Medial axes and symmetry sets of isophote curves

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For a (usually closed) curve in the plane the symmetry set and the medial axis are measures of the local symmetry of the curve. The medial axis plays the role of a skeleton, something that goes down the middle of the shape enclosed by the curve. It has the structure of a branched 1-dimensional set (a tree). It is used in shape analysis, comparison and recognition. For a 2-dimensional image (like a photograph of a 3D scene), at each point there is a brightness, measured say between 0 and 1. The brightness can be thought of as a third coordinate, after the first two which specify the point in the image. Lines of equal brightness are called isophotes and they are curves in the plane of the image. These curves are studied because they give much information about the image. We plan to carry out an investigation of the medial axes and symmetry sets of these curves, to obtain information which has not been available before.The principal investigator (Giblin) has carried out many similar investigations before but this problem has proved difficult with existing techniques. However recent work of the Visiting Researcher (Uribe-Vargas) introduces techniques which should make the investigation possible.