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

P.D. Ledger∗, B.A. Wilson†, A.A.S. Amad† and W.R.B. Lionheart‡
∗School of Computing & Mathematics, Keele University
†Zienkiewicz Centre for Computational Engineering, Swansea University
‡Department for Mathematics, The University of Manchester
Corresponding author<EMAIL_ADDRESS>
## Abstract

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.

Keywords: Finite element method; Magnetic polarizability tensor; Machine
learning; Metal detection; Object classification; Reduced order model;
Spectral; Validation.
MSC CLASSIFICATION: 65N30; 65N21; 35R30; 35B30

##  1 Introduction

The purpose of this paper is to provide a demonstration of computing object
characterisations for training machine learning algorithms, with the end goal
being able to classify objects for metal detection. One key application is the
discrimination between threat and non-threat objects. With increased gun and
knife crime in many countries (e.g. U.K. knife crime has seen a significant
increase in the last 8-9 years [17]), the early detection of such weapons has
the potential to reduce the number of related attacks and improve safety and
security. The ability to discriminate between metallic clutter and threat
objects is also important for improving the identification of hidden anti-
personal mines and unexploded ordnance (UXO) in areas of former conflict in
the developing world, offering improvements to the safety and security of
local inhabitants and allowing the land to be returned to normal use. As well
as safety and security, the ability to detect and discriminate between
different hidden metallic objects has important applications in archaeological
searches and treasure hunts, non-destructive testing as well as detecting fake
coins in vending machines and automatic checkouts.

The magnetic polarizability tensor (MPT) has been shown to offer an economical
characterisation of conducting permeable objects and explicit formulae for the
computation of its $6$ independent complex coefficients, which are a function
of the exciting frequency, the object’s size, its shape as well as its
conductivity and permeability, have been obtained [6, 28, 30, 32]. The
behaviour of the MPT’s coefficients as a function of frequency, known as its
spectral signature, has been studied theoretically [32] and an efficient
method for the computation of the spectral signature using a reduced order
model based on proper orthogonal decomposition (POD) has been proposed [55].
These computations employ the established Open Source finite element (FE)
package, NGSolve, and the recently derived alternative explicit expressions
formulae for the MPT coefficients [32]. The use of NGSolve [48, 47] ensures
that the solutions to the underlying full order (eddy current type)
transmission problems are accurately computed using high order
$\bm{H}(\text{curl})$ conforming (high order edge element) discretisations
(see [34, 49, 56] and references therein) and the POD technique ensures their
rapid computation over sweeps of frequency. Included in this approach [55] are
a-posteriori error bounds that can be computed at runtime with negligible
additional cost, which ensure reliability with the respect to the FE solutions
and allow the MPT coefficients obtained with POD to be certified.

With the goal of identifying hidden metallic targets in mind, the MPT spectral
signature has been previously used for simple library classification [7, 33],
a $k$ nearest neighbours (KNN) classification algorithm [35] and other machine
learning approaches [52]. In addition, existing examples of practical MPT
classification of objects include in airport security screening [39, 35],
waste sorting [24] and anti-personal landmine detection [44]. In such
situations, induced voltages are measured over a range of frequencies by a
metal detector from which the MPT spectral signature of the hidden object is
obtained and then a classifier applied [58, 57, 45]. With the exception of [7,
33], all previous studies used measured MPT spectral signature information to
build the classifier. As pointed out in [55], such libraries of measured MPT
coefficients contain unavoidable errors if the object is placed in a non-
uniform background field that varies significantly over the object as well as
other errors and noise associated with capacitive coupling with other low-
conducting objects or soil in the background. There will also be other
generals noise (e.g. from amplifiers, parasitic voltages and filtering) [37].
This means the accuracy is about of measured MPT coefficients is about 1% to
5% [15, 35, 37], depending on the application. It is costly (time consuming)
to produce a large library from measuring coefficients and using such
libraries of measured MPT coefficients can limit the performance of machine
learning classifiers (if they are used on a metal detector which has greater
accuracy than that of MPT coefficients in the library). We will instead use
the newly developed POD approach [55] to build a library for object
classification. This means that 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 classifier

From previous studies of the simpler Póyla-Szegö tensor characterisation of an
object for a fixed conductivity contrast in electrical impedance tomography
(EIT), it is known that shape and material contrast information cannot be
separated [8] and, it is generally accepted, that using a MPT characterisation
at a single frequency also only provides limited information. By studying
nanoparticles and their shape reconstruction from plasmonic resonances, Ammari
et al.  [9] show that such resonances are related to the Neumann-Poincaré
operator, used in the computation of the Póyla-Szegö tensor and generalised
polarization (polarizability) tensors with contrasts $k(\omega)$ being a
function of frequency $\omega$. In addition, for electro-sensing, where it has
been postulated that electric fish characterise objects by Pólya-Szegö tensors
[25, 26], spectral information, with frequency dependent contrasts
$k(\omega)$, have been used for the successful classifications of objects [5].
Although there has been success in using the MPT spectral signature for object
classification [7, 33, 35, 39, 24, 44] and a theoretical study of the MPT
spectral signature [32] has been made, the additional information the spectral
signature provides about the object’s shape and its materials remains open.
Furthermore, in such classifications, an MPT’s eigenvalues are commonly used
as the object’s features, as they are invariant under object rotation,
although there are other possibilities and the benefits of these will be
explored in this work.

The main novelties of this work are, firstly, to establish that an MPT
characterisation at a fixed (limiting) frequency only characterises an object
upto an equivalent ellipsoid, which is not surprising given the known related
result for the simpler Póyla-Szegö tensor [8]. Secondly, that an MPT’s
spectral signature, provides a sequence of different equivalent ellipsoids at
each frequency of excitation where, in general, using any single equivalent
ellipsoid provides a different spectral signature to that of the MPT spectral
signature of the original object. Thirdly, to list, and demonstrate,
appropriate sets of tensor invariants, obtained from MPT coefficients, for the
object features in object classification. Fourth, and finally, to provide a
series of practically motivated MPT spectral signature characterisations
computed for realistic threat and non-threat objects.

The paper is organised as follows: In Section 2, the MPT characterisation of a
conducting permeable object is briefly reviewed. This section also briefly
reviews approaches for the efficient computation of their spectral signature
and measurement of the tensor coefficients in metal detection. Then, in
Section 3, the relationship between an MPT characterisation of an object and
an equivalent ellipsoid at a fixed frequency is discussed along with the added
benefits of using an MPT spectral signature for object characterisation.
Section 4 considers alternative MPT spectral signature invariants that are
invariant to object rotation for the purpose of training a machine learning
algorithm for object classification. In Section 5, we present computational
examples of MPT spectral signature characterisations of realistic exemplar
threat and non-threat objects with the purpose being to build a training data
set for object classification. The paper closes with some concluding remarks.

##  2 Review of the MPT for object characterisation

In this section we briefly recall the economical characterisation of a small
conducting permeable isolated object $B_{\alpha}$ with conductivity
$\sigma_{*}$ and permeability $\mu_{*}$ by an MPT if the background medium is
non-conducting ($\sigma=0\,\text{S/m}$) and has permeability of free space
$\mu_{0}=4\pi\times 10^{-7}\text{H/m}$. As mentioned above, this describes the
situation of metal detection for applications including security screening
e.g., [39, 35], waste sorting e.g. [24] and buried landmines and UXOs e.g.
[44] (where the soil’s conductivity is much lower than that of object so that
it can be neglected). We use the description $B_{\alpha}=\alpha B+{\bm{z}}$,
which means that the object is described by a size parameter $\alpha\ll 1$, a
unit sized object $B$ containing the origin and a translation vector
${\bm{z}}$. Then, using the asymptotic formula obtained by Ammari, Chen, Chen,
Garnier and Volkov [6], Ledger and Lionheart [28] have derived the simplified
form

$\displaystyle(\bm{H}_{\alpha}-\bm{H}_{0})(\bm{x})_{i}=(\bm{D}_{\bm{x}}^{2}G(\bm{x},\bm{z}))_{ij}(\mathcal{M})_{jk}(\bm{H}_{0}(\bm{z}))_{k}+O(\alpha^{4}),$
(1)

for the magnetic field perturbation caused by the presence of the object,
which holds as $\alpha\to 0$ and makes the MPT explicit. In the above,
${\bm{H}}_{\alpha}$ is the magnetic interaction field, ${\bm{H}}_{0}$ is the
background magnetic field (in absence of the object),
$G(\bm{x},\bm{z}):=1/({4\pi|\bm{x}-\bm{z}|})$ is the free space Laplace
Green’s function, $(\bm{D}^{2}_{x}G)_{ij}=\partial_{x_{j}}\partial_{x_{i}}G$
denote the elements of the Hessian of $G$ and Einstein summation convention of
the indices is implied in (1). In addition,
${\mathcal{M}}=({\mathcal{M}})_{jk}{\bm{e}}_{j}\otimes{\bm{e}}_{k}$, where
$\bm{e}_{j}$ denotes the $j$th orthonormal unit vector, is the complex
symmetric rank 2 MPT, which describes the shape and material properties of the
object $B_{\alpha}$ and is frequency dependent, but is independent of the
object’s position $\bm{z}$.

We state below the explicit formulae for the computation of the coefficients
$(\mathcal{M})_{ij}$ of ${\mathcal{M}}$ using the splitting,
$(\mathcal{M})_{ij}:=(\mathcal{N}^{0})_{ij}+(\mathcal{R})_{ij}+\mathrm{i}(\mathcal{I})_{ij}$
[32]

$\displaystyle(\mathcal{N}^{0}[\alpha B,\mu_{r}])_{ij}$
$\displaystyle:=\alpha^{3}\delta_{ij}\int_{B}(1-\mu_{r}^{-1})\mathrm{d}\bm{\xi}+\frac{\alpha^{3}}{4}\int_{B\cup
B^{c}}\tilde{\mu}_{r}^{-1}\nabla\times\tilde{\bm{\theta}}_{i}^{(0)}\cdot\nabla\times\tilde{\bm{\theta}}_{j}^{(0)}\
\mathrm{d}\bm{\xi},$ (2a) $\displaystyle(\mathcal{R}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}$
$\displaystyle:=-\frac{\alpha^{3}}{4}\int_{B\cup
B^{c}}\tilde{\mu}_{r}^{-1}\nabla\times\bm{\theta}_{j}^{(1)}\cdot\nabla\times\overline{\bm{\theta}_{i}^{(1)}}\
\mathrm{d}\bm{\xi},$ (2b) $\displaystyle(\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}$
$\displaystyle:=\frac{\alpha^{3}}{4}\int_{B}\nu\Big{(}\bm{\theta}_{j}^{(1)}+(\tilde{\bm{\theta}}_{j}^{(0)}+\bm{e}_{j}\times\bm{\xi})\Big{)}\cdot\Big{(}\overline{\bm{\theta}_{i}^{(1)}+(\tilde{\bm{\theta}}_{i}^{(0)}+\bm{e}_{i}\times\bm{\xi})}\Big{)}\
\mathrm{d}\bm{\xi},$ (2c)

where $\mathrm{i}:=\sqrt{-1}$, $\omega$ is the angular frequency,
$\nu:=\alpha^{2}\omega\mu_{0}\sigma_{*}$, $\delta_{ij}$ is the Kronecker
delta, ${\bm{\xi}}:=(\xi_{1},\xi_{2},\xi_{3})$ is measured from an origin in
$B$, and, with the exception of the expression below (where it indicates the
closure), the overbar denotes the complex conjugate throughout this work.
Furthermore, the square brackets denote the tensors’ dependence on size,
geometry, materials and exciting frequency. In addition, note that

$\displaystyle\tilde{\mu}_{r}(\bm{\xi}):=\left\\{\begin{array}[]{ll}\mu_{r}:=\mu_{*}/\mu_{0}&\bm{\xi}\in{B}\\\
1&\bm{\xi}\in B^{c}:={\mathbb{R}}^{3}\setminus\overline{B}\end{array}\right..$
(5)

The computation of the tensor coefficients in (2) rely on the real vector
field solution ${\bm{\theta}}_{i}^{(0)}$, $i=1,2,3$, of the transmission
problem [32]

$\displaystyle\nabla\times\tilde{\mu}_{r}^{-1}\nabla\times\bm{\theta}_{i}^{(0)}$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{in }B\cup B^{c},$ (6a)
$\displaystyle\nabla\cdot\bm{\theta}_{i}^{(0)}$ $\displaystyle=0$
$\displaystyle\textrm{in }B\cup B^{c},$ (6b)
$\displaystyle[{\bm{n}}\times\bm{\theta}_{i}^{(0)}]_{\Gamma}$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{on }\Gamma,$ (6c)
$\displaystyle[{\bm{n}}\times\tilde{\mu}_{r}^{-1}\nabla\times\bm{\theta}_{i}^{(0)}]_{\Gamma}$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{on }\Gamma,$ (6d)
$\displaystyle\bm{\theta}_{i}^{(0)}-{\bm{e}}_{i}\times\bm{\xi}$
$\displaystyle=\bm{O}(|\bm{\xi}|^{-1})$ $\displaystyle\textrm{as
}|\bm{\xi}|\rightarrow\infty,$ (6e)

where $\Gamma:=\partial B$ and the complex vector field solution
${\bm{\theta}}_{i}^{(1)}$, $i=1,2,3$, of the transmission problem

$\displaystyle\nabla\times{\mu}_{r}^{-1}\nabla\times\bm{\theta}_{i}^{(1)}-\mathrm{i}\nu(\bm{\theta}_{i}^{(0)}+\bm{\theta}_{i}^{(1)})$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{in }B,$ (7a)
$\displaystyle\nabla\times\nabla\times\bm{\theta}_{i}^{(1)}$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{in }B^{c},$ (7b)
$\displaystyle\nabla\cdot\bm{\theta}_{i}^{(1)}$ $\displaystyle=0$
$\displaystyle\textrm{in }B^{c},$ (7c)
$\displaystyle[{\bm{n}}\times\bm{\theta}_{i}^{(1)}]_{\Gamma}$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{on }\Gamma,$ (7d)
$\displaystyle[{\bm{n}}\times\tilde{\mu}_{r}^{-1}\nabla\times\bm{\theta}_{i}^{(1)}]_{\Gamma}$
$\displaystyle=\bm{0}$ $\displaystyle\textrm{on }\Gamma,$ (7e)
$\displaystyle\bm{\theta}_{i}^{(1)}$ $\displaystyle=\bm{O}(|\bm{\xi}|^{-1})$
$\displaystyle\textrm{as }|\bm{\xi}|\rightarrow\infty.$ (7f)

Note also that we have chosen to introduce
$\tilde{\bm{\theta}}_{i}^{(0)}\vcentcolon=\bm{\theta}_{i}^{(0)}-{\bm{e}}_{i}\times\bm{\xi}$,
which can be shown to satisfy the same transmission problem as (6), except
with a non-zero jump condition for
$[{\bm{n}}\times\tilde{\mu}_{r}^{-1}\nabla\times\tilde{\bm{\theta}}_{i}^{(0)}]_{\Gamma}$
and the decay condition
$\tilde{\bm{\theta}}_{i}^{(0)}(\bm{\xi})=\bm{O}(|\bm{\xi}|^{-1})$ as
$|\bm{\xi}|\rightarrow\infty$.

