AF: Small: Extending algorithms for topological notions of similarity

你如何衡量两个签名或两个泥人的接近程度？ 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跟踪数据与道路图进行比较，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。虽然一些比较方法简单地看物体有多接近，但这个项目的目的是设计更复杂的措施，考虑到曲线和曲面的底层结构（或拓扑）。 PI计划与使用这些测量算法的应用领域合作，以开发最适合这些领域的测量方法，此外还将继续在形状分析和计算拓扑社区的更大网络中开展工作，包括组织研讨会，重点指导这些领域的初级女性。该项目将相似性的概念从曲线推广到图形，网格或三维流形。 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相结合。 特别是，PI建议在计算最佳同伦、同调或同位素的基础上，在某些环境空间中检查曲线和曲面之间相似性的基本拓扑概念。每一种可能性都提供了不同的概念，说明两个物体在周围空间中是等价的，每一种可能性都可以根据面积、“高度”或“宽度”的概念进行优化。“虽然这些措施的几个初始算法结果已经公布，但仍然存在许多悬而未决的问题。此外，最近的apticaldevelopments表明，许多潜在的易处理和可行的areasthat尚未从算法的角度进行探索。这些措施中的一些可能是很难计算的，这是理论界感兴趣的，近似或固定参数易处理的算法可能在应用领域证明是实用的。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.