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

Stanford University, Stanford CA

Investigators

Abstract

DMS-0354543 Gunnar E. Carlsson, Persi Diaconis, Leonidas J. Guibas, and Susan Holmes This 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.