Furthermore, the rank 2 tensors $\mathcal{N}^{0}[\alpha B,\mu_{r}]$,
$\mathcal{R}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and $\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$, which make up the splitting
$\mathcal{M}=\mathcal{N}^{0}+\mathcal{R}+\mathrm{i}\mathcal{I}$, are real,
symmetric and each have real eigenvalues [32]. They are related to the MPT by

$\displaystyle\text{Re}({\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}])=$
$\displaystyle\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]=\mathcal{N}^{0}[\alpha
B,\mu_{r}]+\mathcal{R}[\alpha B,\omega,\sigma_{*},\mu_{r}],$
$\displaystyle\text{Im}({\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}])=$
$\displaystyle\mathcal{I}[\alpha B,\omega,\sigma_{*},\mu_{r}].$

We call the MPT’s coefficients as a function of $\omega$ its spectral
signature. Note that the above formulation for the MPT and associated
transmission problems is for the case of a single homogenous object $B$, the
extension to multiple inhomogeneous objects can be found in [33, 32].

###  2.1 Efficient computation of the MPT spectral signature

An efficient procedure for computing the MPT spectral signature of a
conducting permeable object has been proposed in [55] and will be used
throughout the work to generate the numerical results. In this approach,
discrete approximations to $\bm{\theta}_{i}^{(1)}(\omega_{n})$, $i=1,2,3$ are
computed accurately at a small number of logarithmically spaced frequencies
$\omega_{n}$, $n=1,\ldots,N$ using the high order finite element solver
NGSolve and a ${\bm{H}}(\text{curl})$ conforming discretisation [49, 48, 56,
47] on unstructured tetrahedral grids. These solutions are called the
representative full order model solution snapshots. A proper orthogonal
decomposition approach using projection (PODP) [21] is then applied to predict
the discrete approximations to $\bm{\theta}_{i}^{(1)}(\omega)$ at other
frequencies $\omega=\omega_{m}$, $m=1,\ldots,M$ with $M\gg N$ and then
resulting MPT coefficients $({\mathcal{M}}^{PODP}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}$, which make up the MPT’s spectral
signature, follow by simple post-processing. The a-posteriori error estimates

$\displaystyle|({\mathcal{R}}^{PODP}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}-({\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}|\leq$ $\displaystyle(\Delta[\omega])_{ij},$
(8a) $\displaystyle|({\mathcal{I}}^{PODP}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}-({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}|\leq$ $\displaystyle(\Delta[\omega])_{ij},$
(8b)

which bound the accuracy of the real and imaginary parts of the PODP MPT
predication with respect to the full order solution, have been derived and can
be computed at low-computational cost during the online stage of the reduced
order model for each $\omega=\omega_{m}$, $m=1,\ldots,M$. They allow the
reduced order predictions to be certified without having compute additional
full order model solutions and, if desired, can be used to adaptively choose
additional frequencies for the representative full order model solution
snapshots so as to improve the accuracy of the PODP prediction of the MPT.
Note that in subsequent results we focus on PODP solutions and drop the PODP
superscript unless confusion may arise. We refer to [55] for details of the
approach.

####  2.1.1 Limiting frequency of the MPT spectral signature

The eddy current model, on which the MPT description is based, is a low
frequency approximation to the full Maxwell system and is commonly accepted to
be valid if the quasi-static assumption applies (dimension $D\approx\alpha$ of
the object $B_{\alpha}$ is small compared to the wavelength) and the
conductivities are high ($\sigma_{max}\gg\omega\epsilon_{max}$, where
$\epsilon_{max}$ denotes the object’s maximum permittivity, here assumed to be
$\epsilon_{max}=\epsilon_{0}\approx 8.854\times 10^{-12}\text{F/m}$). However,
the topology of the object has important role to play in determining the
limiting frequency at which the approximation remains valid. Schmidt, Sterz
and Hiptmair [46] have obtained the following estimates

$\displaystyle C_{1}\epsilon_{max}\mu_{max}\omega^{2}D^{2}$ $\displaystyle\ll
1,$ (9a) $\displaystyle C_{2}\epsilon_{max}\omega\sigma_{min}^{-1}$
$\displaystyle\ll 1,$ (9b)

that are required to hold to ensure the validity of the eddy current model. In
the above, $C_{1}=C_{1}(B)$ and $C_{2}=C_{2}(B)$ are constants that depend on
the object’s topology. In particular, an object with a long thin extension or
with a small gap (eg hoarse-shoe shaped conductor) lead to capacitive coupling
and have large $C_{1}(B)$ and $C_{2}(B)$ limiting the frequency
$\omega_{limit}$ at which the MPT spectral signature remains valid compared to
using the quasi-static and high conductivity conditions alone. Schmidt et al.
have describe a numerical procedure that allows the constants to be estimated
numerically for different objects by solving a low-dimensional eigenvalue
problem. Once the constants have been found, the limiting the frequency
$\omega_{limit}$ can be estimated from (9). We apply this procedure to the
numerical examples in this work.

####  2.1.2 Illustrative example of the MPT spectral signature for an
irregular tetrahedron

To illustrate the method proposed in [55] for computing the MPT spectral
signature, as well the procedure for obtaining $\omega_{limit}$ described in
Section 2.1.1, we consider the case where $B$ is the irregular tetrahedron
with vertices

${\bm{v}}_{1}=\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},\qquad{\bm{v}}_{2}=\begin{pmatrix}7\\\ 0\\\
0\end{pmatrix},\qquad{\bm{v}}_{3}=\begin{pmatrix}5.5\\\ 4.6\\\
0.0\end{pmatrix},\qquad{\bm{v}}_{4}=\begin{pmatrix}3.3\\\ 2.0\\\
5.0\end{pmatrix},$ (10)

size $\alpha=0.01\text{ m}$ and material properties $\mu_{r}=2$ and
$\sigma_{*}=5.96\times 10^{6}\text{ S/m}$. This object has been chosen as it
does not have rotational or reflectional symmetries and so the associated MPT
has $6$ independent complex coefficients at each frequency.

To describe the object and surrounding truncated unbounded region (set as the
$[-100,100]^{3}$ box), a mesh of $21\,427$ unstructured tetrahedra was
generated and elements of order $p=3$ were used so that the representative
full order model solution snapshots (corresponding to discrete approximations
of ${\bm{\theta}}_{i}^{(1)}(\omega_{n})$, $i=1,2,3$), obtained at $N=21$
logarithmically spaced frequencies, $\omega_{n}$, were found to be converged.
The PODP approach was then applied with a tolerance of $TOL=10^{-4}$ to obtain
the approximations to the solutions ${\bm{\theta}}_{i}^{(1)}(\omega)$,
$i=1,2,3$, at other frequencies and, hence, obtain the spectral signature
shown in Figure 1. In this plot, the crosses are used to indicate the
frequencies $\omega_{n}$ used to compute the representative full order model
solution snapshots, the dashed lines show the certification of the output
$(\cdot)\pm(\Delta[\omega])_{ij}$ and vanish to the predicted MPT
coefficients, at all but the highest frequencies, indicating the high accuracy
of the PODP solution with respect to the full order solution. Note that
acceptable PODP solutions can also be obtained with $N=13$ representative full
order model solutions, but $N=21$ was chosen to ensure $(\Delta[\omega])_{ij}$
is small. The plot also includes a vertical dashed line, which indicates the
limiting frequency $\omega_{limit}$. For $\omega>\omega_{limit}$ we expect the
MPT to no longer provide a valid characterisation of the object due to the
eddy current model being invalid.

$\begin{array}[]{cc}\includegraphics[width=231.26378pt]{Fig1RealTensorCoeficientsError.pdf}&\includegraphics[width=231.26378pt]{Fig1ImaginaryTensorCoeficientsError.pdf}\\\
(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}&({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}\end{array}$

Figure 1: Irregular tetrahedron $B$ with vertices as in (10),
$\alpha=0.01\text{ m}$ $\mu_{r}=2$ and $\sigma_{*}=5.96\times 10^{6}\text{
S/m}$: PODP prediction of the spectral signature for
$(\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$ and
$({\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$ including the
certification of the output $(\cdot)\pm(\Delta[\omega])_{ij}$ and the limiting
frequency $\omega_{limit}$.

This figure illustrates the typical characteristic behaviour of the MPT
spectral signature of an object with homogeneous material parameters in which
the diagonal coefficients of $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and $\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ dominate over their off-diagonal coefficients.
The coefficients of $\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$
illustrate the typical sigmoid behaviour with $\log\omega$ and the
coefficients of $\mathcal{I}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ have a
single local maximum with $\log\omega$, which is also well understood
theoretically [32] (See also Section 3.4). Note that behaviour of these
coefficients is different for objects with inhomogeneous materials [33].

###  2.2 Measurement of MPT coefficients

In the electrical engineering community, the measurement of MPT coefficients
as a function of frequency is sometimes called MPT spectroscopy and procedures
have been developed for anti-personal landmine detection [42, 4, 16, 3] as
well as for MPT measurement and real time classification for security
screening [38, 39, 35, 36, 58] and other applications. The basic idea is that
background fields ${\bm{H}}_{0}^{(m)}$, $m=1,\ldots,M_{e}$, from $M_{e}$
different exciters are generated, in turn, and measurements of the resulting
magnetic field perturbation $({\bm{H}}_{\alpha}-{\bm{H}}_{0}^{(m)})({\bm{x}})$
caused by the presence of a hidden conducting permeable object are made at
sufficiently many positions ${\bm{x}}$ around the object for a range of
exciting frequency. Considering a single frequency excitation, the perturbed
magnetic field is usually measured as an induced voltage in the form

${\rm
V}_{nm}^{\text{ind}}=\int_{C^{(n)}}{\bm{n}}\cdot({\bm{H}}_{\alpha}-{\bm{H}}_{0}^{(m)})({\bm{x}})\mathrm{d}{\bm{x}},$
(11)

where $n=1,\ldots,M_{r}$ correspond to the different receiver coils and
$C^{(n)}$ is an appropriate surface related to coil $n$ [30]. In light of (1),
and the MPT object characterisation, we see that the leading order term gives
an approximation to ${\rm V}_{nm}^{\text{ind}}$ and the accuracy of the
approximation will depend on the uniformity of ${\bm{H}}_{0}({\bm{z}})$ as
well as the object size. As explained in Section 1, the measurements ${\rm
V}_{nm}^{\text{ind, measured}}$ will also have unavoidable errors and noise
from a range of sources. Accepting these errors and noise, the task of
determining the MPT of a hidden object separates in to first determining the
position of the object ${\bm{z}}$, which can be done for example using a MUSIC
approach [6, 7] or some other localisation based approach, and secondly
determining the MPT’s coefficients from an over determined linear system of
the form

${\mathbf{A}}{\mathbf{u}}={\mathbf{b}},$ (12)

using least squares where bold Roman fonts are used for matrices and vectors
in linear systems and, for $i=(n-1)M_{e}+m$, $m=1,\ldots,M_{e}$,
$n=1,\ldots,M_{r}$ $j=3(q-1)+r$, $q,r=1,2,3$, they have the entries

$\displaystyle({\mathbf{u}})_{j}=$ $\displaystyle{\rm
u}_{j}=({\mathcal{M}})_{qr},$ $\displaystyle({\mathbf{A}})_{ij}=$
$\displaystyle{\rm
A}_{ij}=({\bm{H}}_{0}^{(m)}({\bm{z}}))_{r}\int_{C^{(n)}}({\bm{D}}_{\bm{x}}^{2}G({\bm{x}},{\bm{z}}){\bm{n}})_{q}\mathrm{d}{\bm{x}},$
$\displaystyle({\mathbf{b}})_{i}=$ $\displaystyle{\rm
b}_{i}=\int_{C^{(n)}}{\bm{n}}\cdot({\bm{H}}_{\alpha}-{\bm{H}}_{0}^{(m)})({\bm{x}})\mathrm{d}{\bm{x}}\approx
V_{nm}^{\text{ind, measured}}.$

In addition, the known symmetry of ${\mathcal{M}}$ can be used to reduce the
number of complex unknowns in (12) from $9$ to $6$. Clearly, we need the
product $M_{e}M_{r}>6$ in order for the system to be over determined, however,
it is not only important that we have sufficiently many measurements, but also
that the location of emitting and receiving coils are correctly chosen so that
all of the coefficients of ${\mathcal{M}}$ can be determined. For further
details, and an algorithm of how this can be automated, see [30]. The
discrete, measured spectral signature of ${\mathcal{M}}$ follows by repeating
the above process using ${\rm V}_{mn}^{\text{ind, measured}}$ at different
excitation frequencies $\omega_{m}$, $m=1,\ldots,M$ and depending on the
system and application errors of around 1-5% can be expected with current
systems [15, 35, 37]. Note that using a higher order expansion of (1), derived
in [31], will lead to an improved approximation of ${\rm V}_{nm}^{\text{ind}}$
in (11), particularly if ${\bm{H}}_{0}({\bm{z}})$ is non-uniform, and allow
the coefficients of generalised MPTs to be found.

##  3 MPTs and equivalent ellipsoids

In this section, we examine the extent to which a MPT uniquely characterises
an object. First, we consider eigenvalue decompositions of the real and
imaginary parts of the MPT for a fixed frequency and, secondly, for spectral
data. Thirdly, we prove how equivalent ellipsoids can be constructed at
limiting frequencies and discuss the construction of equivalent ellipsoids at
other frequencies.

###  3.1 Eigenvalue decomposition at fixed frequency

Consider the characterisation of an object $B_{\alpha}=\alpha B$ by an MPT
(recall the characterisation is independent of ${\bm{z}}$), which can be
expressed by the splitting

$\displaystyle\mathcal{M}[\alpha B,\omega,\sigma_{*},\mu_{r}]=$
$\displaystyle\mathcal{N}^{0}[\alpha B,\mu_{r}]+\mathcal{R}[\alpha
B,\omega,\sigma_{*},\mu_{r}]+\mathrm{i}\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}],$ $\displaystyle=$
$\displaystyle\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]+\mathrm{i}\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}],$ (13)

where our interest in this section is for a fixed frequency
$0\leq\omega=\Omega<\infty$. For simplicity in this section, we assume that
the parameter dependent coefficients of the rank 2 tensors are arranged as
$3\times 3$ matrices and use the same notation for both. Thus, the associated
matrices $\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ are symmetric,
${\mathcal{N}}^{0}[\alpha B,\mu_{r}]$ is positive definite if $\mu_{r}>1$,
${\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ is negative definite and
${\mathcal{I}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ is positive definite and
they can be diagonalised as follows

$\displaystyle\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]=$
$\displaystyle{\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}{\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}({\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]})^{T},$ (14a) $\displaystyle{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]=$
$\displaystyle{\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}{\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}({\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]})^{T},$ (14b)

where ${\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]}$
is an orthogonal matrix whose columns are the eigenvectors of
$\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ and
${\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]}$
is a diagonal matrix whose diagonal entries are the eigenvalues of
$\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ and $T$ denotes the
transpose. The matrices ${\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}$ and ${\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}$ contain the eigenvectors and eigenvalues of
${\mathcal{I}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$, respectively.
Furthermore, if the object has reflectional or rotational symmetries, the
number of independent coefficients in $\tilde{\mathcal{R}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]$ are reduced. In the case that
$\tilde{\mathcal{R}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\Omega,\sigma_{*},\mu_{r}]$ have at most $3$
independent coefficients, this reduction means that
${\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}={\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]}={\mathbf{Q}}(B)$ where we emphasise that
${\mathbf{Q}}$ only depends on $B$. Moreover, when $\tilde{\mathcal{R}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\Omega,\sigma_{*},\mu_{r}]$ are diagonal, due to canonical choice of $B$ and
the object’s reflectional and rotational symmetries [28], then
${\mathbf{Q}}={\bm{\mathbb{I}}}$ is the identity matrix.

