|CHRISTOPHER G. SMALL, Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada|
|The analysis of random shapes|
The shape of an object or image, regarded as a subset of , can be defined as the total of all properties which are invariant under similarity transformations of the object in . In practice, the shape of an object is encoded from the configuration of a finite set of points called landmarks chosen at important locations on the object. In this talk, we shall survey two of the main approaches to shape analysis due to F. L. Bookstein and D. G. Kendall. Small (1996) provided an extension of the Bookstein representation by representing the shapes of p-simplexes on manifolds. These manifolds are quite distinct from those proposed by D. G. Kendall based upon Procrutes distances. A formula for geodesic distance in simplex shape space permits the implementation of multidimensional scaling methods for the statistical shape analysis of 2- and 3-dimensional objects.
We apply this representation of simplex shapes to a shape analysis of Iron Age brooches from modern day Switzerland. The brooches are then grouped into five classes based upon chronological ordering. It is shown that these five classes form relatively distinct groups when the first two principal coordinates of shape variation are displayed. This supports the assumptions used in archaeological seriation where artifacts are given a rough chronological ordering based upon stylistic features.