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 difference
with the original one, the new Fourier-based metrics are well-defined also for
probability distributions with different 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 W2​, or to the
Kantorovich-Wasserstein distance W1​, 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.Comment: 18 pages, 2 figures, 1 tabl