###  3.2 Eigenvalue decomposition in the spectral case

We now consider the case of the characterisation of an object by an MPT with
varying frequency $\omega$ so that the real and imaginary parts of an MPT
expressed by (13) are available at discrete frequencies
$0\leq\omega_{m}<\infty$, $m=1\ldots,M$. A similar decomposition to (14) again
applies, except ${\mathbf{Q}}^{\tilde{\mathcal{R}}}$,
${\mathbf{\Lambda}}^{\tilde{\mathcal{R}}}$, ${\mathbf{Q}}^{{\mathcal{I}}}$ and
${\mathbf{\Lambda}}^{{\mathcal{I}}}$ are functions of $\omega_{m}$. If $B$ has
reflectional or rotational symmetries,
${\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]}={\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]}={\mathbf{Q}}(B)$ and, in the limiting case
where $\tilde{\mathcal{R}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$ are diagonal,
${\mathbf{Q}}={\mathbb{I}}$ is the identity matrix. If
${\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]}={\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]}={\mathbf{Q}}(B)$ , the only dependence of
the MPT’s coefficients on $\alpha$, $\omega_{m}$, $\mu_{r}$ and $\sigma_{*}$
is through ${\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]}$ and
${\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]}$.

###  3.3 Equivalent Ellipsoids

For an ellipsoidal object $E_{\alpha}=\alpha E$ of size $\alpha$ with material
parameters $\mu_{r}$, $\sigma_{*}$ and aligned with coordinate axes such that
$E$ is defined by

$\displaystyle\left(\frac{\xi_{1}}{a}\right)^{2}+\left(\frac{\xi_{2}}{b}\right)^{2}+\left(\frac{\xi_{3}}{c}\right)^{2}=1,$
(15)

with $a\geq b\geq b\geq c$ then, for a fixed frequency $\omega=\Omega$, its
MPT $\mathcal{M}[\alpha E,\Omega,\sigma_{*},\mu_{r}]$, as well as its real and
imaginary parts, are diagonal. Similarly, in the spectral case, the MPTs
$\mathcal{M}[\alpha E,\omega_{m},\sigma_{*},\mu_{r}]$, $m=1,\ldots,M$ are
diagonal.

We now show that for the cases of $\omega=0$ or $\omega\to\infty$ that
equivalent ellipsoids $E(0)$ and $E(\infty)$ can be found, which have the same
MPT as $\mathcal{M}[\alpha B,0,\sigma_{*},\mu_{r}]$ and $\mathcal{M}[\alpha
B,\infty,\sigma_{*},\mu_{r}]$, respectively, and comment on the construction
of ellipsoids for other fixed frequencies and the spectral case.

####  3.3.1 Equivalent ellipsoid $E(0)$

For the limiting case of $\omega=0$ it is known that [32]

${\mathcal{M}}[\alpha B,\omega=0,\sigma_{*},\mu_{r}]={\mathcal{N}}^{0}[\alpha
B,\mu_{r}]={\mathcal{T}}[\alpha B,\mu_{r}],$ (16)

where ${\mathcal{T}}[\alpha B,k]=({\mathcal{T}}[\alpha
B,k])_{ij}{\bm{e}}_{i}\otimes{\bm{e}}_{j}$ is the Pólya-Szegö tensor [8]. This
tensor is simpler than the MPT and characterises small homogeneous conducting
objects with shape $B$ in electrical impedance tomography
($k=\sigma_{*}/\sigma_{0}$) and small permeable homogeneous objects with shape
$B$ in magnetostatics ($k=\mu_{r}=\mu_{*}/\mu_{0})$, it is symmetric and is
positive (negative) definite provided that the contrast $k>1$ ($0\leq k<1$)
[8]. Given $\alpha$ and $\mu_{r}$, we wish to show there is an equivalent
ellipsoid $E(0)$ such that

$\displaystyle{\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha
B,0,\sigma_{*},\mu_{r}]}=$ $\displaystyle\tilde{\mathcal{R}}[\alpha
E(0),0,\sigma_{*},\mu_{r}]={\mathcal{T}}[\alpha E(0),\mu_{r}],$
$\displaystyle{\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha
B,0,\sigma_{*},\mu_{r}]}=$ $\displaystyle{\mathcal{I}}[\alpha
E(0),0,\sigma_{*},\mu_{r}]=0,$

holds and to determine its radii $a$, $b$ and $c$. Note that the second
expression does not provide any additional information and is automatically
satisfied.

We recall that for the ellipsoid defined by (15) an analytical expression is
available for ${\mathcal{T}}[\alpha E,k]$ in the form

${\mathcal{T}}[\alpha
E,k]=\alpha^{3}(k-1)|E|\left(\begin{array}[]{ccc}\frac{1}{1-A_{1}+kA_{1}}&0&0\\\
0&\frac{1}{1-A_{2}+kA_{2}}&0\\\
0&0&\frac{1}{1-A_{3}+kA_{3}}\end{array}\right),$ (17)

where $|E|:=\frac{4}{3}\pi abc$ and $A_{1}$, $A_{2}$, $A_{3}$ are the
elliptical integrals

$\displaystyle A_{1}:=$
$\displaystyle\frac{bc}{a^{2}}\displaystyle\int_{1}^{\infty}\frac{1}{t^{2}\sqrt{t^{2}-1+\left(\frac{b}{a}\right)^{2}}\sqrt{t^{2}-1+\left(\frac{c}{a}\right)^{2}}}\mathrm{d}t,$
$\displaystyle A_{2}:=$
$\displaystyle\frac{bc}{a^{2}}\displaystyle\int_{1}^{\infty}\frac{1}{\left(t^{2}-1+\left(\frac{b}{a}\right)^{2}\right)^{3/2}\sqrt{t^{2}-1+\left(\frac{c}{a}\right)^{2}}}\mathrm{d}t,$
$\displaystyle A_{3}:=$
$\displaystyle\frac{bc}{a^{2}}\displaystyle\int_{1}^{\infty}\frac{1}{\sqrt{t^{2}-1+\left(\frac{b}{a}\right)^{2}}\left(t^{2}-1+\left(\frac{c}{a}\right)^{2}\right)^{3/2}}\mathrm{d}t,$

that are a function of $a,b,c$ [8]. These integrals can also be shown to be
equivalent to the alternative expressions in terms of incomplete elliptic
integrals given by Osborn [43], which can be computed using standard
libraries.

Given that we know $A_{1}+A_{2}+A_{3}=1$ from [43], then, if $\alpha$ and
$\mu_{r}$ are known (or at least if
$\frac{1}{\alpha^{3}}\left(1+\frac{3}{\mu_{r}-1}\right)$ is known), we can
determine $|E(0)|$ from

$\displaystyle|E(0)|=\frac{1}{\alpha^{3}L}\left(1+\frac{3}{\mu_{r}-1}\right),\qquad
L=\sum_{i=1}^{3}\frac{1}{({\mathbf{\Lambda}}^{{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]})_{ii}},$ (18)

using the eigenvalues of ${\mathcal{N}}^{0}[\alpha B,\mu_{r}]$. As $|E(0)|$
represents the volume of the unit sized equivalent ellipsoid, we expect
$|E(0)|=O(1)$ as $\alpha\to 0$. Once $|E(0)|$ is known, then we can determine

$\displaystyle
A_{i}=\frac{\alpha^{3}|E(0)|}{({\mathbf{\Lambda}}^{{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]})_{ii}}-\frac{1}{\mu_{r}-1},\qquad i=1,2,3,$ (19)

and compute $A_{2}/A_{1}$ and $A_{3}/A_{1}$. Figure 2, which was obtained by
numerical evaluation of the elliptic integrals in MATLAB, shows that
$(A_{2}/A_{1},A_{3}/A_{1})\to(b/a,c/a)$ is injective and shows the range of
the map (numerical evaluation of Osborn’s alternative expressions in terms of
incomplete elliptic integrals gives the same result). Thus, given
$(A_{2}/A_{1},A_{3}/A_{1})$, we know the solution $(b/a,c/a)$ exists and is
unique. Hence, we can determine $(b/a,c/a)$ and find a unique equivalent
ellipsoid defined by $a$, $b$ and $c$ using $|E(0)|=\frac{4}{3}\pi abc$. In
addition, by ordering $({\mathbf{\Lambda}}^{{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]})_{11}\geq({\mathbf{\Lambda}}^{{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]})_{22}\geq({\mathbf{\Lambda}}^{{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]})_{33}$, we will obtain $A_{3}\geq A_{2}\geq A_{1}$. Since
$0<A_{i}<1$ then, $A_{3}/A_{1}\geq A_{2}/A_{1}$ and, hence from Figure 2, we
find $b/a\geq c/a$. Thus, $a\geq b\geq c$ defines the equivalent ellipsoid.
This means that ${\mathcal{M}}[\alpha B,\omega=0,\sigma_{*},\mu_{r}]$ does not
provide a unique object characterisation as there is an equivalent ellipsoid
that has the same MPT.

Figure 2: Contours of $A_{2}/A_{1}$ and $A_{3}/A_{1}$ as function of $b/a$ and
$c/a$.

Note that a numerical approach to finding the equivalent ellipsoid has been
previously proposed by [27] and involves solving the minimisation problem

$\min_{\mathbf{u}}\left|\sum_{i=1}^{3}({\mathbf{\Lambda}}^{{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]})_{ii}-({\mathcal{T}}[\alpha
E(0,{\mathbf{u}}),\mu_{r}])_{ii}\right|^{2},$ (20)

for ${\mathbf{u}}:=(a,b,c)$ so as to find the equivalent ellipsoid $E(0)$.
Since $a>0$, $b>0$ and $c>0$ it can be shown that the associated non-linear
system is continuous with respect to the unknown variables, hence the
approximate solution of this non-linear system is well-posed [27], but it was
unclear if the solution $a$, $b$ and $c$ exists and is unique. Our proof and
alternative approach above overcomes this difficulty. In practice, applying
(20) gives the same result as our procedure for the examples presented in
Section 3.3.4.

####  3.3.2 Equivalent ellipsoid $E(\infty)$

For the case of $\sigma_{*}\to\infty$ it is known that [32]

$\lim_{\sigma_{*}\to\infty}{\mathcal{M}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]={\mathcal{M}}^{\infty}[\alpha
B]\equiv\lim_{\omega\to\infty}{\mathcal{M}}[\alpha
B,\omega,\sigma_{*},\mu_{r}],$ (21)

where, importantly, the latter equivalence must be viewed with care as
$\omega\to\infty$ would violate the eddy current assumption and, instead, this
limit should be viewed as the limiting frequency for which the eddy current
model is valid. If the topology of $B$ is such that its Betti number
$\beta_{1}(B)=0$ then [32]

$\lim_{\sigma_{*}\to\infty}{\mathcal{M}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]={\mathcal{M}}^{\infty}[\alpha
B]={\mathcal{T}}[\alpha B,0]\equiv\lim_{\omega\to\infty}{\mathcal{M}}[\alpha
B,\omega,\sigma_{*},\mu_{r}],$ (22)

and the unique equivalent ellipsoid $E(\infty)$ can also be found analogously
to the approach in Section 3.3.1. Similarly, we find that
$\lim_{\omega\to\infty}{\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ does
not provide a unique object characterisation. For the definition of Betti
numbers and their implications for MPTs see [29] and references therein.

####  3.3.3 Equivalent ellipsoids $E(\Omega)$ and $E(\omega_{m})$

For the non-limiting fixed frequency case $\omega=\Omega$, $\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ is non-vanishing and we can no longer express
${\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha B,0,\sigma_{*},\mu_{r}]}$ in
terms ${\mathcal{T}}[\alpha E,k]$. However, a semi-analytical solution is
available for the MPT of an ellipsoid [19, 12, 11], which would allow an
analogous numerical procedure to (20) to be applied to find an equivalent
ellipsoid. But, given the non-explicit nature of this solution, it is not
possible to show existence or uniqueness in this case. Nonetheless, we
conjecture that for $\beta_{1}(B)=0$ such an equivalent ellipsoid exists so
that the MPT characterisation at a fixed frequency does not uniquely
characterise the object’s shape and materials. To be able do this alot more
data would be required.

If this is extended to the spectral case then, as results for $E(0)$ and
$E(\infty)$ illustrate, there is no longer a single equivalent ellipsoid, but
instead a sequence of equivalent ellipsoids $E(\omega_{m})$, $m=1,\ldots,M$,
which have different dimensions and, consequently, the MPT spectral signature
constructed from an equivalent ellipsoid at one frequency does not match the
object’s MPT eigenvalue spectral signature. To illustrate this, we consider
the practical application to the earlier irregular tetrahedron described in
Section 2.1.2.

####  3.3.4 Equivalent ellipsoid for an irregular tetrahedron

To illustrate that the spectral signature of the MPT for an object contains
richer information than the spectral signature of an equivalent ellipsoid
obtained at a fixed frequency, we compare ${\mathcal{M}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{M}}[\alpha
E(0),\omega,\sigma_{*},\mu_{r}]$ in Figure 3 using method summarised in
Section 2.1 for the case where $B$ is the irregular tetrahedron as described
in Section 2.1.1. The resulting equivalent ellipsoid $E(0)$ has $a=1.4426$,
$b=1.8797$ and $c=2.4243$ (to 4dp). We observe that the eigenvalues of the
real and imaginary parts of ${\mathcal{M}}[\alpha
E(0),\omega,\sigma_{*},\mu_{r}]$ agrees well with those of
${\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ for small $\omega$, but
the spectral signature differs for large $\omega$.

$\begin{array}[]{cc}\includegraphics[width=231.26378pt,keepaspectratio]{Tet_E0_Real.pdf}&\includegraphics[width=231.26378pt,keepaspectratio]{Tet_E0_Imag.pdf}\\\
\lambda_{i}(\tilde{\mathcal{R}})&\lambda_{i}({\mathcal{I}})\end{array}$

Figure 3: Irregular tetrahedron $B$ with vertices as stated in (10),
$\alpha=0.01\text{ m}$, $\mu_{r}=2$ and $\sigma_{*}=5.96\times 10^{6}\text{
S/m}$. Comparison of $\lambda_{i}(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])$ and $\lambda_{i}(\tilde{\mathcal{R}}[\alpha
E(0),\omega,\sigma_{*},\mu_{r}])$ as well as $\lambda_{i}({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])$ and $\lambda_{i}({\mathcal{I}}[\alpha
E(0),\omega,\sigma_{*},\mu_{r}])$ using an equivalent ellipsoid $E(0)$.

