| In this paper, we propose a novel and rigorous framework for region-based active contours that combines the Wasserstein distance between statistical distributions in arbitrary dimension and shape derivative tools. To the best of our knowledge, this is the first variational image segmentation algorithm that involves region-dependent multi-dimensional descriptors based on the optimal transport theory. The distributions are represented owing to non-parametric kernel density estimators (e.g. Parzen), and the exact evolution speed corresponding to the Wasserstein-based segmentation energy is provided. To speed-up the computation and be able to handle high-dimensional features and large-scale data, we introduce a sliced Wasserstein approximation of the original Wasserstein distance. The framework is flexible enough to allow either minimization of the Wasserstein distance to known fixed distributions, or maximization of the distance between the distributions of the regions to be segmented (region competition). Numerical results reported to show the advantages of the proposed optimal transport distance with respect to alternative metrics (such as the Kullback-Leibler divergence). These traditional metrics cannot deal properly with distributions having localized supports, and do not take into account the distance between the modes of the histograms. Additionally, our framework handles distributions in arbitrary dimension, which is crucial to segment color images. |
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