Singularities as features of deformation grids

Abstract

Biological shape differences often are represented as diffeo-morphisms of a Cartesian coordinate grid. This paper suggests that their spatially discrete, localized features, for instance the details that suggest underlying developmental or pathological processes, can often be identified with variants of the singularity (x, y)-> (x, x2y + y3). This is an unfamiliar singularity, generic of codimension 1, at which a pair of cusps appears as a function of a parameter for “extrapolation.” I introduce canonical coordinates for such singularities and show how they may be used to produce objective reports of grids encountered in an empirical context. An example is shown involving the corpus callosum in Fetal Alcohol Syndrome. These features appear to be robust under relaxation of bending energy against Euclidean distance, the analogue to multiscale analysis for discrete punctate data.