2014/05/01 by Gunnar Carlsson · 242 citations
Computer Science · Medicine · Mathematics · #Topological and Geometric Data Analysis #Advanced Neuroimaging Techniques and Applications #Homotopy and Cohomology in Algebraic Topology #Persistent homology #Topological data analysis #Point cloud #Homology (biology) #Computer science #Metric space #Computational topology #Topological space #Algebraic topology #Topology (electrical circuits) #Algebraic number #Computation #Theoretical computer science #Mathematics #Artificial intelligence #Pure mathematics #Algorithm #Biology #Combinatorics #Homotopy
paper · pdf · doi:10.1017/s0962492914000051
published in Acta Numerica 23, 289-368 (Cambridge University Press)
openalex publication_date 2014/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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.