AF: Small: Extending algorithms for topological notions of similarity

你如何衡量两个签名或两个粘土雕像的接近程度？该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跟踪数据与路线图进行比较，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。虽然一些比较方法只是简单地观察物体的距离，但该项目旨在设计更复杂的测量方法，考虑到曲线和表面的底层结构（或拓扑结构）。PI计划与使用这些测量算法的应用领域合作，开发最适合该领域的测量方法，此外，她还将继续在更大的形状分析和计算拓扑社区网络中开展工作，包括组织研讨会，重点指导这些领域的年轻女性。这个项目将相似性的概念从曲线推广到图形、网格或3流形。主要的问题和技术来自新兴的计算拓扑领域，它将算法和计算几何与拓扑领域的数学基础和工具相结合。特别地，PI建议在计算最优同伦、同构或同位素的基础上，检查一些环境空间中曲线和曲面之间相似性的基本拓扑概念。每种可能性都提供了一种不同的概念，即两种事物在环境空间中相等意味着什么，每种可能性都可以基于面积、“高度”或“宽度”的概念进行优化。虽然关于这些措施的一些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，从算法的角度来看，许多潜在的可处理和可行的领域尚未被探索。其中一些测量可能难以计算，这是理论界感兴趣的，近似或固定参数可追踪算法可能在应用领域被证明是实用的。该项目还将包括在这些措施的应用领域的合作，以便更好地评估它们的效用。

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.