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A New Approximation Algorithm for the Matching Distance in Multidimensional Persistence

A New Approximation Algorithm for the Matching Distance in Multidimensional Persistence
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Year:    2020

Author:    Andrea Cerri, Patrizio Frosini

Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 291–309

Abstract

Topological Persistence has proven to be a promising framework for dealing with problems concerning shape analysis and comparison. In this context, it was originally introduced by taking into account 1-dimensional properties of shapes, modeled by real-valued functions. More recently, Topological Persistence has been generalized to consider multidimensional properties of shapes, coded by vector-valued functions. This extension has led to introduce suitable shape descriptors, named the multidimensional persistence Betti numbers functions, and a distance to compare them, the so-called multidimensional matching distance.
In this paper we propose a new computational framework to deal with the multidimensional matching distance. We start by proving some new theoretical results, and then we use them to formulate an algorithm for computing such a distance up to an arbitrary threshold error.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1809-m2018-0043

Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 291–309

Published online:    2020-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Multidimensional persistent topology Matching distance Shape comparison.

Author Details

Andrea Cerri

Patrizio Frosini

  1. Topological Dynamics and Topological Data Analysis

    A Brief Introduction to Multidimensional Persistent Betti Numbers

    Cerri, Andrea | Frosini, Patrizio

    2021

    https://doi.org/10.1007/978-981-16-0174-3_18 [Citations: 0]

  2. On the geometrical properties of the coherent matching distance in 2D persistent homology

    Cerri, Andrea | Ethier, Marc | Frosini, Patrizio

    Journal of Applied and Computational Topology, Vol. 3 (2019), Iss. 4 P.381

    https://doi.org/10.1007/s41468-019-00041-y [Citations: 6]

  3. Geometry of the matching distance for 2D filtering functions

    Ethier, Marc | Frosini, Patrizio | Quercioli, Nicola | Tombari, Francesca

    Journal of Applied and Computational Topology, Vol. 7 (2023), Iss. 4 P.815

    https://doi.org/10.1007/s41468-023-00128-7 [Citations: 0]