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

arXiv: https://arxiv.org/abs/2005.06530 · HTML: https://mlsanity.com/paper/2005.06530/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2020.

## Built on by (most cited first, 3 of 3)

- [The Fourier Discrepancy Function](https://mlsanity.com/paper/2102.02979/) (2021)
- [A kinetic approach to consensus-based segmentation of biomedical images](https://mlsanity.com/paper/2211.05226/) (2022)
- [I-SAFE: Wasserstein Coherence Metrics for Structural Auditing of Scientific AI Models](https://mlsanity.com/paper/2605.21731/) (2026)

## Abstract (arXiv)

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.