thin, with a deficit in motor function. The two combine in a very promising bipolar discrimination of the exposed from the unexposed in this sample. Thus there is considerable information in callosal form for prognosis of neuropsychological deficits in this frequently encountered birth defect.

Brain damage consequent to prenatal alcohol exposure can be detected by measurements of the corpus callosum in the midline magnetic resonance (MR) brain image in adolescents and adults. The present article extends this finding into the neonatal period, when the power of detection to ameliorate the quality of the child's future life is greatest. The midline corpus callosum of the very young infant can be located reliably in multiple frames of clinical transfontanelle ultrasound. We studied a sample of 18 children aged 17 weeks or less, 7 of whom were exposed to high levels of alcohol prenatally and 11 of whom were not exposed or only minimally exposed. The midline callosum of each child was imaged up to 50 times by a standard clinical device, and coplanar subsets of these series were averaged with reference to fiducial image structures. On each average image four semilandmark points were set and their configuration quantified by standard landmark methods. The angle between the terminal bulb of splenium and the long axis of the callosal outline classifies four of the seven exposed infants as different from all 11 of the unexposed infants. This simple angle measurement upon averaged ultrasound images of the human neonatal midline corpus callosum, perhaps a version of the long-sought “biomarker of prenatal alcohol damage,” may be able to discriminate baby brains affected by prenatal alcohol exposure from those that were unaffected.

This chapter contains sections titled: The description of shape Shape‐preserving transformations Shape transformation and homogeneous coordinates The general 2‐D affine transformation Affine transformation in homogeneous coordinates The Procrustes transformation Procrustes alignment The projective transform Nonlinear transformations Warping: the spatial transformation of an image Overdetermined spatial transformations The piecewise warp The piecewise affine warp Warping: forward and reverse mapping