Advances in Geometric Morphometrics
 Philipp Mitteroecker,
 Philipp Gunz
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Abstract
Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thinplate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.
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 Title
 Advances in Geometric Morphometrics
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Evolutionary Biology
Volume 36, Issue 2 , pp 235247
 Cover Date
 20090601
 DOI
 10.1007/s116920099055x
 Print ISSN
 00713260
 Online ISSN
 19342845
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Asymmetry
 Form space
 Kendall shape space
 Procrustes
 Semilandmarks
 Thinplate spline
 Authors

 Philipp Mitteroecker ^{(1)} ^{(2)}
 Philipp Gunz ^{(3)}
 Author Affiliations

 1. Konrad Lorenz Institute for Evolution and Cognition Research, Adolf Lorenz Gasse 2, Altenberg, A3422, Austria
 2. Department of Theoretical Biology, University of Vienna, Althanstrasse 14, Vienna, A1091, Austria
 3. Department of Human Evolution, MaxPlanckInstitute for Evolutionary Anthropology, Deutscher Platz 6, Leipzig, D04103, Germany