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Optimal transportation (OT) finds the most economical way to transport one measure to another and plays an important role in geometric modeling and processing.
Based on the Brenier theorem, the OT problem is equivalent to the Alexandrov problem, which is the dual to the Pogorelov problem.
Although solving the Alexandrov/Pogorelov problem are both equivalent to solving the Monge-Amp\`{e}re equation, the former requires second type boundary condition and the latter requires much simpler Dirichlet boundary condition. Hence, we propose to use the Pogorelov map to approximate the OT map. The Pogorelov problem can be solved by a convex geometric optimization framework, in which we need to ensure the searching inside the admissible space. In this work, we prove the discrete Alexandrov maximum principle, which gives an apriori estimate of the searching.
Our experimental results demonstrate that the Pogorelov map does approximate the OT map well with much more efficient computation.

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