Fast alignment of heterogeneous images in sliced Wasserstein distance

Many applications of computer vision rely on the alignment of similar but non-identical images. We present a fast algorithm for aligning heterogeneous images based on optimal transport. Our approach combines the speed of fast Fourier methods with the robustness of sliced probability metrics and allows us to efficiently compute the alignment between two $L \times L$ images using the sliced 2-Wasserstein distance in $O(L^2 \log L)$ operations. We show that our method is robust to translations, rotations and deformations in the images.