AF: Small: Synopsis Data Structures for Data Analysis in Shape Spaces

科学研究产生了许多不同的形状。蛋白质骨架被表示为曲线，核磁共振扫描为我们提供了大脑的三维表示，材料中的应力张量可以用曲线流形中的点集表示。由于数据采集和传感技术的进步，我们现在拥有了大型的形状数据库，并有机会通过对这些形状收集进行详细的分析来更深入地了解科学过程。因此，分析Shape集合时的一个核心挑战是找到这些集合的简明描述或摘要。这样的概要为我们提供了数据的快照，描述了人口中的局部和全球变化，并为高级和严格的数据分析提供了基础。然而，形状具有复杂的表示形式，形状所在的空间在数学上是错综复杂的。计算具有可证明质量的紧凑概要需要新的几何和算法思想。在这个项目中，PI将利用在形状匹配、数据分析、概要数据结构和非欧几里德空间几何方面的经验，开发一套算法技术来产生形状上的概要结构，包括广义均值和中值、聚类、子空间表示以及近似直方图和分布。这项研究为形状空间中严格的数据分析奠定了基础。它开发了算法和几何构件，将产生对这些空间的更深层次的理解，这样做将为形状所在的非欧几里德空间的计算产生许多新的想法。更广泛地说，将形状作为深入科学研究的途径的计算研究利用了计算方法的力量以及日益增长的数据存储，并为新兴的数据驱动型科学领域创造了新的机会。

Scientific inquiries generate many different kinds of shapes. Protein backbones are represented as curves, MRI scans give us a three-dimensional representation of the brain, and stress tensors in materials can be represented by point sets in a curved manifold. Because of improvements in data gathering and sensing technologies, we now have large databases of shapes, and have the chance to gain a deeper understanding of scientific processes by performing detailed analysis of these shape collections.Shape databases are large and complex. Thus, a core challenge when analyzing shape collections is finding succinct descriptions, or synopses, of these collections. Such synopses give us snapshots of the data, describe local and global variations in a population, and provide the building blocks for advanced and rigorous data analysis. However, shapes have complex representations, and the spaces that shapes inhabit are mathematically intricate. Computing compact synopses with provable quality requires new geometric and algorithmic ideas.In this project, the PI, using experience in the areas of shape matching, data analysis, synopsis data structures, and the geometry of non-Euclidean spaces, will develop a suite of algorithmic techniques for producing synopsis structures on shapes, including generalized means and medians, clusterings, subspace representations, and approximate histograms and distributions. This research lays the foundations for rigorous data analysis in shape spaces. It develops the algorithmic and geometric building blocks that will yield a deeper understanding of these spaces, and in doing so will yield many new ideas for computations in the non-Euclidean spaces that shapes inhabit. More broadly, the computational study of shape as a pathway to deeper scientific inquiry takes advantage of the power of computational methods as well as growing stores of data, and creates new opportunities for the emerging area of data-driven science.