The Earth Mover's Distance under Transformation Sets

The Earth Mover’s Distance (EMD) is a distance measure between distributions with applications in image retrieval and matching. We consider the problem of computing a transformation of one distribution which minimizes its EMD to another. The applications discussed here include estimation of the size at which a color pattern occurs in an image, lighting-invariant object recognition, and point feature matching in stereo image pairs. We present a monotonically convergent iteration which can be applied to a large class of EMD under transformation problems, although the iteration may converge to only a locally optimal transformation. We also provide algorithms that are guaranteed to compute a globally optimal transformation for a few specific problems, including some EMD under translation problems.