We present a numerical method to solve the optimal transport problem with a
quadratic cost when the source and target measures are periodic probability densities.
This method relies on a numerical resolution of the corresponding Monge-Ampère
equation. We use an existing Newton-like algorithm that we generalize to the case of
a non uniform final density. The main idea consists of designing an iterative scheme
where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we discretize and solve at each iteration, in two different ways: a second order finite difference scheme and a Fourier transform (FT) method. The FT method, made possible thanks to a preconditioning step based on the coefficient-averaged equation, results in an overall O(P LogP )-operations algorithm, where P is the number of discretization points. We prove that the generalized algorithm converges to the solution of the optimal transport problem, under suitable conditions on the initial and final densities. Numerical experiments demonstrating the robustness
and efficiency of the method on several examples of image processing, including an
application to multiple sclerosis disease detection, are shown. We also demonstrate by
numerical tests that the method is competitive against some other methods available.
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/3157 |
| Date | 13 December 2010 |
| Creators | Saumier Demers, Louis-Philippe |
| Contributors | Agueh, Martial, Khouider, Boualem |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | Available to the World Wide Web |
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