The equivalence of Fourier-based and Wasserstein metrics on imaging problems

  • Gennaro Auricchio

    Università degli Studi di Pavia, Italy
  • Andrea Codegoni

    Università degli Studi di Pavia, Italy
  • Stefano Gualandi

    Università degli Studi di Pavia, Italy
  • Giuseppe Toscani

    Università degli Studi di Pavia, Italy
  • Marco Veneroni

    Università degli Studi di Pavia, Italy
The equivalence of Fourier-based and Wasserstein metrics on imaging problems cover

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Abstract

We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At di¤erence with the original one, the new Fourier-based metrics are well-defined also for probability distributions with di¤erent centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fourier-based metrics are equivalent either to the Euclidean–Wasserstein distance , or to the Kantorovich–Wasserstein distance , with explicit constants of equivalence. Numerical results then show that in benchmark problems of image processing, Fourier metrics provide a better runtime with respect to Wasserstein ones.

Cite this article

Gennaro Auricchio, Andrea Codegoni, Stefano Gualandi, Giuseppe Toscani, Marco Veneroni, The equivalence of Fourier-based and Wasserstein metrics on imaging problems. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 3, pp. 627–649

DOI 10.4171/RLM/908