Creases as local features of deformation grids

To help identify underlying developmental or pathological processes, biological shape differences often are represented as diffeomorphisms of a Cartesian coordinate grid. The problem addressed here is the extraction of spatially discrete, localized, organized features of such transformation grids. Some features can be identified with variants of the singularity (x, y)→(x, x2y+y3) that are visually evident as creases in suitably enhanced grid diagrams. The crease is a non-generic singularity at which a pair of cusps appears as a function of a parameter for extrapolation. The examples here show how these representations extract statistically informative and scientifically helpful features from deformations that help characterize two brain diseases, schizophrenia and Fetal Alcohol Syndrome, in two dimensions. Under smoothing by relaxation of bending energy against Euclidean distance, one analogue to multiscale analysis for discrete punctate data, creases are robust in location and orientation.