🍩 Database of Original & Non-Theoretical Uses of Topology

(found 4 matches in 0.000908s)

  1. Topological Pattern Recognition for Point Cloud Data* (2014)

    Gunnar Carlsson
    Abstract In this paper we discuss the adaptation of the methods of homology from algebraic topology to the problem of pattern recognition in point cloud data sets. The method is referred to as persistent homology, and has numerous applications to scientific problems. We discuss the definition and computation of homology in the standard setting of simplicial complexes and topological spaces, then show how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object. We present several different cases where persistent homology is used, to illustrate the different ways in which the method can be applied.

  2. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition (2007)

    Gurjeet Singh, Facundo Mémoli, Gunnar Carlsson
    Abstract We present a computational method for extracting simple descriptions of high dimensional data sets in the form of simplicial complexes. Our method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data. The proposed method is not dependent on any particular clustering algorithm, i.e. any clustering algorithm may be used with Mapper. We implement this method and present a few sample applications in which simple descriptions of the data present important information about its structure.

  3. Topology Across Scales on Heterogeneous Cell Data (2025)

    Maria Torras-Perez, Iris H.R. Yoon, Praveen Weeratunga, Ling-Pei Ho, Helen M. Byrne, Ulrike Tillmann, Heather A. Harrington

  4. Euler Characteristic Surfaces: A Stable Multiscale Topological Summary of Time Series Data (2024)

    Anamika Roy, Atish J. Mitra, Tapati Dutta
    Abstract We present Euler Characteristic Surfaces as a multiscale spatiotemporal topological summary of time series data encapsulating the topology of the system at different time instants and length scales. Euler Characteristic Surfaces with an appropriate metric is used to quantify stability and locate critical changes in a dynamical system with respect to variations in a parameter, while being substantially computationally cheaper than available alternate methods such as persistent homology. The stability of the construction is demonstrated by a quantitative comparison bound with persistent homology, and a quantitative stability bound under small changes in time is established. The proposed construction is used to analyze two different kinds of simulated disordered flow situations.