Medial axes and symmetry sets of isophote curves

对于平面中的（通常是闭合的）曲线，对称集和中轴是曲线局部对称性的度量。中轴线扮演骨架的角色，它沿着曲线包围的形状的中间延伸。它具有分支一维集合（树）的结构。它用于形状分析、比较和识别。对于二维图像（如3D场景的照片），在每个点处都有一个亮度，例如在0和1之间测量。亮度可以被认为是在前两个坐标之后的第三个坐标，它指定了图像中的点。等亮度的线称为等照度线，它们是图像平面上的曲线。研究这些曲线是因为它们提供了关于图像的许多信息。我们计划进行调查的中轴线和对称集的这些曲线，以获得信息还没有提供before.The首席研究员（Giblin）进行了许多类似的调查之前，但这个问题已经证明很难与现有的技术。然而，访问研究员（Uribe-Vargas）最近的工作介绍了应该使调查成为可能的技术。

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.