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.

The aim of change detection (CD) is to detect changes occurred in the same area by comparing two images of that place taken at different times. The challenging part of the CD is to keep track of the changes the user wants to highlight, such as new buildings, and to ignore changes due to external factors such as environmental, lighting condition, fog or seasonal changes. Recent developments in the field of deep learning enabled researchers to achieve outstanding performance in this area. In particular, different mechanisms of space-time attention allowed to exploit the spatial features that are extracted from the models and to correlate them also in a temporal way by exploiting both the available images. The downside is that the models have become increasingly complex and large, often unfeasible for edge applications. These are limitations when the models must be applied to the industrial field or in applications requiring real-time performances. In this work we propose a novel model, called TinyCD, demonstrating to be both lightweight and effective, able to achieve performances comparable or even superior to the current state of the art with 13-150X fewer parameters. In our approach we have exploited the importance of low-level features to compare images. We introduce a novel mixing block capable of cross correlating features in both space and time domains. Finally, to fully exploit the information contained in the computed features, we define the PW-MLP block able to perform a pixel wise classification.

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