Optimal Transportation for Electrical Impedance Tomography

This work establishes a framework for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance ($W_2$). A general form of the Fréchet gradient is systematically derived by optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on ${\mathbb{S}}^1$ is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to $O(N)$ from $O(N^3)$ of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography (EIT) problem. Numerical examples are presented to illustrate the effectiveness of our method.