The corresponding results comparing ${\mathcal{M}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{M}}[\alpha
E(\infty),\omega,\sigma_{*},\mu_{r}]$ are shown in Figure 4, where the
equivalent ellipsoid $E(\infty)$ is defined by $a=1.3693$, $b=1.9090$ and
$c=2.9404$ (to 4dp). We observe that the eigenvalues of the real and imaginary
parts of ${\mathcal{M}}[\alpha E(\infty),\omega,\sigma_{*},\mu_{r}]$ agree
well with those of ${\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ for
small and large $\omega$, but the spectral signature differs considerably for
other $\omega$.

$\begin{array}[]{cc}\includegraphics[width=231.26378pt,keepaspectratio]{Tet_E8_Real.pdf}&\includegraphics[width=231.26378pt,keepaspectratio]{Tet_E8_Imag.pdf}\\\
\lambda_{i}(\tilde{\mathcal{R}})&\lambda_{i}({\mathcal{I}})\end{array}$

Figure 4: Irregular tetrahedron $B$ with vertices as stated in (10),
$\alpha=0.01\text{ m}$ $\mu_{r}=2$ and $\sigma_{*}=5.96\times 10^{6}\text{
S/m}$: Comparison of $\lambda_{i}(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])$ and $\lambda_{i}(\tilde{\mathcal{R}}[\alpha
E(\infty),\omega,\sigma_{*},\mu_{r}])$ as well as
$\lambda_{i}({\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}])$ and
$\lambda_{i}({\mathcal{I}}[\alpha E(\infty),\omega,\sigma_{*},\mu_{r}])$ using
an equivalent ellipsoid $E(\infty)$.

These results motivate the advantages of using spectral MPT signature over
using MPT information from a single frequency, since, rather than a single
equivalent ellipsoid, there are multiple equivalent ellipsoids corresponding
to different $\omega$ providing greater information for the classification.

###  3.4 Understanding the spectral signature of MPTs

Building on the above, we now provide a theoretical justification as to why an
object’s MPT spectral signature offers an improved object characterisation
over an MPT obtained at a single frequency. Recalling that
$\nu:=\alpha^{2}\omega\mu_{0}\sigma_{*}$, then for an object $B$, the
following spectral representations have been derived [32] for the tensor
coefficients

$\displaystyle({\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$
$\displaystyle=\frac{\alpha^{3}}{4}\sum_{n=1}^{\infty}\text{Re}(\beta_{n}^{B}(\nu))\lambda_{n}^{B}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}_{i}^{(0)}\right>_{L^{2}(B)}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}_{j}^{(0)}\right>_{L^{2}(B)},$
(23a) $\displaystyle({\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$
$\displaystyle=\frac{\alpha^{3}}{4}\sum_{n=1}^{\infty}\text{Im}(\beta_{n}^{B}(\nu))\lambda_{n}^{B}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}_{i}^{(0)}\right>_{L^{2}(B)}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}_{j}^{(0)}\right>_{L^{2}(B)},$
(23b)

where the notation
$\left<{\bm{u}},{\bm{v}}\right>_{L^{2}(B)}:=\int_{B}{\bm{u}}\cdot\overline{\bm{v}}\mathrm{d}{\bm{\xi}}$
denotes the $L^{2}$ inner product over $B$ and

$\beta_{n}^{B}(\nu)=\text{Re}(\beta_{n}^{B}(\nu)+\mathrm{i}\text{Im}(\beta_{n}^{B}(\nu))=-\frac{\nu^{2}}{\nu^{2}+(\lambda_{n}^{B})^{2}}+\mathrm{i}\frac{\nu\lambda_{n}^{B}}{\nu^{2}+(\lambda_{n}^{B})^{2}},$
(24)

are functions whose real part is sigmoid with $\log\nu$ (or $\nu$) and whose
imaginary part has a single local maximum with $\log\nu$ (or $\nu$). In the
above, $(\lambda_{n}^{B},{\bm{\phi}}_{n}^{B})$ are real eigen–solution pairs
for the problem

$\displaystyle\nabla\times\mu_{r}^{-1}\nabla\times{\bm{\phi}}_{n}^{B}$
$\displaystyle=\lambda_{n}{\bm{\phi}}_{n}^{B}$ in $B$ (25a)
$\displaystyle\nabla\times\nabla\times{\bm{\phi}}_{n}^{B}$
$\displaystyle={\bm{0}}$ in $B^{c}$ (25b)
$\displaystyle[{\bm{n}}\times{\bm{\phi}}_{n}^{B}]$ $\displaystyle={\bm{0}}$ on
$\partial B$ (25c)
$\displaystyle[{\bm{n}}\times\mu_{r}\nabla\times{\bm{\phi}}_{n}^{B}]$
$\displaystyle={\bm{0}}$ on $\partial B$ (25d)
$\displaystyle{\bm{\phi}}_{n}^{B}$ $\displaystyle=O(|{\bm{\xi}}|^{-1}|)$
$\displaystyle\text{as $|{\bm{\xi}}|\to\infty$}.$ (25e)

and the real weights
$\lambda_{n}^{B}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}_{i}^{(0)}\right>_{L^{2}(B)}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}_{j}^{(0)}\right>_{L^{2}(B)}$
together with the complex functions $\beta_{n}^{B}(\nu)$ ultimately determine
the shape of an MPT’s spectral signature. For small $\omega$, the shape of the
MPT spectral signature is well captured by a single dominant mode in the
summations (23) so that, if the dominant mode is the first mode,

$\displaystyle({\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$
$\displaystyle\approx\frac{\alpha^{3}}{4}\text{Re}(\beta_{1}^{B}(\nu))\lambda_{1}^{B}\left<{\bm{\phi}}_{1}^{B},{\bm{\theta}}_{i}^{(0)}\right>_{L^{2}(B)}\left<{\bm{\phi}}_{1}^{B},{\bm{\theta}}_{j}^{(0)}\right>_{L^{2}(B)},$
(26a) $\displaystyle({\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$
$\displaystyle\approx\frac{\alpha^{3}}{4}\text{Im}(\beta_{1}^{B}(\nu))\lambda_{1}^{B}\left<{\bm{\phi}}_{1}^{B},{\bm{\theta}}_{i}^{(0)}\right>_{L^{2}(B)}\left<{\bm{\phi}}_{1}^{B},{\bm{\theta}}_{j}^{(0)}\right>_{L^{2}(B)},$
(26b)

which makes the role of $\text{Re}(\beta_{1}^{B}(\nu))$
$\text{Im}(\beta_{1}^{B}(\nu))$ explicit in determining the shape of the MPT
frequency spectral signature. For further details see [32].

Similar representations to (23) can be obtained for $({\mathcal{R}}[\alpha
E,\omega,\sigma_{*},\mu_{r}])_{ij}$ and $({\mathcal{I}}[\alpha
E,\omega,\sigma_{*},\mu_{r},])_{ij}$ for an ellipsoidal object
$E_{\alpha}=\alpha E$, which is aligned with the coordinate axes such that the
associated tensor is diagonal. The non-zero coefficients for $i=j$ are
expressed in terms of real eigen–solutions
$(\lambda_{n}^{E},{\bm{\phi}}_{n}^{E})$ where

$\displaystyle\nabla\times\mu_{r}^{-1}\nabla\times{\bm{\phi}}_{n}^{E}$
$\displaystyle=\lambda_{n}{\bm{\phi}}_{n}^{E}$ $\displaystyle\text{in $E$},$
(27a) $\displaystyle\nabla\times\nabla\times{\bm{\phi}}_{n}^{E}$
$\displaystyle={\bm{0}}$ $\displaystyle\text{in $E^{c}$},$ (27b)
$\displaystyle[{\bm{n}}\times{\bm{\phi}}_{n}^{E}]$ $\displaystyle={\bm{0}}$
$\displaystyle\text{on $\partial E$},$ (27c)
$\displaystyle[{\bm{n}}\times\mu_{r}\nabla\times{\bm{\phi}}_{n}^{E}]$
$\displaystyle={\bm{0}}$ $\displaystyle\text{on $\partial E$},$ (27d)
$\displaystyle{\bm{\phi}}_{n}^{E}$ $\displaystyle=O(|{\bm{\xi}}|^{-1}|)$
$\displaystyle\text{as $|{\bm{\xi}}|\to\infty$}.$ (27e)

For simplicity, we consider objects $B$ where the MPT is diagonal such that
the eigenvalues of ${\mathcal{N}^{0}},{\mathcal{R}}$ and ${\mathcal{I}}$
coincides with the corresponding non-zero tensor coefficients in the following
two subsections.

####  3.4.1 Equivalent ellipsoids at $\omega=0$

Comparing ${\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{M}}[\alpha E(0),\omega,\sigma_{*},\mu_{r}]$ we have, for an
equivalent ellipsoid constructed at $\omega=0$, that

${\mathcal{M}}[\alpha B,\omega=0,\sigma_{*},\mu_{r}]={\mathcal{N}}^{0}[\alpha
B,\mu_{r}]={\mathcal{N}}^{0}[\alpha E(0),\mu_{r}]={\mathcal{M}}[\alpha
E(0),\omega=0,\sigma_{*},\mu_{r}],$ (28)

but, there is no equivalence between the eigensolutions
$(\lambda_{n}^{B},{\bm{\phi}}_{n}^{B})$ and
$(\lambda_{n}^{E},{\bm{\phi}}_{n}^{E})$ and, hence, we expect
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$,
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ to be different from
$\tilde{\mathcal{R}}[\alpha E(0),\omega,\sigma_{*},\mu_{r}]$,
${\mathcal{I}}[\alpha E(0),\omega,\sigma_{*},\mu_{r}]$ away from $\omega=0$.

####  3.4.2 Equivalent ellipsoids as $\omega\to\infty$

Comparing ${\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{M}}[\alpha E(\infty),\omega,\sigma_{*},\mu_{r}]$ we have, for an
equivalent ellipsoid constructed for $\omega\to\infty$, that

$\displaystyle{\mathcal{M}}^{\infty}[\alpha B]=$
$\displaystyle{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]+\lim_{\omega\to\infty}({\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]+\mathrm{i}{\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])$ $\displaystyle=$
$\displaystyle{\mathcal{N}}^{0}[\alpha
B,\mu_{r}]+\lim_{\omega\to\infty}{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ $\displaystyle=$
$\displaystyle{\mathcal{N}}^{0}[\alpha
E(\infty),\mu_{r}]+\lim_{\omega\to\infty}{\mathcal{R}}[\alpha
E(\infty),\omega,\sigma_{*},\mu_{r}]={\mathcal{M}}^{\infty}[\alpha
E(\infty)].$ (29)

In addition,

$\displaystyle\lim_{\omega\to\infty}({\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}=-\frac{\alpha^{3}}{4}\sum_{n=1}^{\infty}\lambda_{n}^{B}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(B)}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}^{(0)}_{j}\right>_{L^{2}(B)},$
$\displaystyle\lim_{\omega\to\infty}({\mathcal{R}}[\alpha
E(\infty),\omega,\sigma_{*},\mu_{r}])_{ij}=-\frac{\alpha^{3}}{4}\sum_{n=1}^{\infty}\lambda_{n}^{E}\left<{\bm{\phi}}_{n}^{E},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(E(\infty))}\left<{\bm{\phi}}_{n}^{E},{\bm{\theta}}^{(0)}_{j}\right>_{L^{2}(E(\infty))}.$

##### Case of $\mu_{r}=1$

If $\mu_{r}=1$ then $({\mathcal{N}}^{0}[\alpha
B,\mu_{r}])_{ij}=({\mathcal{N}}^{0}[\alpha E(\infty),\mu_{r}])_{ij}=0$ and,
thus, for $i=j$ we have

$\sum_{n=1}^{\infty}\lambda_{n}^{B}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(B)}^{2}=\sum_{n=1}^{\infty}\lambda_{n}^{E}\left<{\bm{\phi}}_{n}^{E},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(E(\infty))}^{2}.$
(30)

However, only in the case where $B=E(\infty)$ can we expect the solutions
${\bm{\theta}}^{(0)}_{i}$ to the transmission problems (6) for the objects $B$
and $E(\infty)$ to be same. Otherwise, we expect the solutions
${\bm{\theta}}^{(0)}_{i}$ to be different and
$(\lambda_{n}^{B},{\bm{\phi}}_{n}^{B})\neq(\lambda_{n}^{E},{\bm{\phi}}_{n}^{E})$.
Expression (30) only guarantees that the sum is the same and, consequently,
${\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ will be different to
${\mathcal{M}}[\alpha E(\infty),\omega,\sigma_{*},\mu_{r}]$ away from the
limiting cases of $\omega=0$ and $\omega\to\infty$ for $\mu_{r}=1$.

##### Case of $\mu_{r}\neq 1$

If $\mu_{r}\neq 1$ then $({\mathcal{N}}^{0}[\alpha
B,\mu_{r}])_{ij}\neq({\mathcal{N}}^{0}[\alpha E(\infty),\mu_{r}])_{ij}$ and,
therefore, for $i=j$ we have

$\sum_{n=1}^{\infty}\lambda_{n}^{B}\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(B)}^{2}\neq\sum_{n=1}^{\infty}\lambda_{n}^{E}\left<{\bm{\phi}}_{n}^{E},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(E)}^{2}.$
(31)

Thus, $\lambda_{n}^{E}\neq\lambda_{n}^{B}$ and
$\left<{\bm{\phi}}_{n}^{B},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(B)}^{2}\neq\left<{\bm{\phi}}_{n}^{E},{\bm{\theta}}^{(0)}_{i}\right>_{L^{2}(E(\infty))}^{2}$
for all $n$. Consequently, ${\mathcal{M}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$
will be different to ${\mathcal{M}}[\alpha
E(\infty),\omega,\sigma_{*},\mu_{r}]$ away from the limiting cases of
$\omega\to\infty$ for $\mu_{r}\neq 1$.

##  4 MPT Spectral signature invariants for object classification

Bishop [13] describes the process of classification as taking an input vector
${\mathbf{x}}$ and assigning it to one of $K$ discrete classes $C_{k}$,
$k=1,\ldots,K$. For example, in security screening, the simplest form of
classification with $K=2$ involves only the classes threat ($C_{1}$) and non-
threat ($C_{2}$), and one with a higher level of fidelity might include the
classes of metallic objects such as key ($C_{1}$), coin ($C_{2}$), gun
($C_{3}$), knife ($C_{4}$) … where the class numbers are assigned as desired.
He recommends that it is convenient (in probabilistic methods of
classification) to use a 1-of-$K$ coding system in which the entries in a
vector ${\mathbf{t}}\in{\mathbb{R}}^{K}$ take the form

${\rm t}_{i}:=\left\\{\begin{array}[]{ll}1&\text{if $i=k$}\\\
0&\text{otherwise}\end{array}\right.,$

if the correct class is $C_{k}$. Requiring that we always have
$\sum_{k=1}^{K}{\rm t}_{k}=1$, then this approach has the advantage that ${\rm
t}_{k}$ can be interpreted as the probability that the correct class is
$C_{k}$. In this section, we focus on alternative choices of the $F$ features
in the input vector ${\mathbf{x}}\in{\mathbb{R}}^{F}$ for the classifier. In
future work we will compare the performance of different classifiers based for
these alternatives. We focus on suitable features that are invariant to
rotation of the object. Note that the rank 2 MPT, and hence the invariants
considered below, are invariant to the position of the object.

