FRG: Algebraic topology as a tool in feature location, feature classification, shape recognition, and shape description

FRG：代数拓扑作为特征定位、特征分类、形状识别和形状描述的工具

• 批准号: 0354543
• 负责人: Gunnar Carlsson
• 金额: --
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354543&HistoricalAwards=false
• 关键词: FRG; Algebraic; topology; tool; feature

DMS-0354543 Gunnar E. Carlsson, Persi Diaconis, Leonidas J. Guibas, and Susan HolmesThis is a DMS Focused Reseach Group award under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm. The principal investigators are Gunnar E. Carlsson, Persi Diaconis, Leonidas J. Guibas, and Susan Holmes at Stanford University. This project will develop topological tools for understanding qualitative properties of data sets. We will use homology as applied to data sets directly and to derived complexes to define invariants or signatures that distinguish between the underlying geometric objects. Important goals will include the identification, location, and classification of qualitative features of the data set, such as the presence of corners, edges, cone points, etc. and the use of homology applied to canonically defined blowups and tangent complexes to distinguish between two dimensional shapes in three dimensional Euclidean space. We will use the recently developed techniques of persistence and landmarking to make homology a stable and readily computable invariant. We will also develop the theory of multidimensional persistence, in which one studies spaces that are equipped with several parameters, in order to better understand data sets in which there are several different parameters describing different geometric properties of the space. The overall goal is to continue to develop and improve the available tools for studying qualitative information about geometric objects.The goal of this project is to develop tools for understanding data sets that are not easy to understand using standard methods of statistics and analysis. This kind of data might include singular points, or might be strongly curved. The data is also high dimensional, in the sense that each data point has many coordinates. For instance, we might have a data set whose points each of which is an image, which has one coordinate for each pixel. Many standard tools rely on linear approximations, which do not work well in strongly curved or singular problems. The kind of tools we have in mind are in part topological, in the sense that they measure more qualitative properties of the spaces involved, such as connectedness, or the number of holes in a space, and so on. For example, the project takes the point of view that it is better to understand qualitative properties before attempting to do more precise quantitative analysis and better to distinguish shapes by understanding them qualitatively rather than doing data base comparisons. Thus, methods will be developed to compute, in a timely, robust, and trustworthy manner, the fundamental geometric properties that any realistic mathematical model associated to a given data set must contain. Then statistical and analytic techniques may be applied to the geometrically correct models in order to extract the detailed information desired by practitioners.

DMS-0354543 贡纳·E卡尔松，珀西Diaconis，列奥尼达斯J。吉巴斯，和苏珊霍姆斯这是一个DMS重点研究组奖下征求http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm。 主要研究人员是Gunnar E。卡尔松、佩尔西·迪亚科尼斯、列奥尼达斯·J·吉巴斯和苏珊·霍姆斯在斯坦福大学。 本项目将开发用于理解数据集定性特性的拓扑工具。 我们将使用直接应用于数据集的同调，并将其应用于派生的复形，以定义区分底层几何对象的不变量或签名。 重要的目标将包括数据集的定性特征的识别、定位和分类，例如角、边、锥点等的存在，以及应用于规范定义的爆破和切复体的同源性的使用，以区分三维欧几里得空间中的二维形状。 我们将使用最近开发的持久性和地标技术，使同源性稳定，易于计算的不变量。我们还将发展多维持久性理论，其中一个研究配备了几个参数的空间，以便更好地理解数据集，其中有几个不同的参数描述空间的不同几何属性。 总体目标是继续开发和改进现有的工具，用于研究几何物体的定性信息，本项目的目标是开发工具，用于理解使用标准统计和分析方法不容易理解的数据集。这类数据可能包含奇点，或者可能是强烈弯曲的。数据也是高维的，在这个意义上，每个数据点有许多坐标。例如，我们可能有一个数据集，它的每个点都是一个图像，每个像素都有一个坐标。许多标准工具依赖于线性近似，这在强弯曲或奇异问题中不能很好地工作。我们所考虑的这类工具在某种程度上是拓扑的，因为它们测量所涉及的空间的更多定性性质，例如连通性，或空间中的孔的数量，等等。该项目的观点是，在试图进行更精确的定量分析之前，最好先了解定性性质，并通过了解它们来区分形状而不是进行数据库比较。 因此，将开发方法来计算，在一个及时的，强大的，和值得信赖的方式，基本的几何属性，任何现实的数学模型相关联的一个给定的数据集必须包含。 然后，可以将统计和分析技术应用于几何上正确的模型，以提取从业者所需的详细信息。