Fast transport optimization on the circle Export

@article{citeulike:4083140, abstract = {Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts. This theory is applied to a transportation problem arising in image processing: for two sets of point masses, both of which have the same total mass, find an optimal transport plan with respect to a given cost function that satisfies the Monge condition. For the case of N real-valued point masses we present an O(N log epsilon) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.}, archivePrefix = {arXiv}, author = {Delon, Julie and Salomon, Julien and Sobolevskii, Andrei}, citeulike-article-id = {4083140}, citeulike-linkout-0 = {http://arxiv.org/abs/0902.3527}, citeulike-linkout-1 = {http://arxiv.org/pdf/0902.3527}, day = {20}, eprint = {0902.3527}, keywords = {1d, ansobolarxiv, weak-kam}, month = {Feb}, posted-at = {2009-02-23 07:35:29}, priority = {2}, title = {Fast transport optimization on the circle}, url = {http://arxiv.org/abs/0902.3527}, year = {2009} }

TY - JOUR ID - citeulike:4083140 L3 - citeulike-article-id:4083140 N2 - Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts. This theory is applied to a transportation problem arising in image processing: for two sets of point masses, both of which have the same total mass, find an optimal transport plan with respect to a given cost function that satisfies the Monge condition. For the case of N real-valued point masses we present an O(N log epsilon) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations. TI - Fast transport optimization on the circle KW - 1d KW - ansobolarxiv KW - weak-kam AU - Delon, Julie AU - Salomon, Julien AU - Sobolevskii, Andrei PY - 2009/Feb/20/ UR - http://arxiv.org/abs/0902.3527 ER -