Optimal Transportation for Electrical Impedance Tomography

TL;DR

This paper introduces a novel optimal transportation-based framework for electrical impedance tomography, featuring a new efficient algorithm on the circle and demonstrating improved computational performance and effectiveness in numerical tests.

Contribution

It develops a new OT-based approach with a fast algorithm for EIT, reducing computational complexity and enabling more effective inverse boundary problem solutions.

Findings

Reduced algorithm complexity to O(N) from O(N^3)

Demonstrated effectiveness through numerical examples

Provided a systematic Fréchet gradient derivation using OT theory

Abstract

This work establishes a framework for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance ($W_{2}$). A general form of the Fr\'echet 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.