On the Shape of Plane Images

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Abstract
This paper studies set patterns in R2 and gives a definition of shape in terms of transformation groups, where the transformations need not be rigid and linear, but mom structured than arbitrary homeomorphisms. The resulting concept of shape classes and shape groups is studied analytically. On such shape classes, prior measures intended for Bayesian image processing are introduced. The priors are given as Markov processes of the Gibbs type on certain graphs and where the generators are simple geometric objects in R2, for example, line segments or other arcs. These continuum-based models differ from earlier, lattice-based ones in that they incorporate more shape information in the prior measures. This leads to algorithms for pattern synthesis and image processing, which have been implemented by APL code and applied in an extensive series of computer experiments. Since the algorithms are computer intensive, a limit theorem for the prior measures is presented that is intended to speed up the computations drastically.

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