###  4.1 Tensor eigenvalues

Recall that the diagonal matrices
${\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]}$ and
${\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]}$
contain the eigenvalues of
$\tilde{\mathcal{R}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$, respectively, and
satisfy the object rotation invariant property

$\displaystyle\lambda_{i}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])=({\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]})_{ii}=$
$\displaystyle({\mathbf{\Lambda}}^{\tilde{\mathcal{R}}[\alpha
R(B),\omega_{m},\sigma_{*},\mu_{r}]})_{ii}=\lambda_{i}(\tilde{\mathcal{R}}[\alpha
R(B),\omega_{m},\sigma_{*},\mu_{r}]),$
$\displaystyle\lambda_{i}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])=({\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]})_{ii}=$
$\displaystyle({\mathbf{\Lambda}}^{{\mathcal{I}}[\alpha
R(B),\omega_{m},\sigma_{*},\mu_{r}]})_{ii}=\lambda_{i}({\mathcal{I}}[\alpha
R(B),\omega_{m},\sigma_{*},\mu_{r}]),$

at each discrete frequencies $\omega_{m}$, $m=1,\ldots,M$ in the MPT spectral
signature. Thus, one option is to select the features for the classifier as

$({\mathbf{x}})_{i}={\rm
x}_{i}=\left\\{\begin{array}[]{ll}\lambda_{j}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=j+(m-1)M\\\
\lambda_{j}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=j+(m+2)M\end{array}\right.,$

where $j=1,2,3$ and $m=1,\ldots,M$ so that $F=6M$. This is particularly
attractive, since any hidden object is likely to be in some unknown rotated
configuration compared to canonical choice of the corresponding object in the
library and, as the eigenvalues are invariant under object rotation, we do not
need knowledge of the orthogonal rotation matrix ${\mathbf{R}}$ to perform the
classification. Furthermore, in practice, measurements lead to noisy tensor
coefficients in the form
${\mathcal{M}}+{\mathcal{E}}_{r}+\mathrm{i}{\mathcal{E}}_{\mathrm{i}}$ where
${\mathcal{E}}_{r}+\mathrm{i}{\mathcal{E}}_{\mathrm{i}}$ is a complex
symmetric rank 2 tensor and represents the noise. To understand the effects of
noise, consider for simplicity a symmetric real matrix ${\mathbf{A}}$
corrupted by a real symmetric ${\mathbf{E}}$, applying results on eigenvalue
perturbations [18] we find that

$\sum_{i=1}^{3}(\lambda_{i}({\mathbf{A}}+{\mathbf{E}})-\lambda_{i}({\mathbf{A}}))^{2}\leq\|{\mathbf{E}}\|_{F}^{2},$

so that the eigenvalues $\lambda_{i}$ of ${\mathbf{A}}$ are similar to those
of ${\mathbf{A}}+{\mathbf{E}}$ provided ${\mathbf{E}}$ represents the low-
moderate noise. However, for an eigenvalue-eigenvector pair
$\lambda_{1},\mathbf{q}_{1}$ [18]

$\text{dist}(\mathbf{q}_{1}({\mathbf{A}}),\mathbf{q}_{1}({\mathbf{A}}+{\mathbf{E}}))\leq\frac{4}{d}C({\mathbf{E}}),$

where $d=\min_{\mu\in\lambda_{i}({\mathbf{A}})}|\lambda_{i}-\mu|>0$ and
$C({\mathbf{E}})$ is a constant depending on ${\mathbf{E}}$. In other words,
if the eigenvalues are close (so $d$ is small), the eigenvectors will be badly
effected by the noise. The same applies to the real and imaginary parts of
${\mathcal{M}}+{\mathcal{E}}_{r}+\mathrm{i}{\mathcal{E}}_{\mathrm{i}}$ when
the coefficients are arranged as matrices.

###  4.2 Tensor invariants

While $\lambda_{i}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\mu_{r},\sigma_{*}])$, $\lambda_{i}({\mathcal{I}}[\alpha
B,\omega_{m},\mu_{r},\sigma_{*}])$, $m=1,\ldots,M$, are invariant under object
rotation, their behaviour is well understood and they behave well for noisy
measurements, classifying objects on the basis of these may still cause
practical issues. Firstly, care is needed with the ordering of the eigenvalues
since choosing a simple rule such as
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ may lead to confusing results. For
example, if the object has rotational and/or reflectional symmetries, we might
find there are only $2$ independent eigenvalues at each frequency in the real
and imaginary parts of the MPT, then, applying the aforementioned rule
independently to $\tilde{\mathcal{R}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$
and ${\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$ could lead to
$\lambda_{2}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])=\lambda_{3}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])$ and $\lambda_{1}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])=\lambda_{2}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])$. Secondly, there is a danger that different
ordering rules are applied in the creation of the training library for the
classifier compared to that used for testing some new candidate object. To
overcome this, tensor invariants can be used, which are independent of how
$\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are assigned. One possibility
are the principal tensor invariants, which, for a rank 2 tensor
${\mathcal{A}}$, are (e.g. [14])

$\displaystyle I_{1}({\mathcal{A}}):=$
$\displaystyle\hbox{tr}\,({\mathcal{A}})=\lambda_{1}({\mathcal{A}})+\lambda_{2}({\mathcal{A}})+\lambda_{3}({\mathcal{A}}),$
(32a) $\displaystyle I_{2}({\mathcal{A}}):=$
$\displaystyle\frac{1}{2}\left(\hbox{tr}\,({\mathcal{A}})^{2}-\hbox{tr}\,({\mathcal{A}}^{2})\right)=\lambda_{1}({\mathcal{A}})\lambda_{2}({\mathcal{A}})+\lambda_{1}({\mathcal{A}})\lambda_{3}({\mathcal{A}})+\lambda_{2}({\mathcal{A}})\lambda_{3}({\mathcal{A}}),$
(32b) $\displaystyle I_{3}({\mathcal{A}}):=$
$\displaystyle\hbox{det}\,({\mathcal{A}})=\lambda_{1}({\mathcal{A}})\lambda_{2}({\mathcal{A}})\lambda_{3}({\mathcal{A}}),$
(32c)

which contain the same information as the tensor’s eigenvalues
$\lambda_{i}({\mathcal{A}})$ and can also be computed from (32). They satisfy

$\displaystyle\lambda^{3}-I_{1}({\mathcal{A}})\lambda^{2}+I_{2}({\mathcal{A}})\lambda-
I_{3}({\mathcal{A}})=0,$ (33)

are rotationally invariant and, like the eigenvalues, are less-susceptible to
noise than the tensor’s eigenvectors.

Borrowing notation from continuum mechanics (e.g. [14]),
$I_{1}({\mathcal{A}})$ is related to the hydrostatic part of ${\mathcal{A}}$
given by ${\mathcal{H}}=\frac{1}{3}\hbox{tr}({\mathcal{A}}){\mathbb{I}}$ and
is associated with the extent to which the operation ${\mathcal{H}}{\bm{v}}$
stretches or shrinks the magnitude of ${\bm{v}}$. The invariant
$I_{2}({\mathcal{A}})$ is often, but not exclusively, related to the
deviatoric part of ${\mathcal{A}}$ given by
${\mathcal{S}}={\mathcal{A}}-{\mathcal{H}}$ describing the extent to which
${\mathcal{S}}{\bm{v}}$ distorts the components of ${\bm{v}}$. The invariant
$I_{3}({\mathcal{A}})$ describes the extent of coupling of the two
aforementioned cases and whether or not the tensor ${\mathcal{A}}$, when
arranged as a $3\times 3$ matrix, is singular or not. In addition, when
applied to (limiting cases) of $\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]$, it has a further physical interpretation:
Recall that the product $\alpha B$ implies that there are an infinite number
of ways to choose $\alpha\ll 1$ and $B$, which still result in the same
$\alpha B$. For example, if $|B|$ is chosen such that
$\hbox{det}\,({\mathcal{N}}^{0}[\alpha
B,\mu_{r}])=I_{3}({\mathcal{N}}^{0}[\alpha B,\mu_{r}])=\alpha^{9}$ then this
invariant provides object size information, while, in general
$I_{3}({\mathcal{N}}^{0}[\alpha B,\mu_{r}])$, will be a function of $|B|$,
$\alpha$ and $\mu_{r}$. Similarly, $I_{3}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])$ and $I_{3}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])$ will be functions of $|B|$, $\alpha$,
$\omega_{m}$, $\sigma_{*}$ and $\mu_{r}$. Thus,
$I_{3}(\tilde{\mathcal{R}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}])$ and
$I_{3}({\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}])$, for fixed
$\alpha$, $\omega_{m}$, $\sigma_{*}$ and $\mu_{r}$, will scale like $|B|$.
Using principal invariants, we could then select the features as

${\rm x}_{i}=\left\\{\begin{array}[]{ll}I_{j}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=j+(m-1)M\\\ I_{j}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=j+(m+2)M\end{array}\right.,$ (34)

where $j=1,2,3$ and $m=1,\ldots,M$ so that $F=6M$.

As an alternative to the principal invariants stated in (32), the alternative
set of invariants (e.g. [14])

$\displaystyle I_{1}({\mathcal{A}}):=$
$\displaystyle\hbox{tr}\,({\mathcal{A}})=\lambda_{1}({\mathcal{A}})+\lambda_{2}({\mathcal{A}})+\lambda_{3}({\mathcal{A}}),$
(35a) $\displaystyle J_{2}({\mathcal{A}}):=$
$\displaystyle\frac{1}{2}\hbox{tr}\,({\mathcal{S}}^{2})=\frac{1}{3}I_{1}({\mathcal{A}})^{2}-I_{2}({\mathcal{A}})=\frac{1}{2}(s_{1}({\mathcal{S}})^{2}+s_{2}({\mathcal{S}})^{2}+s_{3}({\mathcal{S}})^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{2}((\lambda_{1}({\mathcal{A}})-I_{1}({\mathcal{A}})/3)^{2}+(\lambda_{2}({\mathcal{A}})-I_{1}({\mathcal{A}})/3)^{2}+(\lambda_{3}({\mathcal{A}})-I_{1}({\mathcal{A}})/3)^{2}),$
(35b) $\displaystyle J_{3}({\mathcal{A}}):=$
$\displaystyle\hbox{det}\,({\mathcal{S}})=\frac{2}{27}I_{1}({\mathcal{A}})^{3}-\frac{1}{3}I_{1}({\mathcal{A}})I_{2}({\mathcal{A}})+I_{3}({\mathcal{A}})=\frac{1}{3}s_{1}({\mathcal{S}})s_{2}({\mathcal{S}})s_{3}({\mathcal{S}})$
$\displaystyle=$
$\displaystyle(\lambda_{1}({\mathcal{A}})-I_{1}({\mathcal{A}})/3)(\lambda_{2}({\mathcal{A}})-I_{1}({\mathcal{A}})/3)(\lambda_{3}({\mathcal{A}})-I_{1}({\mathcal{A}})/3),$
(35c)

where $\lambda_{i}({\mathcal{A}})=s_{i}({\mathcal{A}})+I_{1}({\mathcal{A}})/3$
can be used. These invariants satisfy

$s^{3}-J_{2}({\mathcal{A}})s-J_{3}({\mathcal{A}})=0,$

and the roots of this equation are the eigenvalues $s_{i}$ of ${\mathcal{S}}$.
The invariants $J_{2}({\mathcal{A}})$ and $J_{3}({\mathcal{A}})$ are both
related to the extent to which ${\mathcal{S}}{\bm{v}}$ distorts the components
of ${\bm{v}}$. In this case, we can select the features as

${\rm x}_{i}=\left\\{\begin{array}[]{ll}I_{1}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=1+(m-1)M\\\
J_{j}(\tilde{\mathcal{R}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=j+(m-1)M\\\ I_{1}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=1+(m+2)M\\\ J_{j}({\mathcal{I}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}]),&i=j+(m+2)M\end{array}\right.,$ (36)

where $j=2,3$ and $m=1,\ldots,M$ and $F=6M$. One potential advantage of using
(35) as a set of features is that, for the case where
$\tilde{\mathcal{R}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega_{m},\sigma_{*},\mu_{r}]$ are both multiples of
identify (such as for the MPT characterisation of a cube or sphere), $J_{2}$
and $J_{3}$ vanish.

###  4.3 Eigenvalues of the commutator of $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$

The off-diagonal enteries $(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}$ and $({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}$ with $i\neq j$ are much smaller than those
on the diagonal with $i=j$ as the results

$\displaystyle|(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}|\leq$
$\displaystyle|\text{tr}(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])|=\left|\sum_{k=1}^{3}\lambda_{k}(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])\right|,$ (37a)
$\displaystyle({\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}\leq$
$\displaystyle\text{tr}({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])=\sum_{k=1}^{3}\lambda_{k}({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]),$ (37b)

obtained in Lemma 6.1 of [32] show. This implies that the tensor’s
eigenvalues, and hence its principal invariants (32) (as well as the
alternative invariants (35)), are dominated by the behaviour of its diagonal
coefficients. To illustrate this, we show, in Figure 5, the comparison between
$\lambda_{i}(\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])$ and
$(\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$ for the
irregular tetrahedron discussed in Section 3.2. We observe that the behaviour
of the eigenvalues is dominated by the diagonal coefficients of
${\tilde{\mathcal{R}}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$, similar arguments
also apply to $\lambda_{i}({{\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]})$ and $({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}$.

$\begin{array}[]{cc}\includegraphics[width=216.81pt]{Fig5RealTensorCoeficients.pdf}&\includegraphics[width=216.81pt]{Fig5RealEigenvalues.pdf}\\\
(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}&\lambda_{i}(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])\end{array}$

Figure 5: Irregular tetrahedron $B$ with vertices (10), $\alpha=0.01\text{ m}$
$\mu_{r}=2$ and $\sigma_{*}=5.96\times 10^{6}\text{ S/m}$: Comparison of
$(\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$ and
$\lambda_{i}(\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])$.

To improve the discrimination between objects whose tensors
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ have different
eigenvectors, we consider their commutator, which has coefficients

$\displaystyle({\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ij}:=(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ik}({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{kj}-({\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{ik}(\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])_{kj}$ (38)

where Einstein summation convention of the indices is implied. The commutator
measures how different the eigenspaces of $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ are. It vanishes when the
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ are simultaneously
diagonalisable (i.e. the eigenvectors of
${\mathbf{Q}}^{\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]}={\mathbf{Q}}^{{\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]}={\mathbf{Q}}(B)$ are the same). In Lemma 8.11 of
[32], $|({\mathcal{Z}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}|$ has been
shown to grow at most linearly with $\omega$. In addition, the coefficients of
${\mathcal{Z}}$ transform as a rank 2 tensor and so the eigenvalues of
${\mathcal{Z}}[\alpha B]$ and ${\mathcal{Z}}[\alpha{\mathbf{R}}(B)]$ are the
same.

