Magnetic Induction Tomography: Separation of the Ill-Posed and Non-Linear Inverse Problem into a Series of Isolated and Less Demanding Subproblems

Abstract

Magnetic induction tomography (MIT) is based on remotely excited eddy currents inside a measurement object. The conductivity distribution shapes the eddies, and their secondary fields are detected and used to reconstruct the conductivities. While the forward problem from given conductivities to detected signals can be unambiguously simulated, the inverse problem from received signals back to searched conductivities is a non-linear ill-posed problem that compromises MIT and results in rather blurry imaging. An MIT inversion is commonly applied over the entire process (i.e., localized conductivities are directly determined from specific signal features), but this involves considerable computation. The present more theoretical work treats the inverse problem as a non-retroactive series of four individual subproblems, each one less difficult by itself. The decoupled tasks yield better insights and control and promote more efficient computation. The overall problem is divided into an ill-posed but linear problem for reconstructing eddy currents from given signals and a nonlinear but benign problem for reconstructing conductivities from given eddies. The separated approach is unsuitable for common and circular MIT designs, as it merely fits the data structure of a recently presented and planar 3D MIT realization for large biomedical phantoms. For this MIT scanner, in discretization, the number of unknown and independent eddy current elements reflects the number of ultimately searched conductivities. For clarity and better representation, representative 2D bodies are used here and measured at the depth of the 3D scanner. The overall difficulty is not substantially smaller or different than for 3D bodies. In summary, the linear problem from signals to eddies dominates the overall MIT performance.