Image segmentation via <inline-formula><tex-math id="M1">\begin{document}$ L_1 $\end{document}</tex-math></inline-formula> Monge-Kantorovich problem

Abstract

This paper provides a fast approach to apply the Earth Mover's Distance (EMD) (a.k.a optimal transport, Wasserstein distance) for supervised and unsupervised image segmentation. The model globally incorporates the transportation costs (original Monge-Kantorovich type) among histograms of multiple dimensional features, e.g. gray intensity and texture in image's foreground and background. The computational complexity is often high for the EMD between two histograms on Euclidean spaces with dimensions larger than one. We overcome this computational difficulty by rewriting the model into a \begin{document}$ L_1 $\end{document} type minimization with the linear dimension of feature space. We then apply a fast algorithm based on the primal-dual method. Compare to several state-of-the-art EMD models, the experimental results based on image data sets demonstrate that the proposed method has superior performance in terms of the accuracy and the stability of the image segmentation.