A direct sampling method for magnetic induction tomography

TL;DR

The paper tackles the ill-posed inverse problem of magnetic induction tomography (MIT) by proposing a direct sampling method that uses a novel class of explicit point spread functions (PSFs) to locate conductive inclusions from magnetic field data collected on a surface. The approach builds an index function solely from inner products with carefully designed PSFs, leveraging a duality product on the measurement surface and an analytic representation of the scattered field. The authors prove the PSFs decay away from inclusions and provide explicit PSF expressions in special cases, enabling fast, non-iterative imaging with offline computation and minimal numerical differentiation. Numerical experiments with multiple synthetic inclusions and up to 20% noise demonstrate accurate localization, robustness, and significant reductions in online computation time, indicating strong potential for rapid MIT imaging in biomedical and nondestructive testing applications.

Abstract

This paper proposes a direct sampling method for the inverse problem of magnetic induction tomography (MIT). Our approach defines a class of point spread functions with explicit expressions, which are computed via inner products, leading to a simple and fast imaging process. We then prove that these point spread functions decay with distance, establishing the theoretical basis of the algorithm. Specific expressions for special cases are also derived to visually demonstrate their attenuation pattern. Numerical experimental results further confirm the efficiency and accuracy of the proposed algorithm.

Figures (8)

• Figure 1: Values of $K_{(\boldsymbol y, \boldsymbol \alpha)}(\cdot, \boldsymbol\beta)$ on cross-section $\{x^2+y^2\le1,\,z=0\}$ with different $\gamma$s. Left: $\gamma=0$. Right: $\gamma=4$. In both cases $\boldsymbol y=(0.3, 0.3, 0.0)$ and $\boldsymbol\alpha=\boldsymbol\beta=(1,0,0)$.
• Figure 2: Graphs of normalized $D_0^{-1/2}$ and $D_2^{-1/2}$. They show the growth rate of $K_{(\boldsymbol{0},\boldsymbol{\alpha})}(\boldsymbol{z},\boldsymbol{\beta})$ with different $\gamma$s, $\gamma=0$ and $\gamma=2$, respectively.
• Figure 3: Placement of the 20 coils at the vertices of a regular dodecahedron.
• Figure 4: Description of the geometry. Left: The driving coil and the local coordinate system $x'y'z'$. Right: Distribution of the current density in the driving coil under local coordinate system.
• Figure 5: Reconstructed medium image of example 1. Index function $\tilde{I}$ and $\tilde{I}^4$ on cross-section $\{x^2+y^2\le 1,\,z=0\}$. Left: $\tilde{I}$ with $\epsilon = 0$. Middle: $\tilde{I}^4$ with $\epsilon=0$. Right: $\tilde{I}^4$ with $\epsilon= 20\%$.

Theorems & Definitions (15)

• Theorem 2.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1