Deformation modeling via thin-plate splines can combine discrete location information and differential information (e.g., edge directions, surface normals) in a single computation. For encoding prior knowledge about these elements, there is a new biometrical technology for means, variances, correlations, and the like. The algebra of this formalism fuses classic multivariate biometrics with the ancient thrust of visualization of shape change as "Cartesian transformation." Elements of this statistical space stand for linear combinations of deformation operators, having specific scale and location, that alter all pictorial elements together. The new methods make possible a conceptual orthogonalization of the operations of image understanding into the "horizontal," those that reassign pixel coordinates, and the "vertical," those that deal with pixels and pixel neighborhoods in a standardized geometry.