Geometric flows for shape analysis

本研究计划将集中于形状分析领域的两个最新发展。第一个与向外流动有关，通过向外流动，物体的边界从中间流形中恢复。最近的数学结果表明，可以通过对边界曲面相应的中间流形进行适当的计算来恢复边界曲面的微分几何。此外，我的团队在hamilton-jacobi骨架上的工作，特别是我们基于平均向外通量的公式，已被证明是更一般的内侧积分的特殊情况。事实上，从理论上讲，现在可以直接用可测量函数在表面或体积上的对应的中间积分来求积分。我将专注于将这些理论结果转化为计算设置，以我自己在获取和操纵媒体表示的算法开发方面的专业知识为基础。

This research program will concentrate on two recent developments in the field of shape analysis. The first has to do with an outward flow by which the boundary of an object is recovered from a medial manifold. Very recent results in mathematics show that it is possible to recover the differential geometry of a bounding surface from appropriate calculations on its corresponding medial manifold. Furthermore, my group's work on hamilton-jacobi skeletons, and in particular our average-outward flux based formulation, has been shown to be a special case of a more general medial integral. In fact, it is now possible in theory to obtain integrals of measurable functions on surfaces or volumes directly in terms of their medial integral counterparts. I will focus on translating these theoretical results to a computational setting, grounded in my own expertise in the development of algorithms for obtaining and manipulating medial representations.