The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems
This work addresses computational efficiency in image processing by providing faster alternatives to Wasserstein metrics, though it is incremental as it builds on existing Fourier-based methods.
The paper tackles the problem of extending Fourier-based probability metrics to handle distributions with different centers of mass and discrete measures, showing equivalence to Wasserstein distances with explicit constants, and demonstrates that Fourier metrics achieve better runtime in image processing benchmarks.
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 $W_2$, or to the Kantorovich-Wasserstein distance $W_1$, 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.