Earth Mover Distance on superpixels

Earth Mover Distance (EMD) is a popular distance to compute distances between Probability Density Functions (PDFs). It has been successfully applied in a wide selection of problems of image processing. This success comes from two reasons, a physical one, since it computes a physical cost to transport an element of mass between two images or two histograms, and a statistical one, since it is a cross-bin metric (as opposed to a bin-wise metric). In computer vision, these features are useful since small variation of illuminance can shift the histogram. However, histograms are not a sufficient statistic to discriminate images since they ignore all geometric correlations. In addition, transport also called flow of an histogram loose the information of geometric flow to warp one image on to an other. This paper proposes a new construction of EMD between images. This construction approximates the EMD between two images, by computing a pixel-wise transport at the complexity cost of computing an EMD between 1-D Histograms and preserves the geometrical and topological structure of the image. This construction simply relies on a segmentation of the image (also called superpixelization of the image). Results on matching on images shows the stability of the method even when the superpixelizations are highly inconsistent across images.