arXiv subjects

P. D. Ledger

Publications and source records attributed to P. D. Ledger.

13 recordsLinked to original sources

arXiv2025

Characterising buried objects in metal detection

Current mathematical models for identifying highly conducting buried objects in metal detection assume that the soil is non-conducting and has the same permeability as free space. However, although the electrical conductivity of soil is low, it is not negligible and depends on factors such as soil type and salinity. Moreover, the magnetic permeability of soil varies with its iron content and is often described as uncooperative. Depending on the ground conditions, these soil's properties can influence the induced voltages in the measurement coils of metal detectors and becomes of increasing importance as the frequency of the exciting current source is increased. In this work, we develop a new asymptotic expansion for the perturbed magnetic field due to the presence of a highly conducting magnetic buried object as its size tends to zero, which takes account of the ground conditions. The leading order term of this expansion can be expressed in terms of a complex symmetric rank-2 magnetic polarizability tensor, which characterises the object, can assist in its identification, and we provide conditions under which this tensor's coefficients can be computed independently of the ground conditions. We demonstrate the improved accuracy of our new result, which takes account of the ground conditions, over the situation where the soil's conductivity and permeability are not considered.

math.AP↗

arXiv2025

How far are two symmetric matrices from commuting? With an application to object characterisation and identification in metal detection

Examining the extent to which measurements of rotation matrices are close to each other is challenging due measurement noise. To overcome this, data is typically smoothed and Riemannian and Euclidean metrics are applied. However, if rotation matrices are not directly measured and are instead formed by eigenvectors of measured symmetric matrices, this can be problematic if the associated eigenvalues are close. In this work, we propose novel semi-metrics that can be used to approximate the Riemannian metric for small angles. Our new results do not require eigenvector information and are beneficial for measured datasets. There are also issues when using comparing rotational data arising from computational simulations and it is important that the impact of the approximations on the computed outputs is properly assessed to ensure that the approximations made and the finite precision arithmetic are not unduly polluting the results. In this work, we examine data arising from object characterisation in metal detection using the complex symmetric rank two magnetic polarizability tensor (MPT) description, we rigorously analyse the effects of our numerical approximations and apply our new approximate measures of distance to the commutator of the real and imaginary parts of the MPT to this application. Our new approximate measures of distance provide additional feature information, which is invariant of the object orientation, to aid with object identification using machine learning classifiers. We present Bayesian classification examples to demonstrate the success of our approach.

math.NA↗

arXiv2022

Characterising small objects in the regime between the eddy current model and wave propagation

Being able to characterise objects at low frequencies, but above the limiting frequency of the eddy current approximation of the Maxwell system, is important for improving current metal detection technologies. Importantly, the upper frequency limit of the eddy current model depends on the object topology and on its materials, with the maximum frequency being much smaller for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds for frequencies just beyond the eddy current limit. The leading order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects.

math.AP↗

arXiv2022

Minimal Object Characterisations using Harmonic Generalised Polarizability Tensors and Symmetry Groups

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.

math.NA↗

arXiv2022

Properties of Generalised Magnetic Polarizability Tensors

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.

math.AP↗

arXiv2022

Benchmark computations for the polarization tensor of small conducting objects

The characterisation of small low conducting inclusions in an otherwise uniform background from low-frequency electrical field measurements has important applications in medical imaging using electrical impedance tomography as well as in geological imaging using electrical resistivity tomography. It is known that such objects can be characterised by a Póyla-Szegö (polarizability) tensor. Such characterisations have attracted interest as they can provide object features in a machine learning classification algorithm and provide an alternative imaging solution. However, to be able train machine learning algorithms, large dictionaries are required and it is essential that the characterisations are accurate. In this work, we obtain accurate numerical approximations to the tensor coefficients, by applying an adaptive boundary element method. The goal being to provide a sequence of benchmark {computations} for the tensor coefficients to allow other software developers check the accuracy of their codes.

