Standard Formula for the Uniform Shape Component in Landmark Data

Abstract

In describing how the shapes of landmark configurations vary, an important role is played by the uniform component of shape variation. Geometrically, the uniform component parameterizes shape changes that leave parallel lines parallel throughout the form and so show neither a spatial gradient nor any spatial localization. Algebraically, the shapes produced from a mean form in this way make up a linearized subspace, U, of our multivariate shape space in the vicinity of that mean, so that the particular component that corresponds to a particular specimen or group difference can be computed as a pair of scores (for two-dimensional data) or a quintet of scores (for three-dimensional data) by a Procrustes projection onto U. The versions of this crucial projection that are offered by currently available program packages are inconsistent in several important aspects. This paper suggests a resolution of the contradictions among these approaches. I introduce a nearly exact Procrustes version of this component and show that it is identical to two separate versions previously suggested in the literature and that it represents a more precise estimate for yet another. A somewhat different version of this component is useful for multivariate statistical testing. The computations are demonstrated using the growing rat skull data set familiar from several other canonical demonstrations of the new geometric methods.