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。蛋白质骨架用曲线表示，核磁共振扫描给我们大脑的三维表示，材料中的应力张量可以用弯曲流形中的点集表示。由于数据收集和传感技术的改进，我们现在有了大型的形状数据库，并有机会通过对这些形状集合进行详细分析来更深入地了解科学过程。形状数据库庞大而复杂。因此，分析形状集合时的一个核心挑战是找到这些集合的简洁描述或概要。这样的概要为我们提供了数据的快照，描述了人口的局部和全球变化，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，形状所处的空间在数学上是复杂的。计算具有可证明性质的紧凑概要需要新的几何和算法思想。在这个项目中，PI将利用形状匹配、数据分析、概要数据结构和非欧几里得空间几何等领域的经验，开发一套用于生成形状概要结构的算法技术，包括广义均值和中位数、聚类、子空间表示以及近似直方图和分布。本研究为形状空间的严谨数据分析奠定了基础。它开发了算法和几何构建块，将产生对这些空间的更深层次的理解，这样做将产生许多关于形状所居住的非欧几里得空间的计算的新想法。更广泛地说，形状的计算研究作为深入科学探究的途径，利用了计算方法的力量以及不断增长的数据存储，并为数据驱动科学的新兴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.