傅立叶变换与Wasserstein变换在成像问题上的等价性
The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems
摘要：
我们研究一类基于傅立叶的概率距离的一些扩展的性质，最初引入研究收敛到平衡的空间齐次玻尔兹曼方程的解决方案。在与原来的不同，新的基于傅立叶的度量也定义良好的概率分布与不同的质量中心，和离散的概率测度支持一个规则的网格。在其他属性中，它示出，在离散设置中，这些新的基于傅立叶的度量是等价的欧几里得-Wasserstein距离$W_2$，或Kantorovich-Wasserstein距离$W_1$，具有明确的常数的等价。数值结果表明，在图像处理的基准问题，傅立叶度量提供了更好的运行时间相对于Wasserstein的。
英文摘要：
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.
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