math.NA↗

arXiv2021

Identification of Metallic Objects using Spectral Magnetic Polarizability Tensor Signatures: Object Classification

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.

cs.LG↗

arXiv2020

Identification of Metallic Objects using Spectral MPT Signatures: Object Characterisation and Invariants

The early detection of terrorist threats, 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 that can be computed for different threat and non-threat objects and has an established theoretical background, which shows that the induced voltage is a function of the hidden object's MPT coefficients. In this paper, we describe the additional characterisation information that measurements of the induced voltage over a range of frequencies offer compared to measurements at a single frequency. We call such object characterisations its MPT spectral signature. Then, we present a series of alternative rotational invariants for the purpose of classifying hidden objects using MPT spectral signatures. Finally, we include examples of computed MPT spectral signature characterisations of realistic threat and non-threat objects that can be used to train machine learning algorithms for classification purposes.

cs.CR↗

arXiv2020

Efficient Computation of the Magnetic Polarizability Tensor Spectral Signature using POD

Our interest lies in the identification of hidden conducting permeable objects from measurements of the perturbed magnetic field in metal detection taken over range of low frequencies. The magnetic polarizability tensor (MPT) provides a characterisation of a conducting permeable object using a small number of coefficients, has explicit formula for their calculation and a well understood frequency behaviour, which we call its spectral signature. However, to compute such signatures, and build a library of them for object classification, requires repeated solution of a direct (full order) problem, which is typically accomplished using a finite element discretisation. To overcome this issue, we propose an efficient reduced order model (ROM) using a proper orthogonal decomposition (POD) for the rapid computation of MPT spectral signatures. Our ROM benefits from output certificates, which give bounds on the accuracy of the predicted outputs with respect to the full order model solutions. To further increase the efficiency of the computation of the MPT spectral signature, we provide scaling results, which enable an immediate calculation of the signature under changes in the object size or conductivity. We illustrate our approach by application to a range of homogenous and inhomogeneous conducting permeable objects.

math.NA↗

arXiv2019

The Spectral Properties of the Magnetic Polarizability Tensor for Metallic Object Characterisation

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.

math.AP↗

arXiv2018

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

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 $N$ 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.

math.AP↗

arXiv2018

Generalised Magnetic Polarizability Tensors

We present a new complete asymptotic expansion for the low frequency time-harmonic magnetic field perturbation caused by the presence of a conducting (permeable) object as its size tends to zero for the eddy current regime of Maxwell's equations. The new asymptotic expansion allows the characterisation of the shape and material properties of such objects by a new class of generalised magnetic polarizability tensors and we provide an explicit formula for their calculation. Our result will have important implications for metal detectors since it will improve small object discrimination and, for situations where the background field varies over the inclusion, this information will be useable, and indeed useful, in identifying their shape and material properties. Thus, improving the ability of metal detectors to locate landmines and unexploded ordnance, sort metals in recycling processes, ensure food safety as well as enhancing security screening at airports and public events.

math.AP↗

arXiv2015

Understanding the Magnetic Polarizability Tensor

The aim of this paper is provide new insights into the properties of the rank 2 polarizability tensor $\check{\check{\mathcal M}}$ proposed in (P.D. Ledger and W.R.B. Lionheart Characterising the shape and material properties of hidden targets from magnetic induction data, IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv015) for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy current regime. In particular, we explore its connection with the magnetic polarizability tensor and the Pólya-Szegö tensor and how, by introducing new splittings of $\check{\check{\mathcal M}}$, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the Pólya-Szegö tensor and expressions for the low frequency and high conductivity limiting coefficients of $\check{\check{\mathcal M}}$. We show, for the high conductivity case (and for frequencies at the limit of the quasi-static approximation), that it is important to consider whether the object is simply or multiply connected but, for the low frequency case, the coefficients are independent of the connectedness of the object. Furthermore, we explore the frequency response of the coefficients of $\check{\check{\mathcal M}}$ for a range of simply and multiply connected objects.

physics.optics↗