Abstract
Computing the Kantorovich distance for images is equivalent to solving a very large transportation problem. The cost-function of this transportation problem depends on which distance-function one uses to measure distances between pixels.
In this paper we present an algorithm, with a computational complexity of roughly order \(\mathcal{O}\)(N2), where N is equal to the number of pixels in the two images, in case the underlying distance-function is the L1-metric, an approximation of the L2-metric or the square of the L2-metric; a standard algorithm would have a computational complexity of order \(\mathcal{O}\)(N3). The algorithm is based on the classical primal-dual algorithm.
The algorithm also gives rise to a transportation plan from one image to the other and we also show how this transportation plan can be used for interpolation and possibly also for compression and discrimination.
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Kaijser, T. Computing the Kantorovich Distance for Images. Journal of Mathematical Imaging and Vision 9, 173–191 (1998). https://doi.org/10.1023/A:1008389726910
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DOI: https://doi.org/10.1023/A:1008389726910