Visualizing biological shape differences

Recent developments in morphometrics, the measurement of biological shape and shape change, are reviewed. After a roster of named, recognizably corresponding locations, or landmarks, have been located in each of a sample of images, the description of group differences and other effects upon form may usefully proceed in terms of a finite series of components of deformation of those images at different geometric scales. The component of largest scale is the uniform shear; remaining components are vector multiples of principal warps, which are eigenfunctions (with respect to net Euclidean displacement) of a convenient analog to physical bending energy. The mathematics of this decomposition, based in the algebra of thin-plate splines in two dimensions, is discussed