Fast Transport Optimization for Monge Costs on the Circle
Delon, Julie; Salomon, Julien; Sobolevski, Andrei (2010), Fast Transport Optimization for Monge Costs on the Circle, SIAM Journal on Applied Mathematics, 70, 7, p. 2239-2258. http://dx.doi.org/10.1137/090772708
Type
Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00362834/en/Date
2010Journal name
SIAM Journal on Applied MathematicsVolume
70Number
7Publisher
SIAM
Pages
2239-2258
Publication identifier
Metadata
Show full item recordAbstract (EN)
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.Subjects / Keywords
Monge-Kantorovich problem; Aubry-Mather (weak KAM) theory; Optimization and ControlRelated items
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