Characterizing the shape and material properties of hidden targets from magnetic induction data

Abstract

The aim of this paper is to show that, for the eddy current model, the leading order term for the perturbation in the magnetic field, due to the presence of a small conducting magnetic inclusion, can be expressed in terms of a symmetric rank 2 polarization tensor. This tensor contains information about the shape and material properties of the object and is independent of position. We apply a recently derived asymptotic formula for the perturbed magnetic field, due to the presence of a conducting inclusion, which is expressed in terms of a new class of rank 4 polarization tensors (

Generalised magnetic polarizability tensors

“…In this work, we provide a new complete asymptotic expansion of ( H α − H 0 )( x ) for a highly conducting (possibly permeable and multiply connected) object as α →0, thus extending the result in Ammari et al and Ledger and Lionheart, which provided only the leading order term. We write our result in terms of a new class of (higher rank) generalised MPTs (GMPTs), which characterise the object's shape and its material characteristics.…”

“…The explicit expression for their coefficients is with respect to the standard orthonormal basis rather than the space of harmonic polynomials. They are functions of B , α , σ ∗ , μ ∗ / μ 0 , and ω and can be computed by solving a generalised form of the vectorial transmission problem obtained in Ammari et al and Ledger and Lionheart . Moreover, the leading order term in our new expansion agrees with our previous result, and here, the GMPT agrees with .…”

“…Still further, Ammari and Kang have shown that there are difficulties in separating geometrical information from the material contrast c in and hence the limiting cases of . Finally, if an object has rotational or reflectional symmetries, the number of nonzero independent coefficients in a symmetric rank 2 tensor is greatly reduced making discrimination between objects difficult (eg, the independent nonzero coefficients of for a cylinder and a cone are the same due to rotational and reflectional symmetries that are present in both objects, even though the cylinder has an additional mirror symmetry normal to a rotation axis that is not present in a cone).…”

Classification of materials for conducting spheroids based on the first order polarization tensor

“…In applied mathematics and analysis, previous studies by [1,14,15,16,17,18,19] have shown that the perturbation due to the presence of the conducting objects can be represented by asymptotic formulas and PT is actually the dominant term in the formula. The asymptotic formula and the corresponding PT are generally not the same for different systems.…”

“…Therefore, the PT caused by a conducting object is also refered as the PT for that object. Some examples regarding computation of the PT based on the explicit formula and according to the geometry as well as the material of a conducting object can be found in [17,20,21,22,23]. On the other hand, the PT is sometimes referred as the polarizibility tensor in engineering literatures and can be obtained from field measurements either in laboratory or fieldwork [8,9,10,12,13].…”