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 metrics 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 $W_2$, 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.

中文翻译：

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基于傅立叶和Wasserstein度量的成像问题的等价性

我们研究了一类基于傅立叶的概率度量的某些扩展的性质，最初是为了研究空间齐次Boltzmann方程解的收敛到平衡而引入的。与最初的度量方法不同，对于具有不同质心的概率分布以及在常规网格上支持的离散概率度量，新的基于傅立叶的度量标准也得到了很好的定义。除其他属性外，还显示出，在离散设置下，这些新的基于傅立叶的度量等效于欧几里德-沃瑟斯坦距离$ W_2 $或坎托罗维奇-沃瑟斯坦距离$ W_1 $，具有明显的等效常数。数值结果表明，在图像处理的基准问题中，相对于Wasserstein而言，傅里叶度量提供了更好的运行时间。