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Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions

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Abstract

This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued continuous function. Measuring dissimilarity amounts to minimizing the change in the functions due to the application of homeomorphisms between topological spaces, with respect to the L ∞-norm. In order to obtain the lower bound, a suitable metric between size functions, called matching distance, is introduced. It compares size functions by solving an optimal matching problem between countable point sets. The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance. We also prove that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.

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Authors and Affiliations

  1. ARCES, Università di Bologna, via Toffano 2/2, 40135, Bologna, Italy

    Michele d’Amico & Patrizio Frosini

  2. Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127, Bologna, Italy

    Patrizio Frosini

  3. DISMI, Università di Modena e Reggio Emilia, via Amendola 2, Pad. Morselli, 42100, Reggio Emilia, Italy

    Claudia Landi

Authors
  1. Michele d’Amico
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  2. Patrizio Frosini
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  3. Claudia Landi
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Correspondence to Claudia Landi.

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d’Amico, M., Frosini, P. & Landi, C. Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions. Acta Appl Math 109, 527–554 (2010). https://doi.org/10.1007/s10440-008-9332-1

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  • DOI: https://doi.org/10.1007/s10440-008-9332-1

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