Donal O'Shea

A geometric criterion for shape-based non-rigid correspondence

A geometric criterion is developed for establishing shape based non rigid correspondence between plane curves. Unlike previous efforts, the criterion does not use rigid invariants of shape. Instead, shapes are compared non rigidly from the vantage point of the correspondence. Geometric invariants are proposed for curves whose shapes can be exactly matched by a non rigid correspondence. The invariants are based on angular deviations of convex and concave segments of the curves. Examples of correspondences between curves obtained from medical images are provided.<<ETX>>

Non-Rigid Shape Comparison of Plane Curves in Images

A mathematical theory for establishing correspondences between curves and for non-rigid shape comparison is developed in this paper. The proposed correspondences, called bimorphisms, are more general than those obtained from one-to-one functions. Their topology is investigated in detail.A new criterion for non-rigid shape comparison using bimorphisms is also proposed. The criterion avoids many of the mathematical problems of previous approaches by comparing shapes non-rigidly from the bimorphism.Geometric invariants are calculated for curves whose shapes can be exactly matched with a bimorphism. The invariants are related to the concave and convex segments of a curve and provide justification for parsing the curve into such segments.

Elementary catastrophes, phase transitions and singularities

Abstract The paper discusses the agreement between observed qualitative behavior of phase transitions in liquid mixtures and the results of catastrophe-theoretic modeling. Some mechanisms to explain the agreement are advanced.