It is easy to show that, since $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ are symmetric, ${\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ has vanishing diagonal coefficients and is skew
symmetric. Then, by arranging the coefficients of ${\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ as a $3\times 3$ matrix, we find that its
eigenvalues are zero or purely imaginary

$\displaystyle\lambda_{i}({\mathcal{Z}})\in\left\\{0,\pm\mathrm{i}\sqrt{({\mathcal{Z}})_{12}^{2}+({\mathcal{Z}})_{13}^{2}+({\mathcal{Z}})_{23}^{2}}\right\\},$

and, thus,
$\sqrt{({\mathcal{Z}})_{12}^{2}+({\mathcal{Z}})_{13}^{2}+({\mathcal{Z}})_{23}}=\sqrt{I_{2}({\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])}$ is useful as an additional classifier for
situations where the off-diagonal coefficients of the tensors are amongst its
independent coefficients 111 Note that $I_{1}({\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])=I_{3}({\mathcal{Z}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])=0$. For an object where the only independent
coefficients $(\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$
and $({\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}])_{ij}$ are associated
with $i=j$ then
$\sqrt{({\mathcal{Z}})_{12}^{2}+({\mathcal{Z}})_{13}^{2}+({\mathcal{Z}})_{23}}$
vanishes. This invariant can easily be added to the list of features in (4.1),
(34) or (36) as

${\rm x}_{i}=\sqrt{I_{2}({\mathcal{Z}}[\alpha
B,\omega_{m},\sigma_{*},\mu_{r}])},\qquad i=6M+m,$ (39)

for $m=1,\ldots,M$ bringing the total number of features to $F=7M$.

##  5 Building a training data set for classification

In this section, we provide a series of illustrative examples to demonstrate
how the PODP approach [55] to compute the MPT spectral signatures described in
Section 2.1 can be combined with an appropriate choice of eigenvalues or
tensor invariants in Section 4 and sampling at $M$ frequencies to form a
training dataset for object classification. Forming this dictionary involves
considering different physical objects $B_{\alpha}^{(p)}$, $p=1,\ldots,P$, and
provides the pairs (${\mathbf{x}}_{p}\in{\mathbb{R}}^{F}$,
${\mathbf{t}}_{p}\in{\mathbb{R}}^{K}$) for each object where the entries
${\mathbf{x}}_{p}$ are of the form of (4.1) (or (34) or (36)) and (39) and the
entries of ${\mathbf{t}}_{p}$ are all $0$ except for one entry with value $1$,
corresponding to the class of the object $B_{\alpha}^{(p)}$. For the numerical
examples considered, we use the author’s MPT-Calculator program (commit number
b861dfb) to generate the MPT spectral signatures, which, in turn, uses the
NGSolve package (version 6.2.2004) developed by the group led by Schöberl for
the underlying finite element computations [49, 48, 56, 47]. We note that this
data set can be easily enriched by using the scaling results obtained in Lemma
5.1 and Lemma 5.2 of [55], which enable the MPT spectral signatures of objects
with the same underlying geometry, but with different sizes and different
conductivities to be obtained at no cost.

We separate the results into four subsections the first two relating to non-
threat items (house keys and British coins) and the second two to threat items
(TT-33 semi automatic pistol and knives).

###  5.1 Non-threat items: Keys for pin-tumbler locks

Common materials for keys for pin-tumbler locks include brass, plated brass,
nickel silver, and steel. Amongst these, brass is often chosen due to its low
cost, ease of cutting and its self lubricating characteristics, which avoids
the key getting stuck in a lock. Therefore, for this study, we restrict
ourselves to keys made of brass and we select the material parameters to be
$\mu_{r}=\mu_{*}/\mu_{0}=1$ and $\sigma_{*}=1.5\times 10^{7}\text{ S/m}$ that
correspond to its conductivity value being at $26\%$ of the value for copper
in the International Annealed Copper Standard (IACS) [1]. An illustration of
the cross-section of a key for a pin-tumbler lock is included in Figure 6
where the dimensions are similar to a house key and we indicate the physical
key $B_{\alpha}$ as well as the non-dimensional object $B$ used in the
computations.

$\begin{array}[]{cc}\includegraphics[width=185.0082pt]{key2.pdf}&\includegraphics[width=185.0082pt]{key1.pdf}\\\
\text{(a)}\ B_{\alpha}&\text{(b)}\ B\end{array}$

Figure 6: Typical key for a pin-tumbler lock: (a) Physical object $B_{\alpha}$
and (b) non-dimensional object $B$.

To understand how the effects of small changes in the key geometry affect the
MPT frequency spectral signature for frequencies in the range
$10^{2}\leq\omega\leq 10^{8}\text{rad/s}$, a sequence of $9$ different key
geometries were produced. In each case, we set $\alpha=0.001$m and specify the
dimensions of the different cases for $B$ to be non-dimensional. For example,
for key 1, $B$ is of length $34$, has a width of 6.5 (min)-14 (max) and a
thickness of $2.5$ whereas $B$ for key 4 has a maximum thickness of 2.5 and a
deep blade cut of 0.75 and notches of maximum size 1.75. The meshes of the two
sets of keys are shown in Figures 7 and 8. These meshes have local refinement
towards the edges of the keys and each case the mesh extends out to a
truncation boundary in the form of the $[-1000,1000]^{3}$ rectangular box and
comprise of between $51\,726$ and $108\,523$ unstructured tetrahedra.
Importantly, note that the connectedness of the different key types. Of the
different keys, keys 2, 4, 6, 7, 8 and 9 are multiply connected and have Betti
numbers $\beta_{0}(B)=\beta_{1}(B)=1$ and $\beta_{2}(B)$ the remaining keys
are simply connected with $\beta_{0}(B)=1$ and $\beta_{1}(B)=\beta_{2}(B)$.

(a) Key 1

(b) Key 2

(c) Key 3

(d) Key 4

(e) Key 9

Figure 7: Set 1 of brass house keys : Surface distribution of elements of the
keys in the meshes cases 1-4 and 9.

(a) Key 5

(b) Key 6

(c) Key 7

(d) Key 8

Figure 8: Set 2 of brass house keys : Surface distribution of elements of the
keys in the meshes cases 5-8 (see Figure 7 for key 9 also in this set).

####  5.1.1 Set 1 of brass house keys

Restricting consideration to the set 1 of house keys, we begin by illustrating
that $p$-refinement of the mesh of $56\,241$ unstructured tetrahedra for key 1
using $p=0,1,2,3$ order elements leads to a rapid convergence of the MPT
spectral signature presented in the form of the eigenvalues of
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ 222Note that the
coefficients of ${\mathcal{N}}^{0}$ vanish in this case as $\mu_{r}=1$, but we
keep to the notation of $\tilde{\mathcal{R}}={\mathcal{N}}^{0}+{\mathcal{R}}$
for ease of comparison with later results and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$, namely $\lambda_{i}(\tilde{\mathcal{R}})$ and
$\lambda_{i}\left(\mathcal{I}\right)$, $i=1,2,3$, as illustrated in Figures 9
and 10. Note that, due to the reflectional symmetries for key 1, there are
only two independent coefficients each in $\tilde{\mathcal{R}}$ and
${\mathcal{I}}$ and hence only two distinct eigenvalues in each case. Also, we
have chosen to order the eigenvalues so that those corresponding to $i=1$ and
$i=2$ are the same in these plots.

(a) $\lambda_{1}(\tilde{\mathcal{R}})=\lambda_{2}(\tilde{\mathcal{R}})$ (b)
$\lambda_{3}(\tilde{\mathcal{R}})$

Figure 9: Key 1 from set 1 of brass house keys: $p$-refinement study using
$p=0,1,2,3$ order elements for (a)
$\lambda_{1}(\tilde{\mathcal{R}})=\lambda_{2}(\tilde{\mathcal{R}})$, (b)
$\lambda_{3}(\tilde{\mathcal{R}})$.

(a) $\lambda_{1}(\mathcal{I})=\lambda_{2}(\mathcal{I})$ (b)
$\lambda_{3}(\mathcal{I})$

Figure 10: Key 1 from set 1 of brass house keys: $p$-refinement study using
$p=0,1,2,3$ order elements for (a)
$\lambda_{1}(\mathcal{I})=\lambda_{2}(\mathcal{I})$ and (b)
$\lambda_{3}(\mathcal{I})$.

The role played by a key’s topology and its equivalent ellipsoid at a fixed
frequency is now considered. Previously, in (22), the equivalence between
${\mathcal{M}}^{\infty}[\alpha B]$ and ${\mathcal{T}}[\alpha B,0]$ for the
situation where $\beta_{1}(B)=0$ was highlighted. In Figure 11, we compare
$\lambda_{i}(\tilde{\mathcal{R}})$ and $\lambda_{i}({\mathcal{T}}[\alpha
B,0])$ for key 1 and key 2, the former having $\beta_{1}(B)=0$ and the latter
having $\beta_{1}(B)=1$. As expected, since
$\lim_{\omega\to\infty}({\mathcal{I}})_{ij}=0$, we see good agreement in the
limiting case as $\omega\to\infty$ (up to the limit of the eddy current model)
between $\lambda_{i}(\tilde{\mathcal{R}})$ and
$\lambda_{i}({\mathcal{T}}[\alpha B,0])$ for key 1, but not for key 2. Thus,
highlighting the important role that an object’s topology plays.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig11abettiNumber.pdf}&\includegraphics[scale={0.5}]{Fig11bbettiNumber.pdf}\\\
\text{(a) Key 1}&\text{(a) Key 2}\end{array}$

Figure 11: Set 1 of brass house keys: Comparison of
$\lambda_{i}(\tilde{\mathcal{R}})$ and $\lambda_{i}({\mathcal{T}}[\alpha
B,0])$ for (a) Key 1 and (b) Key 2

In a similar manner to the results shown in Figures 9 and 10, by performing
$p$-refinement on the meshes for the other keys, and considering snapshot
frequencies, the MPT coefficients were found converge using $p=3$ elements.
However, to accelerate the computation of the MPT spectral signature for the
keys, the PODP approach described in [55], and outlined in Section 2.1, was
followed. This involves computing solutions at $N$ representative full order
model solutions at logarithmically spaced frequencies and then extracting a
basis using a tolerance of $TOL=10^{-8}$ and solving reduced sized problems to
approximate ${\bm{\theta}}_{i}^{(1)}(\omega)$ at other frequencies and,
henceforth, predict the MPT coefficients at other frequencies. We illustrate
the process in Figure 12 for key 1 using $N=31$. The a-posteriori error
estimates $(\Delta[\omega])_{ij}$ that are obtained at low-computational cost
at run-time during the online stage of the reduced order model, are used to
certify the reduced order model solutions that have been obtained, are also
shown in this figure. These illustrate that, in this case, the reduced order
model predictions are reliable with respect to the full order model prediction
of the MPT. Note that the PODP solutions are also very acceptable using $N=13$
representative full order model solution snapshots, but we have used $N=31$ to
ensure small $(\Delta[\omega])_{ij}$ is small at all but the highest
frequencies. Still further, the frequency $\omega_{limit}$, obtained using the
method described in Section 2.1.1, at which the eddy current approximation for
this object is predicted to break down is shown. Tighter certificates bounds
could be obtained by increasing $N$, however, this was not deemed to be
necessary as the bounds, which provide confidence that the PODP predictions
are accurate, are already tight for $\omega\leq\omega_{limit}$.

(a) $(\tilde{\mathcal{R}})_{ij}$ (b) $(\mathcal{I})_{ij}$

Figure 12: Key 1 from set 1 of brass house keys: PODP prediction of the
spectral signature showing also the frequencies used for the representative
full order solution snapshots and limiting frequency for (a)
$(\tilde{\mathcal{R}})_{ij}$ and (b) $(\mathcal{I})_{ij}$.

To compare the results for different keys in set 1, we present the MPT
spectral signature using the principal invariants $I_{i}$, $i=1,2,3$, for
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ that have been obtained
using the PODP approach in Figure 13. In this figure, we observe a family of
curves that each show a similar behaviour for all the keys in the set.

The invariant $I_{1}(\tilde{\mathcal{R}})$ is monotonically decreasing with
$\log\omega$, implying the hydrostatic part of $\tilde{\mathcal{R}}$ is
associated with a maximum response at high frequencies, while the invariant
$I_{2}(\tilde{\mathcal{R}})$, which is monotonically increasing with
$\log\omega$, implies the deviatoric part of $\tilde{\mathcal{R}}$ begin to
plays a significant role for $\omega>10^{5}\text{rad/s}$. The invariant
$I_{3}(\tilde{\mathcal{R}})$, which is monotonically decreasing with
$\log\omega$, implies the interaction between the hydrostatic and deviatoric
part of $\tilde{\mathcal{R}}$ begin to plays a significant role for
$\omega>10^{5}\text{rad/s}$. The invariants $I_{i}({\mathcal{I}})$, $i=1,2,3$,
each have a single local maximum and are greater or equal to $0$ for all
$\omega$. The invariant $I_{1}({\mathcal{I}})$ implies that hydrostatic part
of ${\mathcal{I}}$ is associated with a maximum response at $\omega\approx
10^{5}\text{rad/s}$ and has a broad response over the frequency range
$10^{2}\leq\omega\leq 10^{8}\text{rad/s}$ while the invariant
$I_{2}({\mathcal{I}})$ implies the deviatoric part of ${\mathcal{I}}$ is
associated with a maximum response at $\omega\approx 10^{5}\text{rad/s}$ , but
its effects are more limited to the narrower frequency band
$10^{4}\leq\omega\leq 10^{7}\text{rad/s}$. Finally, the invariant
$I_{3}({\mathcal{I}})$ has a maximum at $\omega\approx 10^{5}\text{rad/s}$,
although interaction between hydrostatic and deviatoric parts are more limited
to the $10^{4}\leq\omega\leq 10^{6}\text{rad/s}$. Comparing the keys, the
effects are diminished from keys 1,2,3,4 and 9, in turn and, for example, the
peak value of $I_{3}({\mathcal{I}})$ reduces in sequence of the volumes of the
keys, as expected. Furthermore, the results for $I_{i}$, $i=2,3$ applied to
$\tilde{\mathcal{R}}$ and ${\mathcal{I}}$ are similar when comparing the keys
1 and 2.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig13RealInvarient_1.pdf}&\includegraphics[scale={0.5}]{Fig13ImaginaryInvarient_1.pdf}\\\
\text{(a) }I_{1}(\tilde{\mathcal{R}})&\text{(b) }I_{1}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig13RealInvarient_2.pdf}&\includegraphics[scale={0.5}]{Fig13ImaginaryInvarient_2.pdf}\\\
\text{(c) }I_{2}(\tilde{\mathcal{R}})&\text{(d) }I_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig13RealInvarient_3.pdf}&\includegraphics[scale={0.5}]{Fig13ImaginaryInvarient_3.pdf}\\\
\text{(e) }I_{3}(\tilde{\mathcal{R}})&\text{(f)
}I_{3}(\mathcal{I})\end{array}$

Figure 13: Set 1 of brass house keys: Comparison of tensor invariants. (a)
$I_{1}(\tilde{\mathcal{R}})$, (b) $I_{1}(\mathcal{I})$ (c)
$I_{2}(\tilde{\mathcal{R}})$, (d) $I_{2}(\mathcal{I})$, (e)
$I_{3}(\tilde{\mathcal{R}})$ and (f) $I_{3}(\mathcal{I})$.

