The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems

We investigate properties of some extensions of a class of Fourier-based metrics, originally introduced to study convergence to equilibrium the solution to the spatially homogeneous Boltzmann equation. At difference the original one, the new Fourier-based metrics are well-defined also for distributions with different centers of mass, and for discrete measures supported over a regular grid. Among other properties, it shown that, in the discrete setting, these new Fourier-based metrics are either to the Euclidean-Wasserstein distance W₂, or to the-Wasserstein distance W₁, with explicit constants of equivalence. results then show that in benchmark problems of image processing, metrics provide a better runtime with respect to Wasserstein ones.