Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors

Abstract

Realistic applications in metal detection involve multiple inhomogeneousconducting permeable objects, and the aim of this paper is to characterise such objects by polarizability tensors. We show that, for the eddy current model, the leading order terms for the perturbation in the magnetic field, due to the presence of N small conducting permeable homogeneous inclusions, comprises of a sum of N terms with each containing a complex symmetric rank 2 polarizability tensor. Each tensor contains information about the shape and material properties of one of the objects and is independent of its position. The asymptotic expansion we obtain extends a previously known result for a single isolated object and applies in situations where the object sizes are small and the objects are sufficiently well separated. We also obtain a second expansion that describes the perturbed magnetic field for inhomogeneous and closely spaced objects, which again characterises the objects by a complex symmetric rank 2 tensor.The tensor's coefficients can be computed by solving a vector valued transmission problem, and we include numerical examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical prediction. We also include algorithms for the localisation and identification of multiple inhomogeneous objects.

The spectral properties of the magnetic polarizability tensor for metallic object characterisation

“…The behaviour was found to be similar for objects with inhomogeneous permeability; however, for objects with inhomogeneous conductivity, computational results show that the eigenvalues of the real part have multiple nonstationary infection points, and the imaginary part has multiple local maxima. 12 This was also found to be the case in the measurements of the MPTs of US coins, which are made up of different conducting materials. 17 An insight into the spectral behaviour of MPT coefficients is provided by the analytical solution for a conducting permeable sphere obtained by Wait and Spies, 18 who also provide a description for the transient perturbed magnetic field when the excitation is through a step or impulse function.…”

“…If the object is inhomogeneous, with possibly different piecewise-constant values of * and in different regions of the object, then (4) and (5) still hold if we replace B with B = ∪ N n=1 B (n) and B = B + z by B = B + z to describe the fact that B is made up of N regions. 12 We require that B (and B (n) ) have Lipschitz boundaries and note that…”

“…Numerical results for other object shapes can be found in the previous studies. [12][13][14] The matrices of eigenvalues Λ  * and Λ  * are strongly dependent on = 2 * 0 . The limiting behaviour of − 0 =  * + i * = − * +  * for = 0 and → ∞ has already been investigated, and we recall that…”

“…9 We have extended the work of Ammari et al to provide a complete asymptotic expansion for the perturbed magnetic field, which allows an object to be characterised by generalised MPTs of which the rank 2 MPT is the simplest case. 11 In Ledger et al, 12 we have developed asymptotic expansions for the perturbed magnetic field that describe (a) the leading order term for a small inhomogeneous conducting permeable object and (b) the response in the case of multiple small conducting permeable objects, thus, extending the rank 2 MPT description of each of the objects to these cases. Some properties of the MPT are described in Ledger and Lionheart, 13 and the availability of the explicit formula for the isolated single object case has been communicated to the engineering community.…”

“…Some properties of the MPT are described in Ledger and Lionheart, 13 and the availability of the explicit formula for the isolated single object case has been communicated to the engineering community. 14 The measurement of time-harmonic perturbed field data, at a range of frequencies, has already been shown to be beneficial for object location and object identification with several dictionary-based algorithms being proposed for metal detection 8,12,15 and electro-sensing. 16 These benefits are realised because of the frequency dependence of the tensor coefficients.…”