Next, in Figure 14, we present the MPT spectral signature using the
alternative invariants $J_{i}$, $i=2,3$, for $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ that have been obtained using the PODP approach.
Note that we do not reproduce the invariant $I_{1}$, which also forms part of
this set, as it has already been shown in Figure 13. We observe that, for the
different keys making up set 1, a family of similar curves is obtained and
there are similarities to the behaviour of the invariants $I_{i}$, $i=2,3$,
with frequency for these tensors. However, the following differences are
noteworthy, firstly, the monotonic increase and decrease of
$J_{2}(\tilde{\mathcal{R}})$ and $J_{3}(\tilde{\mathcal{R}})$ with
$\log\omega$, respectively, is initially much more rapid than that of
$I_{2}(\tilde{\mathcal{R}})$ and $I_{3}(\tilde{\mathcal{R}})$. Secondly, the
curves for $J_{i}(\tilde{\mathcal{R}})$ and $J_{i}({\mathcal{I}})$, $i=2,3$,
are very similar when comparing the keys 3 and 4, whereas the corresponding
curves for $I_{i}(\tilde{\mathcal{R}})$ and $I_{i}({\mathcal{I}})$, $i=2,3$,
for these keys are different. On the other hand, the similarities previously
observed between the invariants $I_{i}$, $i=2,3$, for $\tilde{\mathcal{R}}$
and ${\mathcal{I}}$ keys 1 and 2 are also reflected by the invariants $J_{i}$,
$i=2,3$, for these tensors and keys.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig14RealInvarient_2.pdf}&\includegraphics[scale={0.5}]{Fig14ImaginaryInvarient_2.pdf}\\\
\text{(a) }J_{2}(\tilde{\mathcal{R}})&\text{(b) }J_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig14RealInvarient_3.pdf}&\includegraphics[scale={0.5}]{Fig14ImaginaryInvarient_3.pdf}\\\
\text{(c) }J_{3}(\tilde{\mathcal{R}})&\text{(d)
}J_{3}(\mathcal{I})\end{array}$

Figure 14: Set 1 of brass house keys: Comparison of tensor invariants. (a)
$J_{2}(\tilde{\mathcal{R}})$, (b) $J_{2}(\mathcal{I})$, (c)
$J_{3}(\tilde{\mathcal{R}})$ and (d) $J_{3}(\mathcal{I})$.

Finally, we consider comparisons of the invariant
$\sqrt{I_{2}({\mathcal{Z}}[\alpha B,\omega,\sigma_{*},\mu_{r}])}$ as a
function of $\omega$ for keys in set 1, which can provide additional
information about the object’s characterisation if the independent
coefficients of the MPT are not only associated with its diagonal
coefficients. We begin by showing the convergence of
$\sqrt{I_{2}({\mathcal{Z}})}$ to $0$ under $p$-refinement for key 1 in Figure
15. The mirror symmetries for this object imply that
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ each have only $3$
independent coefficients (at each frequency) that lie on the diagonal of the
tensors and so we expect $\sqrt{I_{2}({\mathcal{Z}})}$ to vanish for exact
computations. Alongside this, in the same figure, we show the $p$-convergence
of key 4, which has $6$ independent coefficients each in $\tilde{\mathcal{R}}$
and ${\mathcal{I}}$ (at each frequency) and exhibits rapid convergence of the
invariant $\sqrt{I_{2}({\mathcal{Z}})}$ to the shown curve as a function of
$\omega$. The behaviour of $\sqrt{I_{2}({\mathcal{Z}})}$ for keys 4 and 9 is
shown in Figure 16. For keys $1-3$ we henceforth set
$\sqrt{I_{2}({\mathcal{Z}})}=0$.

(a) Key 1 (b) Key 4

Figure 15: Set 1 of brass keys: $p$-refinement study for
$\sqrt{I_{2}({\mathcal{Z}})}$ for (a) Key 1 and (b) Key 4. Figure 16: Set 1 of
brass keys: Comparison of $\sqrt{I_{2}({\mathcal{Z}})}$ for keys 4 and 9.

####  5.1.2 Set 2 of brass house keys

Turning our attention to set 2 of the brass house keys, we reproduce the
results previously shown in Figure 13 for the invariants $I_{i}$, $i=1,2,3$,
for $\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ 333Note that the
coefficients of ${\mathcal{N}}^{0}$ vanish as $\mu_{r}=1$, but we keep to the
notation of $\tilde{R}={\mathcal{N}}^{0}+{\mathcal{R}}$ for ease of comparison
with later results and ${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ for
the second set of keys and show these in Figure 17.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig17RealInvarient_1.pdf}&\includegraphics[scale={0.5}]{Fig17ImaginaryInvarient_1.pdf}\\\
\text{(a) }I_{1}(\tilde{\mathcal{R}})&\text{(b) }I_{1}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig17RealInvarient_2.pdf}&\includegraphics[scale={0.5}]{Fig17ImaginaryInvarient_2.pdf}\\\
\text{(c) }I_{2}(\tilde{\mathcal{R}})&\text{(d) }I_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig17RealInvarient_3.pdf}&\includegraphics[scale={0.5}]{Fig17ImaginaryInvarient_3.pdf}\\\
\text{(e) }I_{3}(\tilde{\mathcal{R}})&\text{(f)
}I_{3}(\mathcal{I})\end{array}$

Figure 17: Set 2 of brass house keys: Comparison of tensor invariants. (a)
$I_{1}(\tilde{\mathcal{R}})$, (b) $I_{1}(\mathcal{I})$ (c)
$I_{2}(\tilde{\mathcal{R}})$, (d) $I_{2}(\mathcal{I})$, (e)
$I_{3}(\tilde{\mathcal{R}})$ and (f) $I_{3}(\mathcal{I})$

In a similar manner to Figure 13, we see that the results included in Figure
17 form a family of similar curves and that their behaviour follows a similar
pattern to that previous described for the keys in set 1. The results for
$I_{i}$, $i=1,2,3$ for the tensor characterisations of keys 5 and 6 are
similar, which is not surprising given the similarities in these geometries.
In addition, there are only small differences in $I_{i}$, $i=1,2,3$ for the
tensor characterisations of keys 8 and 9. Note that key 8 has a circular head
and key 9 a polygonal head, but the volume of material is similar and the
symmetries of the objects and the number of independent coefficients in
$\tilde{\mathcal{R}}$ and ${\mathcal{I}}$ (for each frequency) are otherwise
the same for these two keys. Keys $5-8$ are associated with a gradual
reduction in the volume of the material for the key and we can see that the
magnitude of the associated $I_{3}({\mathcal{I}})$ and
$I_{3}(\tilde{\mathcal{R}})$, curves for these cases reduces as expected.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig18RealInvarient_2.pdf}&\includegraphics[scale={0.5}]{Fig18ImaginaryInvarient_2.pdf}\\\
\text{(a) }J_{2}(\tilde{\mathcal{R}})&\text{(b) }J_{3}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig18RealInvarient_3.pdf}&\includegraphics[scale={0.5}]{Fig18ImaginaryInvarient_3.pdf}\\\
\text{(c) }J_{3}(\tilde{\mathcal{R}})&\text{(d)
}J_{3}(\mathcal{I})\end{array}$

Figure 18: Set 2 of brass house keys: Comparison of tensor invariants. (a)
$J_{2}(\tilde{\mathcal{R}})$, (b) $J_{2}(\mathcal{I})$, (c)
$J_{3}(\tilde{\mathcal{R}})$ and (d) $J_{3}(\mathcal{I})$

Similarly, in Figure 18, we see the results for the invariants $J_{i}$,
$i=2,3$ form a family of curves with the behaviour of the invariants similar
to that described for the keys in set 1. Note that the results for the keys in
set 2 for $J_{i}$, $i=2,3$ can be grouped into keys 5, 6 and keys 7, 8 and 9
where the results for keys 8 and 9 for $J_{2}({\mathcal{I}})$ (and
$J_{3}({\mathcal{I}})$) are indistinguishable on this scale. The former group
does not contain the notches or the blade cut while the latter set all have
the same notches, keys 8 and 9 have the deep blade cut and key 9 differs from
the others by having a polygonal head rather than a circular head (although
has a similar volume to keys 7 and 8).

Of the keys in set 2, only keys 8 and 9 have independent coefficients in
${\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ that are not only associated with the diagonal
entries of the tensor. The behaviour of $\sqrt{I_{2}({\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])}$ for these keys is shown in Figure 19. For the
other keys in set 2 we set $\sqrt{I_{2}({\mathcal{Z}})}=0$.

Figure 19: Set 2 of brass house keys: Comparison of the tensor invariant
$\sqrt{I_{2}({\mathcal{Z}})}$.

###  5.2 Non-threat Items: British coins

In this section, inspired by the previous article on MPT characterisations of
US coins [15], we present MPT spectral signature characterisations for British
coins in the denominations 1 pence (p), 2p, 5p, 10p, 20p, 50p, £1, £2. We use
the 1982 (20p), 1992 (1p, 2p), 1997 (50p), 1998 (£2), 2012 (5p,10p), 2017 (£1)
issues of these denominations as listed in Table 1, which also summarises the
shape, diameter, thickness, composition based on the information available
from the Royal Mint [40]. The table also sets out the electrical properties,
where the conductivity values for the different material compositions have
been obtained from [22] at room temperature. For the quoted compositions, we
have assumed that $\mu_{r}=1$, however, in practice, some Copper-Nickel
mixtures with a high iron content can have a $\mu_{r}$ slightly above $1$ (e.g
[20]). The later issues of the 1p, 2p, 5p and 10p coins have a significantly
different composition to that presented in Table 1 and, instead of a high
copper content, they are instead copper plated steel. Note that each of the
coins considered are simply connected.

With the exception of the £1 and £2 denominations, the coins are modelled as
homogeneous conductors while the former are each modelled as an annulus with
two different materials. The majority of the coins have a circular face and
only the 20p and 50p differ, being Reuleaux heptagonal discs. The coins with a
circular face are modelled so that their circular region lies in the plane
spanned by ${\bm{e}}_{1}$ and ${\bm{e}}_{2}$ and, hence, they have rotational
symmetry about the ${\bm{e}}_{3}$ axis (for any angle). Consequently, the
independent coefficients of $\tilde{\mathcal{R}}[\alpha
B,\omega,\mu_{r},\sigma_{*}]$ 444Note that the coefficients of
${\mathcal{N}}^{0}$ vanish as $\mu_{r}=1$, but we keep to the notation of
$\tilde{R}={\mathcal{N}}^{0}+{\mathcal{R}}$ for ease of comparison with later
results and ${\mathcal{I}}[\alpha B,\omega,\mu_{r},\sigma_{*}]$ for such coins
are $(\tilde{\mathcal{R}})_{11}=(\tilde{\mathcal{R}})_{22}$,
$(\tilde{\mathcal{R}})_{33}$ and $({\mathcal{I}})_{11}=({\mathcal{I}})_{22}$
and $({\mathcal{I}})_{33}$ (for each frequency). The Reuleaux heptagonal discs
are modelled in a similar way, with a 51.428 (4dp) degree rotational symmetry
about the ${\bm{e}}_{3}$ and, consequently, it also follows that their
independent coefficients of the MPT for such coins are associated with the
same entries.

Coin |  Shape |  Diameter in mm |  Thickness in mm |  Composition |  Relative Permeability ($\mu_{r}$) |  Conductivity ($\sigma_{*}$) in S/m
---|---|---|---|---|---|---
1p (1971-Date) |  Circular Disc |  20.3 |  1.52 |  97% Copper, 2.5% Zinc and 0.5% Tin |  1 |  4.03$\times 10^{7}$
2p (1971-Date) |  Circular Disc |  25.9 |  2.03 |  97% Copper, 2.5% Zinc and 0.5% Tin |  1 |  4.03$\times 10^{7}$
5p (1990-Date) |  Circular Disc |  18 |  1.7 |  75% Copper and 25% Nickel |  1 |  2.91$\times 10^{6}$
10p (1990-Date) |  Circular Disc |  24.5 |  1.85 |  75% Copper and 25% Nickel |  1 |  2.91$\times 10^{6}$
20p (1982-Date) |  Reuleaux Heptagonal Disc |  21.4 |  1.7 |  84% Copper and 16% Nickel |  1 |  5.26$\times 10^{6}$
50p (1997-Date) |  Reuleaux Heptagonal Disc |  27.3 |  1.78 |  75% Copper and 25% Nickel |  1 |  2.91$\times 10^{6}$
£1 (2017-Date) |  Annulus |  15.2 /23.45 (in/out) |  2.8 / 2.8 (in/out) |  Nickel Plated Brass / 70% Copper, 24.5% Zinc and 5.5% Nickel |  1 /1 (in/out) |  1.63$\times 10^{7}$ / 5.26$\times 10^{6}$ (in/out)
£2 (1998-Date) |  Annulus |  21 /28.4 (in/out) |  2.5 /2.5 (in/out) |  75% Copper and 25% Nickel/ 97% Copper, 2.5% Zinc and 0.5% Tin |  1 /1 (in/out) |  2.91$\times 10^{6}$ / 1.93$\times 10^{7}$ (in/out)

Table 1: Set of British Coins 1p, 2p, 5,10p, £1 and £2 : Coin shape,
dimensions and electrical properties.

To model the 1p coin, we considered $B$ to be a circular disc of diameter 20.3
and thickness 1.52 and set $\alpha=0.001$ mm. An unstructured mesh of
$33\,351$ unstructured tetrahedra was generated to model the object and the
region surrounding it out to a truncation boundary in the form of the
rectangular box $[-1000,1000]^{3}$. In a similar way, unstructured meshes of
between $24\,963$ and $36\,957$ tetrahedra were generated to model the other
coins. On these meshes, $p=4$ elements were found to be satisfactory for
accurately computing the representative full order model solution snapshots.
In order to produce the MPT spectral signature for the coins, $N=13$
representative full order solution snapshots were obtained at logarithmically
spaced frequencies over the range $10^{1}\leq\omega\leq 10^{10}\text{rad/s}$
were used in combination with the PODP approach and a tolerance of
$TOL=10^{-6}$.

