Optimal Mass Transport (OMT) distance - cerr/CERR GitHub Wiki
Formulation
Wasserstein distance is a metric for distributions derived from optimal mass transport. The formulation of OMT due to Monge and Kantorovich [3,4] may be expressed as follows: where denotes the set of all the couplings between and (joint distributions whose two marginal distributions are and ; is the cost of moving unit mass from x to y. The optimal gives a transport plan from to . And the optimal value of the object function is called Wasserstein distance.
Application
Compare dose distributions
Compare images
Example
See Jupyter notebook demonstrating OMT distance between two dose distributions, computed using CERRx.
References
- Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3):375{393, 2000.
- Lenaic Chizat, Gabriel Peyre, Bernhard Schmitzer, and Francois-Xavier Vialard. An interpolating distance between optimal transport and Fisher-Rao metrics. Foundations of Computational Mathematics, 10:1{44, 2016.
- Cedric Villani. Topics in Optimal Transportation. American Mathematical Soc., 2003.
- Cedric Villani. Optimal Transport: Old and New, volume 338. Springer Science & Business Media, 2008.