Monotonicity of the Laplace Transform for Tomography in Dissipative Systems

TL;DR

This work develops a monotonicity-based framework for Magnetic Induction Tomography (MIT) in dissipative, parabolic settings. It introduces the Transfer Operator $\mathcal{H}_{\eta}$, defined via Laplace transforms to map the source current density to the measured field, and proves a Monotonicity Principle on the real axis: $\alpha \le \beta$ implies $\mathcal{H}_{\alpha}(\lambda) \preceq \mathcal{H}_{\beta}(\lambda)$ for suitable $\lambda$. Existence of $\mathcal{H}_{\eta}$ is established through coercivity and the Lax-Milgram theorem, enabling a non-iterative, monotonicity-based imaging method that yields upper/lower bounds on anomalies by testing small subdomains. The approach provides real-time capability and resilience to noise, extending monotonicity-based imaging from elliptic problems to parabolic MIT and highlighting the utility of Laplace-domain analysis for inverse obstacle problems in dissipative media.

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.

Figures (3)

• Figure 1: Schematic of a generic system for Magnetic Induction Tomography. A set of electrical currents circulating in the region $\Omega_S$ induces electrical currents in the conducting domain under imaging $\Omega_C$. The spatial distribution of the electrical resistivity $\eta$ affects the reaction field measured in the source region $\Omega_S$.
• Figure 2: Overview of MPs for parabolic PDEs represented in the Laplace plane. In su2023tranfer, the data are embedded in exponentially decaying functions that may suffer from a poor signal-to-noise ratio in real-world settings. In su2017monotonicitytamburrino2021monotonicity, an MP for the time constants of the free response was proved. Measuring many time constants is a challenging problem. In tamburrino2006fast a MP valid in the limit of small angular frequencies (large skin-depth) was proved. In tamburrino2010recent a MP valid in the limit of large angular frequencies (small skin-depth) was proved. In su2023tranfersu2017monotonicitytamburrino2006fasttamburrino2010recent, the MP was proved in a discrete setting, for both the underlying PDE and the measurement system. In this work, MP is proved for (i) quantities that that be robustly measured even in noisy environments (Transfer Operator evaluated on a right real semi-axis), (ii) for an exact model, (iii) for the continuum physical model, and (iv) for an arbitrary measurement system in infinite dimensional space.
• Figure 3: Schematic of a typical eddy current problem: a conducting domain $\Omega_C$ and the source region $\Omega_S$. In $\Omega_S$ a source electrical current density $\mathbf{J}_s$ is applied and the (reaction) electric field $\mathbf{E}_R$ produced by the current density induced in $\Omega_C$ is measured. The operator (Transfer Operator) mapping the applied source current density $\mathbf{J}_s$ into $\mathbf{E}_R$ is the measured data.

Theorems & Definitions (22)

• Proposition 1
• Remark 2
• Proposition 3
• proof
• Proposition 4
• proof
• Lemma 5
• proof
• Remark 6
• Lemma 7