Geometric flows for shape analysis

本研究计划将集中在形状分析领域的两个最新发展。第一个与向外流动有关，通过该向外流动，对象的边界从内侧流形恢复。最近的数学结果表明，它是可能的恢复微分几何的边界表面从适当的计算，其相应的中间流形。此外，我的小组在汉密尔顿-雅可比骨架上的工作，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殊情况。事实上，现在理论上可以直接根据可测函数在曲面或体积上的中间积分来获得它们的积分。我将专注于将这些理论结果转化为计算环境，以我自己在获得和操纵内侧表征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.