Functional models of growth for landmark data

Growth models for shape are investigated for landmark data. First the data are given a Euclidean representation using Procrustes tangent coordinates. Roughness penalties are defined for directions of growth in space and for rates of growth in time. These penalties are then combined to give a joint space-time roughness penalty. Transforming the data to bases defined by principal warps in space and time facilitates model specification and fitting, both for parametric and nonparametric models. A generalized cross validation criterion can be used to choose a smoothing parameter for a nonparametric smoothing spline-type model. Fitted models can be interpreted either just in terms of the finite set of landmarks at the finite set of data times, or in terms of a deformation of space which varies continuously through time. The methods are illustrated on a set of rat data.