A direct sampling method for magnetic induction tomography

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.

💡 Research Summary

This paper presents a novel direct sampling method (DSM) for tackling the highly ill-posed inverse problem associated with Magnetic Induction Tomography (MIT). MIT is a non-invasive imaging technique that aims to reconstruct the internal conductivity distribution of an object by measuring perturbations in magnetic fields caused by eddy currents. Traditional approaches to this inverse problem often rely on iterative optimization methods, which require solving the computationally expensive forward problem multiple times, leading to significant time consumption.

The core innovation of this work lies in the design and theoretical analysis of a new class of point spread functions (PSFs). The method operates by defining an index function I(z) over a sampling domain Ω that encloses the unknown conductive region D. For each sampling point z in Ω, the index I(z) is computed as a normalized inner product between the measured secondary magnetic field data H_s on a measurement surface Γ (typically a sphere enclosing the domain) and a specially constructed probing function. This probing function is derived from the fundamental solution of the Laplace equation and involves an auxiliary unit vector β.

The PSF, denoted as K_{y,α}(z,β), is formally defined using a duality product 〈·,·〉γ on the measurement surface Γ. This product incorporates powers of the Laplace-Beltrami operator (-Δ_Γ)^γ, where γ is an even integer parameter controlling the localization properties. The authors rigorously prove (Theorem 3.1) that for a fixed source point y and direction α, the PSF K{y,α}(z,β) decays to zero as the sampling point z approaches the measurement boundary Γ. This decay property ensures that the inner product in the numerator of the index function I(z) will be relatively large when z lies inside or near the true conductive inclusion D, and small when z is far away from it. The denominator serves as a normalization factor. A significant advantage is that the PSFs can be precomputed offline, making the online image reconstruction process extremely fast, involving only the evaluation of inner products.

The paper further provides explicit estimates for the PSF in special cases (e.g., Proposition 3.3 for γ=0), offering intuitive insight into its attenuation behavior, which is dominated by a term of the form 1/(R^2 - |z|^2)^2. The selection of the optimal direction vector β is discussed, with a practical suggestion to test a finite set of candidates and choose the one that maximizes the index function.

Numerical experiments in a three-dimensional setting demonstrate the effectiveness, accuracy, and computational efficiency of the proposed algorithm. Tests include reconstructing single, multiple, and connected conductive inclusions from simulated near-field magnetic data. The results show that the method can successfully identify the location and approximate shape of the inclusions even in the presence of measurement noise, all within a fraction of the time required by conventional iterative methods. The paper concludes that this direct sampling method, grounded in solid mathematical theory and boasting high computational speed, offers a promising alternative for real-time MIT imaging applications.