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

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 met- rics 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 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 pro- vide a better runtime with respect to Wasserstein ones.