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

Gunnar E. Carlsson, Persi Diaconis, Leonidas J. gu和Susan holmes这是DMS重点研究小组在征求中获得的奖项http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm。主要研究者是斯坦福大学的Gunnar E. Carlsson、Persi Diaconis、Leonidas J. gu和Susan Holmes。该项目将开发用于理解数据集定性属性的拓扑工具。我们将使用同调直接应用于数据集和派生复合体来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，以及应用于标准定义的膨胀和切线复合体的同源性来区分三维欧几里德空间中的二维形状。我们将使用最近开发的持久性和标记技术使同源性成为稳定且易于计算的不变量。我们还将发展多维持久性理论，在该理论中，人们研究具有多个参数的空间，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特性。总体目标是继续开发和改进用于研究几何对象定性信息的可用工具。该项目的目标是开发工具来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异点，或者可能是强弯曲的。数据也是高维的，因为每个数据点都有许多坐标。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。我们想到的这种工具在某种程度上是拓扑的，从某种意义上说，它们测量的是所涉及的空间的更多定性性质，比如连通性，或者空间中孔的数量，等等。例如，该项目采取的观点是，在尝试进行更精确的定量分析之前，最好先了解定性属性，通过定性理解来更好地区分形状，而不是进行数据库比较。因此，将开发方法以及时、健壮和可靠的方式计算与给定数据集相关的任何实际数学模型必须包含的基本几何属性。然后，统计和分析技术可以应用于几何正确的模型，以提取从业者所需的详细信息。

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.