Barcodes: The persistent topology of data

2007/10/26 by Robert Ghrist · 1,304 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Digital Image Processing Techniques #Homotopy and Cohomology in Algebraic Topology #Topological data analysis #Persistent homology #Point cloud #Computational topology #Algebraic topology #Homology (biology) #Topology (electrical circuits) #Computer science #Algebraic number #Feature (linguistics) #General topology #Theoretical computer science #Cloud computing #Mathematics #Artificial intelligence #Algorithm #Topological space #Discrete mathematics #Pure mathematics #Combinatorics #Biology

paper · pdf · doi:10.1090/s0273-0979-07-01191-3

published in Bulletin of the American Mathematical Society 45(1), 61-75 (American Mathematical Society)

openalex publication_date 2007/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets— <bold>persistent homology</bold> —and a novel representation of this algebraic characterization— <bold>barcodes</bold> . We sketch an application of these techniques to the classification of natural images.

Citations

• �ber den h�heren Zusammenhang kompakter R�ume und eine Klasse von zusammenhangstreuen Abbildungen
• Stability of Persistence Diagrams
• Three-dimensional alpha shapes
• Pattern theory: the mathematics of perception
• On the Local Behavior of Spaces of Natural Images
• AN ALGEBRAIC TOPOLOGICAL METHOD FOR FEATURE IDENTIFICATION
• A weak definition of Delaunay triangulation
• Computing Persistent Homology
• Topological Persistence and Simplification

Cited by

• One-dimensional reduction of multidimensional persistent homology
• Visualizing High-Dimensional Data: Advances in the Past Decade
• Persistent Brain Network Homology From the Perspective of Dendrogram
• Stratifying Multiparameter Persistent Homology
• Persistent Homology of Collaboration Networks
• Geometry in the space of persistence modules
• Hall Effect Thruster Forecasting using a Topological Approach for Data Assimilation
• Topological data analysis [wikipedia]