Exploratory methods for the study of incomplete and intersecting shape boundaries from landmark data

Structured spatial point patterns appear in many applications within the natural sciences. Often the points record the location of key features, called landmarks, on continuous object boundaries, such as anatomical features on a human face or on an animal skull. In other situations, the points may simply be arbitrarily spaced marks along a smooth curve, such as on handwritten numbers or letters. Sometimes the points may record the location of clearly visible features from a general structure which has disappeared, such as building foundations at an archaeological site. This paper proposes novel exploratory methods for the identification of structure within point datasets. In particular, points are linked together to form curves which estimate the original shape from which the points are the only recorded information. Nonparametric regression methods are applied to polar coordinate variables obtained from the point locations and periodic modelling allows closed curves to be fitted to circular and elliptical shapes even when data are available on only part of the boundary. Further, the model allows discontinuities to be identified to describe rapid changes in the curves. These generalizations are particularly important when the points represent shapes which are occluded or are intersecting. A range of real-data examples is used to motivate the modelling and to illustrate the flexibility of the approach. The method successfully identifies underlying structure and its output could also be used as the basis for further analysis.