Abstract
Structured spatial point patterns appear in many applications within the natural sciences. The points often record the location of key features, called landmarks, on continuous object boundaries, such as anatomical features on a human face. In other situations, the points may simply be arbitrarily spaced marks along a smooth curve, such as on handwritten numbers. 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 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 the underlying structure and its output could also be used as the basis for further analysis.
Highlights
Many scientific investigations involve the recording of spatially located data
Once the data are collected often the original context is lost and the aim of the analysis is to identify which points are associated with each other and to link the points to reconstruct the original shape. These can be seen as estimates of continuous curves and object outlines
A landmark is defined as a point of correspondence on each object that matches between and within populations [3]
Summary
Many scientific investigations involve the recording of spatially located data. This data might summarize objects within an image as digitized versions of continuous curves. Intersecting curves are described by allowing discontinuities in the fitted curves These procedures are illustrated using simulated data and varied real datasets describing human faces, gorilla skulls, handwritten number 3’s, and an archaeological site. These provide a wide variety of point patterns and reinforce the general usefulness of the proposed methods. For mathematical detailed description and applications of shape-based analysis of points, refer to, for example, Batschelet [1], Bookstein [2], Dryden and Mardia [3], and Lele and Richtsmeier [4] To allow for this wide variety of possible curves a nonparametric fitting approach, such as splines, can be used (see, e.g., [5, 6]).
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