论文标题
基于傅立叶和沃斯坦的等效性在成像问题上
The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems
论文作者
论文摘要
我们研究了一类基于傅立叶的概率指标的某些扩展的性能，该概率指标最初是为了研究与空间均质玻尔兹曼方程的均衡的融合。在与原始的不同，新的基于傅立叶的指标也是明确的，对于具有不同质量中心的概率分布，以及在常规网格上支持的离散概率度量。除其他属性外，还表明，在离散设置中，这些新的基于傅立叶的指标与Euclidean-Wasserstein距离$ W_2 $或Kantorovich-Wasserstein距离$ W_1 $等效，具有显而易见的常数。然后，数值结果表明，在图像处理的基准问题中，傅立叶指标可提供与瓦斯坦（Wasserstein）相对于瓦斯汀（Wasserstein）提供的更好的运行时间。
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.
