Characterisation of Multiple Conducting Permeable Objects in Metal Detection by Polarizability Tensors

P.D. Ledger, W.R.B. Lionheart and A.A.S. Amad

Zienkiewicz Centre for Computational Engineering, College of Engineering,

Swansea University Bay Campus, Swansea. SA1 8EN

School of Mathematics, Alan Turing Building,

The University of Manchester, Oxford Road, Manchester, M13 9PL

20th September 2018

## Abstract

Realistic applications in metal detection involve multiple inhomogeneous conducting 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 small conducting permeable homogeneous inclusions, comprises of a sum of 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.

MSC: 35R30, 35B30

Keywords: Polarizability Tensors; Asymptotic Expansion; Eddy Currents; Metal Detectors; Land Mine Detection

## 1 Introduction

There is considerable interest in being able to locate and characterise multiple conducting permeable objects from measurements of mutual inductance between a transmitting and a receiving coil, where the coupling is inductive. An obvious example is in metal detection where the goal is to identify and locate the multiple objects present in a low conducting background. Applications include security screening, archaeological digs, ensuring food safety as well as the search for landmines and unexploded ordnance and landmines. Other applications include magnetic induction tomography for medical imaging applications and monitoring of corrosion of steel reinforcement in concrete structures.

In all these practical applications, the need to locate and distinguish between multiple conducting permeable inclusions is common. This includes benign situations, such as coins and keys accidentally left in a pocket during a security search or a treasure hunter becoming lucky and discovering a hoard of Roman coins, as well as threat situations, where the risks need to be clearly identified from the background clutter. For example, in the case of searching for unexploded landmines, the ground can be contaminated by ring-pulls, coins and other metallic shrapnel, which makes the process of clearing them very slow as each metallic object needs to be dug up with care. Furthermore, conducting objects are also often inhomogeneous and made up of several different metals. For instance, the barrel of a gun is invariably steel while the frame could be a lighter alloy, jacketed bullets have a lead shot and a brass jacket and modern coins often consist of a cheaper metal encased in nickel or brass alloy. Thus, in practical metal detection applications, it is important to be able to describe both multiple objects and inhomogeneous objects.

Magnetic polarizability tensors (MPTs) hold considerable promise for the low-cost characterisation in metal detection. An asymptotic expansion describing the perturbed magnetic field due to the presence of a small conducting permeable object has been obtained by Ammari, Chen, Chen, Garnier and Volkov [ammarivolkov2013], which characterises the object in terms of a rank 4 tensor. Ledger and Lionheart have shown that this asymptotic expansion simplifies for orthonormal coordinates and allows a conducting permeable object to be characterised by a complex symmetric rank 2 MPT with an explicit expression for its 6 coefficients [ledgerlionheart2014]. Ledger and Lionheart have also investigated the properties of this tensor [ledgerlionheart2016] and they have written the article [ledgerlionheart2018] to explain these developments to the electrical engineering community as well as to show how it applies in several realistic situations. In [ledgerlionheart2017] they have obtained a complete asymptotic expansion of the magnetic field, which characterises the object in terms of a new class of generalised magnetic polarizability tensors (GMPTs), the rank 2 MPT being the simplest case. The availability of an explicit formula for the MPT’s coefficients, and its improved understanding, allows new algorithms for object location and identification to be designed e.g. [ammarivolkov2013b].

Electrical engineers have applied MPTs to a range of practical metal detection applications, including walk through metal detectors, in line scanners and demining e.g. [barrowes2008, shubitidze2004, shubitdze2007, baumbook, marsh2015, dekdouk, zhao2016], see also our article [ledgerlionheart2018] for a recent review, but without knowledge of the explicit formula described above. Engineers have made a prediction of the form of the response for multiple objects e.g. [braunisch2002], but without an explicit criteria on the size or the distance between the objects in order for the approximation to hold. Grzegorczyk, Barrowes, Shubitidze, Fernández and O’Neill have applied a time domain approach to classify multiple unexploded ordinance using descriptions related to MPTs [Gregorczy]. Davidson, Abel-Rehim, Hu, Marsh, O’Toole and Peyton have made measurements of MPTs for inhomogeneous US coins [davidson2018] and Yin, Li, Withers and Peyton have also made measurements to characterise inhomogeneous aluminium/carbon-fibre reinforced plastic sheets [yin2010]. But, in all cases, without an explicit formula.

Our work has the following novelties: Firstly, we characterise rigidly joined collections of different metals (i.e. metals touching or held in that configuration by a non-conducting material) by MPTs overcoming a deficiency of our previous work. Secondly, we find that the frequency spectra of the eigenvalues of the real and imaginary parts of the MPT for an inhomogeneous object exhibit multiple non-stationary inflection points and maxima, respectively, and the number of these gives an upper bound on the number of materials making up the object. To achieve this, we revisit the asymptotic formula of Ammari et al. [ammarivolkov2013] and our previous work [ledgerlionheart2014] and extend it to treat multiple objects by describing the perturbed magnetic field as a sum of terms involving the MPTs associated with each of the inclusions. We also provide a criteria based on the distance between the objects, which determines the situations in which the expression will hold. We derive a second asymptotic expansion that describes the perturbed magnetic field in the case of inhomogeneous objects and, as a corollary, this also describes the magnetic field perturbation in the case of closely spaced objects. In each case, we provide new explicit formulae for the MPTs. We also present algorithms for the localisation and characterisation of objects, which extends those for the isolated object case [ammarivolkov2013].

The paper is organised as follows: In Section 2, the characterisation of a single homogeneous object is briefly reviewed. Section LABEL:sect:main presents our main results for characterising multiple and inhomogeneous objects by MPTs. Sections LABEL:sect:proof1 and LABEL:sect:proof2 contain the details of the proofs for our main results. In Section LABEL:sect:num, we present numerical results to demonstrate the accuracy of the asymptotic formulae and presents results of algorithms for the localisation and identification of multiple (inhomogeneous) objects.

## 2 Characterisation of a Single Conducting Permeable Object

We begin by recalling known results for the characterisation of a single homogenous conducting permeable object. Following [ammarivolkov2013, ledgerlionheart2014] we describe a single inclusion by , which means that it can be thought of a unit-sized object located at the origin, scaled by and translated by . We assume the background is non-conducting and non-permeable and introduce the position dependent conductivity and permeability as σ*in 0in