The early detection of terrorist threat objects, such as guns and knives, through improved metal detection, has the potential to reduce the number of attacks and improve public safety and security. To achieve this, there is considerable potential to use the fields applied and measured by a metal detector to discriminate between different shapes and different metals since, hidden within the field perturbation, is object characterisation information. The magnetic polarizability tensor (MPT) offers an economical characterisation of metallic objects and its spectral signature provides additional object characterisation information. The MPT spectral signature can be determined from measurements of the induced voltage over a range frequencies in a metal signature for a hidden object. With classification in mind, it can also be computed in advance for different threat and non-threat objects. In the article, we evaluate the performance of probabilistic and non-probabilistic machine learning algorithms, trained using a dictionary of computed MPT spectral signatures, to classify objects for metal detection. We discuss the importances of using appropriate features and selecting an appropriate algorithm depending on the classification problem being solved and we present numerical results for a range of practically motivated metal detection classification problems.

We present new alternative complete asymptotic expansions for the time harmonic low-frequency magnetic field perturbation caused by the presence of a conducting permeable object as its size tends to zero for the eddy current approximation of the Maxwell system. Our new alternative formulations enable a natural extension of the well known rank 2 magnetic polarizability tensor (MPT) object characterisation to higher order tensor descriptions by introducing generalised MPTs (GMPTs) using multi-indices. In particular, we identify the magnetostatic contribution, provide new results on the symmetries of GMPTs, derive explicit formulae for the real and imaginary parts of GMPT coefficients and also describe the spectral behaviour of GMPT coefficients. We also introduce the concept of harmonic GMPTs (HGMPTs) that have fewer coefficients than other GMPT descriptions of the same order. We describe the scaling, translation and rotational properties of HGMPTs and describe an approach for obtaining those HGMPT coefficients that are invariant under the action of a symmetry group. Such an approach is one candidate for selecting features in object classification for hidden object identification using HGMPTs.

We introduce a new type of object characterisation, which is capable of accurately describing small isolated inclusions for potential field inverse problems such as in electrostatics, magnetostatics and related low frequency Maxwell problems. Relevant applications include characterising ferrous unexploded ordnance (UXO) from magnetostatic field measurements in magnetometry, describing small conducting inclusions for medical imaging using electrical impedance tomography (EIT), performing geological ground surveys using electrical resistivity imaging (ERT), characterising objects by electrosensing fish to navigate and identify food as well as describing the effective properties of dilute composites. Our object characterisation builds on the generalised polarizability tensor (GPT) object characterisation concept and provides an alternative to the compacted GPT (CGPT). We call the new characterisations harmonic GPTs (HGPTs) as their coefficients correspond to products of harmonic polynomials. Then, we show that the number of independent coefficients of HGPTs needed to characterise objects can be significantly reduced by considering the symmetry group of the object and propose a systematic approach for determining the subspace of symmetric harmonic polynomials that is fixed by the group and its dimension. This enable us to determine the independent HGPT coefficients for different symmetry groups.