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Journal of Probability and Statistics
Volume 2016 (2016), Article ID 1285026, 9 pages
Research Article

Exploratory Methods for the Study of Incomplete and Intersecting Shape Boundaries from Landmark Data

1University of Benghazi, Benghazi, Libya
2University of Leeds, Leeds, UK

Received 25 July 2016; Revised 10 October 2016; Accepted 12 October 2016

Academic Editor: Z. D. Bai

Copyright © 2016 Fathi M. O. Hamed and Robert G. Aykroyd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.