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

Abstract

The measurement of time-harmonic perturbed field data, at a range of frequencies, is beneficial for practical metal detection, where the goal is to locate and identify hidden targets. In particular, these benefits are realised when frequency-dependent magnetic polarizability tensors (MPTs) are used to provide an economical characterisation of conducting permeable objects, and a dictionary-based classifier is employed. However, despite the advantages shown in dictionary-based classifiers, the behaviour of the MPT coefficients with frequency is not properly understood. In this paper, we rigorously analyse, for the first time, the spectral properties of the coefficients of the MPT. This analysis has the potential to improve existing algorithms and design new approaches for object location and identification in metal detection. Our analysis also enables the response transient response from a conducting permeable object to be predicted for more general forms of excitation. KEYWORDSasymptotic analysis, eddy current, inverse problems, magnetic polarizability tensor, metal detection, spectral problems MSC CLASSIFICATION 35R30; 35B30 INTRODUCTIONIn metal detection, there is considerable interest in being able to locate and identify conducting permeable objects from the measurements of mutual inductance between a transmitting and a measurement coil. Applications include security screening, archaeology excavations, ensuring food safety as well as the search for land mines, and unexploded ordnance. There are also closely related topics such as magnetic induction tomography for medical imaging and eddy current testing for monitoring the corrosion of steel reinforcement in concrete structures.Within the metal detection community, magnetic polarizability tensors (MPTs) have attracted considerable interest to assist with the identification of objects when the transmitting coil is excited by a sinusoidal signal, eg, 1-7 Engineers believe that a rank 2 MPT provides an economical characterisation of a conducting permeable object that is invariant of position. An asymptotic formula providing the leading order term for the perturbed magnetic field due to the presence of a small conducting permeable object has been obtained by Ammari, Chen, Chen, Garnier, and Volkov, 8 which characterises the object in terms of a rank 4 tensor. We have shown that this simplifies for orthonormal coordinates and allows an object to be characterised by a complex symmetric rank 2 MPT, with an explicit formula for its coefficients, thus, justifying the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Efficient computation of the magnetic polarizabiltiy tensor spectral signature using proper orthogonal decomposition

“…1 Its coefficients are a function of the exciting frequency, the object's size, its shape as well as its conductivity and permeability. Explicit formulas for computing the tensor coefficients have been derived [1][2][3][4] and validated against exact solutions and measurements. 3,5 Also, the way in which the tensor coefficients vary with the exciting frequency is theoretically well understood 4 offering improved object classification.…”

“…One approach for the automated computation of the MPT spectral signature is to postprocess finite element method (FEM) solutions to eddy current problems obtained using commercial packages (e.g., with ANSYS as in Reference 6); however, improved accuracy, and a better understanding, can be gained by using the available explicit expressions for MPT coefficients, which rely on computing finite element (FE) approximations to a transmission problem. 1,3,4 A further alternative would be to use the boundary element method (BEM) to discretize the transmission problem, which only requires discretization of the conductor's surface for a homogenous conductor and hence has fewer degrees of freedom. However, unlike the sparse matrices in FEM, BEM results in fully populated matrices, and, for general inhomogeneous objects, requires discretization of the conductor's volume and coupling with FEM.…”

“…The present article addresses this issue by proposing a reduced order model, in the form of a (projected) proper orthogonal decomposition (POD) scheme, that relies on full order model solutions 1 computed using the established open source FE package, NGSolve, and the recently derived alternative explicit expressions formulas for the MPT coefficients. 4 The use of NGSolve 23,24 ensures that the solutions to underlying (eddy current type) transmission problems are accurately computed using high order H(curl) conforming (high order edge element) discretizations (see References 25-27 and references therein) and the POD technique ensures their rapid computation over sweeps of frequency. This means the MPT coefficients are obtained with higher accuracy than can currently be achieved from measurements (noise appropriate to the system can be added to the library during a classifier training if desired), the spectral signature is accurately computed for a large frequency range (up to the limit of the eddy current model) rather than obtained at a small number of discrete frequencies and, through scripting, it allows a much larger library of objects and variations of materials to be considered, which is all highly desirable for achieving greater fidelity and accuracy when training machine learning classifiers.…”

Combining Electromagnetic Spectroscopy and Ground-Penetrating Radar for the Detection of Anti-Personnel Landmines

“…Metallic objects are known to respond to magnetic fields in a complex manner as a function of their conductivity, permeability, size, and the frequency of the applied field. Figure 2 shows the nature of these dependencies, where the underlying theory behind these spectroscopic trends has been reported in [26]. Through the implementation of MIS, the sensor used in this study can exploit a distinctive spectral response to characterise metallic targets.…”

High-Frequency Electromagnetic Induction (HFEMI) Sensor Results from IED Constituent Parts

“…High-frequency electromagnetic induction (HFEMI) operates in between the LFEMI and GPR frequency ranges. Loosely defined as the frequency regime from 100 kHz to 20 MHz, HFEMI produces long enough wavelengths (λ l) to still satisfy magnetoquasistatic assumptions, i.e., displacement currents are negligible compared to conduction currents ( εω σ 1) [1,[13][14][15][16][17][18][19][20]. Yet the frequency is high enough that less-conducting, or intermediate electrically conducting materials (IECM), targets respond and produce identifiable signatures.…”