Singularities and the features of deformation grids

Biological shape differences often are represented as diffeomorphisms of a Cartesian coordinate grid. The problem addressed here is the extraction of spatially discrete, localized features of such transformation grids, which often help to identify underlying developmental or pathological processes. This paper shows how some such features can be identified with variants of the singularity (x, y) --> (x, x(2)y + y(3)) that are visually evident as creases in the grid. The crease is a nongeneric singularity at which a pair of cusps appears as a function of a parameter for extrapolation. The paper shows how this representation extracts informative discrete feature sets from deformations characterizing two different brain diseases, schizophrenia and Fetal Alcohol Syndrome, in the midsagittal plane (plane of symmetry). Creases appear to be robust under relaxation of bending energy against Euclidean distance, one analogue to multiscale analysis far discrete punctate data. I suggest that they comprise the simplest words in an eventual grammar of grids.