Geometric flows for shape analysis

这项研究计划将集中在形状分析领域的两个最新发展。第一种与向外流动有关，通过这种向外流动，物体的边界从中间流形中恢复出来。最新的数学结果表明，通过在其相应的中间流形上进行适当的计算，可以恢复边界曲面的微分几何。此外，我的团队关于哈密顿-雅可比骨架的工作，特别是我们基于平均向外通量的公式，已经被证明是更一般的中间积分的特例。事实上，现在在理论上可以直接获得曲面或体积上的可测函数的积分，就是它们的中间积分对应的积分。我将专注于将这些理论结果转化为计算环境，基于我自己在开发获得和操作中间表示的算法方面的专业知识。

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.