Monotonicity of the Laplace Transform for Tomography in Dissipative Systems

TL;DR

This paper establishes a monotonicity principle for the transfer operator in Magnetic Induction Tomography, aiding the solution of the nonlinear, ill-posed inverse problem for imaging dissipative materials.

Contribution

It introduces a novel monotonicity principle for the Laplace transform of the transfer operator in MIT, providing a new theoretical foundation for imaging methods.

Findings

Proves the transfer operator satisfies a monotonicity principle on a real semi-axis.

Provides a description of a real-time imaging method based on this principle.

Addresses mathematical challenges in processing MIT data for dissipative materials.

Abstract

This paper addresses the problem of tomography for the interior of dissipative materials, with a focus on Magnetic Induction Tomography (MIT), a proven technique for imaging the interior of conductive materials using low-frequency electromagnetic fields. Processing MIT data is mathematically challenging because of the non-linear and ill-posed nature of the underlying inverse problem. On the other hand, the Monotonicity Principle is recognized as the basis for developing effective approaches. In this framework, the paper presents a principle of monotonicity for the Transfer Operator in Magnetic Induction Tomography, i.e. the operator mapping the Laplace transform of the applied source onto the Laplace transform of the measured quantity. Specifically, it is proved that the Transfer Operator satisfies a Monotonicity Principle when evaluated on a proper real semi-axis of the complex plane. The description of the related (real-time) imaging method is also given.