AF: Small: Extending algorithms for topological notions of similarity

你会如何衡量两个签名或两个泥人的接近程度？这个项目探索了测量曲线或三维物体之间相似性的数学和计算--无论是将GPS跟踪数据与道路地图进行比较，还是将内部器官的医学扫描与健康器官的参考模型进行比较。虽然一些广为人知的比较方法只是看物体有多近，但这个项目的目标是设计更贴近的测量方法，考虑到曲线和曲面的基本结构(或拓扑)。PI计划与使用这些测量算法的应用领域合作，开发最适合这些领域的测量方法，此外，还将继续她在形状分析和计算拓扑社区的更大网络中的工作，包括组织研讨会，重点指导这些领域的年轻女性。该项目概括了从曲线到图形、网格或三维流形的相似性概念。主要的问题和技术来自新兴的计算拓扑学领域，它将算法和计算几何与拓扑学领域的数学基础和工具结合在一起。特别是，PI建议在计算最佳同伦、同调或同伦的基础上，研究某些环境空间中曲线和曲面之间的相似的基本拓扑概念。每一种可能性都提供了一种不同的概念，即两个东西在环境空间中相等意味着什么，每一种可能性都可以基于面积、“高度”或“宽度”的概念进行优化。虽然关于这些测量的几个初步算法结果已经公布，但仍有许多悬而未决的问题。此外，最近的数学发展表明，许多潜在的可处理和可行的领域尚未从算法的角度进行探索。其中一些度量可能很难计算，这是理论界感兴趣的，而近似或固定参数易处理的算法可能在应用领域被证明是实用的。该项目还将包括这些措施应用领域的合作，以便更好地评估其效用。

How would you measure the closeness of two signatures or two clayfigurines? This project explores the mathematics and computation ofmeasuring similarity between curves or 3-dimensional objects --whether to compare GPS tracking data to road maps, or medical scans of internal organs toreference models of healthy organs. While some well understood comparison methods simplylook at how close the objects are, this project aims to design moresophisticated measures that take into account the underlying structure(or topology) of the curves and surfaces. The PI plans to collaboratewith applications areas that use these measurement algorithms todevelop measures that best fit the areas, in addition to continuingher work in a larger network of shape analysis and computationaltopology communities, including organizing workshops focused atmentoring junior women in these areas.This project generalizes notions of similarity from curves to graphs, meshes, or3-manifolds. The primary problems and techniques come from theemerging field of computational topology, which combines algorithmsand computational geometry with mathematical foundations and toolsfrom the area of topology. In particular, the PI proposes to examinefundamental topological notions of similarity between curves andsurfaces in some ambient space, based on computing optimal homotopies, homologies, or isotopies. Eachpossibility offers a different notion of what it means for two thingsto be equivalent in the ambient space, and each can be optimized basedon the notions of area, "height," or "width." While several initialalgorithmic results on these measures have been published, there aremany open questions that remain. In addition, recent mathematicaldevelopments indicate many potentially tractable and feasible areasthat are yet to be explored from the algorithmic perspective. Some ofthese measures are likely to be hard to compute, which is of interestto the theoretical community, and approximation or fixed parametertractable algorithms may prove practical in applications areas. Theproject will also include collaborations in applications areas forthese measures, in order to better evaluate their utility.