Although the PODP solutions are very acceptable using $N=13$ representative
full order model solution snapshots, in order to achieve smaller a-posteriori
error estimates, results obtained with $N=21$ and $TOL=10^{-8}$ are considered
and shown in Figure 20. Also included in this figure is the limiting frequency
$\omega_{limit}$ predicted by following the approach in Section 2.1.1. The
rotational symmetry of the object implies that the object has just two
independent coefficients each in $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$, which lie on the diagonal of the tensors. Of
these $(\tilde{\mathcal{R}})_{33}$ and $(\mathcal{I})_{33}$ have the largest
magnitude in a direction that is perpendicular to the plane of the disc, which
is as expected for a non-magnetic disc [15]. Note that the 1p coin issued
after 1992, which has a high $\mu_{r}$ value, would have dominant components
$(\tilde{\mathcal{R}})_{11}=(\tilde{\mathcal{R}})_{22}$ and
$(\mathcal{I})_{11}=(\mathcal{I})_{22}$ in the plane of the disc, as expected
for a magnetic disc [15]. A similar study was performed for each of the coins
listed in Table 1 in [54] in order to ensure the results were accurate.

(a) $(\tilde{\mathcal{R}})_{ii}$ (b) $(\mathcal{I})_{ii}$

Figure 20: 1p Coin from set of British coins: PODP prediction of the spectral
signature showing also the frequencies used for the representative full order
solution snapshots and limiting frequency for (a) $(\tilde{\mathcal{R}})_{ii}$
and (b) $(\mathcal{I})_{ii}$.

To compare the results for the different coins, we present the MPT spectral
signature using the principal invariants $I_{i}$, $i=1,2,3$, for
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ that have been obtained
using the PODP approach in Figure 21. In this figure, we have restricted
consideration to frequencies such that $10^{2}\leq\omega\leq
10^{8}\text{rad/s}$ in order to allow comparisons with the earlier key
results. In practice, the eddy current model brakes down at a frequency of
$\omega_{limit}<10^{8}\text{rad/s}$ (or greater) for all the coins considered
and so higher frequencies are physically invalid in any case. Unlike the
corresponding results for the house keys shown in Figures 13 and 17, the
results obtained for the coins shown in Figure 21 do not form a family of
similar curves since both the volumes and materials of the coins vary
significantly motivating the ability to discriminate between different coins,
however, some of the trends previously observed cary over to this case also.
The curves for $I_{1}(\tilde{\mathcal{R}})$ and $I_{3}(\tilde{\mathcal{R}})$
are monotonically decreasing with $\log\omega$, while
$I_{2}(\tilde{\mathcal{R}})$ is monotonically increasing with $\log\omega$.
The curves for $I_{i}({\mathcal{I}})$, $i=1,2,3$, each have a single local
maximum, although the peaks appear at different frequencies for different
coins and the different invariants, however, the width of the frequency band
reduces for all cases, when considering $I_{2}({\mathcal{I}})$ and
$I_{3}({\mathcal{I}})$ compared to $I_{1}({\mathcal{I}})$. On considering the
different coins, we see similarities between the MPT spectral signatures of
the 1p, 2p coins, the 5p, 10p coins and the 20p and 50p coins. This can be
explained as follows: the composition of the coins in these groups is the same
and their dimensions can be approximately obtained by a simple scaling, hence,
the scaling results in Lemma 5.2 of [55] predict that the tensor coefficients
of the larger sized coin can be obtained from the smaller object by a
translation and scaling, which is also observed in the invariants. The 50p
coin has the largest volume and also the highest peak value in
$I_{3}({\mathcal{I}})$, the magnitude of the peaks reduce in sequence of the
volumes of the coins, as expected. The multiple local maxima in the
coefficients of ${\mathcal{I}}$ and the multiple points of inflection in the
coefficients of $\tilde{\mathcal{R}}$, which are known to be associated with
objects with inhomogeneous conductivity [33], are not easily distinguished on
the invariants for the £1 and £2 coins, probably due to the difference in
conductivities being approximately 1 order of magnitude or less.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig21RealInvarient_1.pdf}&\includegraphics[scale={0.5}]{Fig21ImaginaryInvarient_1.pdf}\\\
\text{(a) }I_{1}(\tilde{\mathcal{R}})&\text{(b) }I_{1}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig21RealInvarient_2.pdf}&\includegraphics[scale={0.5}]{Fig21ImaginaryInvarient_2.pdf}\\\
\text{(c) }I_{2}(\tilde{\mathcal{R}})&\text{(d) }I_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig21RealInvarient_3.pdf}&\includegraphics[scale={0.5}]{Fig21ImaginaryInvarient_3.pdf}\\\
\text{(e) }I_{3}(\tilde{\mathcal{R}})&\text{(f)
}I_{3}(\mathcal{I})\end{array}$

Figure 21: British coins: Comparison of tensor invariants. (a)
$I_{1}(\tilde{\mathcal{R}})$, (b) $I_{1}(\mathcal{I})$ (c)
$I_{2}(\tilde{\mathcal{R}})$, (d) $I_{2}(\mathcal{I})$, (e)
$I_{3}(\tilde{\mathcal{R}})$ and (f) $I_{3}(\mathcal{I})$.

The corresponding results obtained for the alternative invariants $J_{i}$,
$i=2,3$, for $\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ are presented in Figure
22. Again, unlike the keys, we see that the plots of these invariants do not
form a family of similar curves as both the volumes and conductivities of the
different coins are different. While for most coins
$J_{2}(\tilde{\mathcal{R}})$ is monotonically increasing with $\log\omega$ and
$J_{3}(\tilde{\mathcal{R}})$ is monotonically decreasing with $\log\omega$, we
see that there are exceptions, most notably with the £2 coin, which can be
explained by its inhomogeneous materials. One might expect a similar behaviour
with the inhomogeneous £1 coin, but it is difficult to observe on this scale.
The behaviour of $J_{2}({\mathcal{I}})$ and $J_{3}({\mathcal{I}})$ with
$\log\omega$ shows a single local maximum for each coin where the presence of
multiple local maxima for the £1 and £2 can’t be observed on this scale. Of
the coins considered, the curves associated with the 2p, 50p and £2 cases have
the largest magnitude, indicating that they have the largest deviatoric
component, which is also expected, given the geometries of these objects.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig22RealDeviatoric_2.pdf}&\includegraphics[scale={0.5}]{Fig22ImaginaryDeviatoric_2.pdf}\\\
\text{(a) }J_{2}(\tilde{\mathcal{R}})&\text{(b) }J_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig22RealDeviatoric_3.pdf}&\includegraphics[scale={0.5}]{Fig22ImaginaryDeviatoric_3.pdf}\\\
\text{(c) }J_{3}(\tilde{\mathcal{R}})&\text{(d)
}J_{3}(\mathcal{I})\end{array}$

Figure 22: British coins: Comparison of tensor invariants. (a)
$J_{2}(\tilde{\mathcal{R}})$, (b) $J_{2}(\mathcal{I})$, (c)
$J_{3}(\tilde{\mathcal{R}})$ and (d) $J_{3}(\mathcal{I})$.

In order to compute the MPT spectral signature, the solution of a reduced
order model is obtained at each frequency of interest. As the frequency
increases, the skin depth reduces and the associated eddy currents become
confined to a thin layer close to the surface of the conductor. In Figure 23,
we show a cut through the 1p coin, on the plane spanned by $\bm{e}_{1}$ and
$\bm{e}_{3}$, in order to illustrate the eddy currents
${\bm{J}}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}$ obtained at the
frequencies $\omega=10^{3}$ rad/s, $\omega=10^{5}$ rad/s and $\omega=10^{7}$
rad/s.

$\begin{array}[]{ccc}\includegraphics[width=231.26378pt,keepaspectratio]{Om_1e3_E3_real_mag.jpeg}&\includegraphics[width=231.26378pt,keepaspectratio]{Om_1e5_E3_real_mag.jpeg}\\\
\textrm{\footnotesize{(a) $|\text{Re}({\bm{J}}^{e})|$ for $\omega=10^{3}$
rad/s.}}&\textrm{\footnotesize{(b) $|\text{Re}({\bm{J}}^{e})|$ for
$\omega=10^{5}$ rad/s.}}\end{array}$

$\begin{array}[]{c}\includegraphics[width=231.26378pt,keepaspectratio]{Om_1e7_E3_real_mag.jpeg}\\\
\textrm{\footnotesize{(c) $|\text{Re}({\bm{J}}^{e})|$ for $\omega=10^{7}$
rad/s.}}\end{array}$

Figure 23: British 1p coins: Contours of the eddy-currents
${\bm{J}}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}$ for different
values of $\omega$, (a) $\omega=10^{3}$ rad/s, (b) $\omega=10^{5}$ rad/s, (c)
$\omega=10^{7}$ rad/s in a cut through the coin, on the plane spanned by
$\bm{e}_{1}$ and $\bm{e}_{3}$.

In Figure 24, we show a contour plot for the eddy-currents
$\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}$ in cut through
the £1 coin, on a plane spanned by $\bm{e}_{1}$ and $\bm{e}_{3}$ for
$\omega=10^{3}\text{rad/s}$. This figure also includes the field lines for
$\text{Re}({\bm{J}}^{e})$.

Figure 24: British £1 coin: Contours of $|\text{Re}({\bm{J}}^{e})|$ in a cut
through the coin, on the plane spanned by $\bm{e}_{1}$ and $\bm{e}_{3}$, where
${\bm{J}}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}$ are the eddy
currents and showing the field lines corresponding to
$\text{Im}({\bm{J}}^{e})$ with $\omega=10^{3}\text{ rad/s}$.

For the coin models, each of the associated MPT frequency spectra have
independent coefficients that are only associated with diagonal entries of the
tensor. Thus, we set $\sqrt{I_{2}({\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])}=0$ in each case.

###  5.3 Threat items: TT-33 Semi-automatic pistol

In this section, we present the MPT spectral signature characterisations for
components of an exemplar semi automatic pistol. We have chosen the Tokarev
TT-33, shown in Figure 25, which was originally designed in the Soviet Union
in the late 1920’s, with production in the USSR between 1930-1954 [50]. It has
also been produced in other countries including China, Hungary, North Korea,
Pakistan, Romania, Vietnam and Yugoslavia and exported to other nations around
world. It is still used by the Bangladeshi and North Korean armed forces and
the police in Pakistan often carry the pistol as a side arm. Under a different
name, it is occasionally supplied to the police and armed forces in China
[23]. We chose to model this gun due to both it’s simplicity and prevalence in
conflict zones and less economically developed countries with about 1.7
million being produced in total [50].

Figure 25: Image of Tokarev TT-33 from [10] reproduced with permission under
Creative Commons License CC-BY

The starting point for our characterisation are the blueprints for the larger
parts of a non-firing replica [2], which we expect to be sufficient for our
purposes. We have chosen to focus our attention on the receiver (with the
magazine and ammunition removed), the reason being that someone wishing to
disguise an semi-automatic pistol might disassemble the main pieces and carry
them separately through a security control check. As the receiver is one of
the larger components, it should be easiest to identify by a metal detector.
The exact materials of the receiver are not known although it is likely to be
made of a carbon steel alloy such as 1020 or 4140, which has a conductivity of
around $\sigma_{*}=4.5\times 10^{6}\text{ S/m}$ to $\sigma_{*}=6.25\times
10^{6}\text{ S/m}$ [41] and, for our simulations, we have chosen
$\sigma_{*}=6.2\times 10^{6}\text{ S/m}$. Such steels are ferrous and exhibit
a non-linear constitutive relationship between the magnetic flux density
${\bm{B}}$ and the magnetic field ${\bm{H}}$, but, if we restrict ourselves to
low field strengths, where the relationship is linear,
$\mu_{*}=\mu_{r}\mu_{0}=|{\bm{B}}|/|{\bm{H}}|$ and the mathematical model
developed in [6, 28] still applies. Values of $\mu_{r}$ obtained
experimentally for different steels vary enormously (eg from $\mu_{r}=100$ to
$\mu_{r}=600$ or larger) as often
$\mu_{*}=\mu_{r}\mu_{0}=|{\bm{B}}|/|{\bm{H}}|$ is applied when the curve is no
longer straight. Numerical simulations using high values of $\mu_{r}$ become
increasingly challenging and so we have chosen $\mu_{r}=5$.

An extreme simplification of the receiver for TT-33 is to model it as simple
L-shape made up of two rectangular regions glued together (the overall
dimensions of the physical L-shape are $148\text{ mm}\times 17.5\text{
mm}\times 10.1\text{ mm}$). A mesh discretising the L-shape, with overall
dimensions $148\times 17.5\times 101$, and the surrounding region out to a
truncation boundary, in the form of a box of dimensions $[-1000,1000]^{3}$,
was generated with $h$-refinement towards the edges, containing $62\,656$
unstructured tetrahedra with $\alpha=0.001\text{ m}$. We then considered a
sequence of geometric improvements on the basic L-shape model, as shown in
Figure 26, which we refer to as TT-33 with a trigger hole, TT-33 with no
internals, TT-33 without chamfers and TT-33 with chamfers, having
discretisations comprising of between $94\,092$ and $175\,217$ tetrahedral
elements, respectively. With the exception of the L-shape, all contain a model
of the trigger guard (a loop of steel where the trigger would be placed) and
assume that the magazine is removed, as the top-view of TT-33 with trigger
hole shown in Figure 27 illustrates. TT-33 with no internals and TT-33 without
chamfers offer further geometric improvements with TT-33 with chamfers being
the closet to the actual blueprint and includes small holes in the receiver
used to fix the other components of the pistol in place. Only the L-shape is
simply connected with $\beta_{1}(B)=0$, the TT-33 with trigger hole
$\beta_{1}(B)=2$, TT-33 with no internals and TT-33 without chamfers each have
$\beta_{1}(B)=4$ while TT-33 without chamfers has $\beta_{1}(B)=13$ each
object has $\beta_{0}(B)=1$ and $\beta_{2}(B)=0$.

$\begin{array}[]{ccc}\includegraphics[width=115.63188pt,keepaspectratio]{TT-33_L_shape_refined.png}&\includegraphics[width=138.75967pt,keepaspectratio]{TT-33_2_blocks.png}&\includegraphics[width=138.75967pt,keepaspectratio]{TT-33_With_Side_Cut.png}\\\
\text{(a)}&\text{(b)}&\text{(c)}\end{array}$

$\begin{array}[]{cc}\includegraphics[width=138.75967pt,keepaspectratio]{TT-33_No_Chamfer_mesh.png}&\includegraphics[width=152.63493pt,keepaspectratio]{TT-33_Chamfer_mesh.png}\\\
\text{(d)}&\text{(e)}\end{array}$

Figure 26: Set of receiver models for TT-33 pistol: surface distribution of
elements for (a) L-shape domain, (b) TT-33 with a trigger hole, (c) TT-33 with
no internals, (d) TT-33 without chamfers and (e) TT-33 with chamfers Figure
27: Set of receiver models for TT-33 pistol: top view of TT-33 with a trigger
hole

In each case, $N=13$ representative solution snapshots to full order problem
at logarithmically spaced frequencies in the range $8\times
10^{-1}\leq\omega\leq 8\times 10^{8}\text{ rad/s}$ were found to converge by
using $p=4$ elements. Then, by applying the PODP approach described in Section
2.1 with a tolerance of $TOL=10^{-6}$, the MPT spectral signature for each of
the receiver models was obtained. A detailed presentation of the results can
be found in [54]. With the exception of the model with chamfers each of the
models of the receiver has an axis of symmetry in the ${\bm{e}}_{2}$ direction
and so there are $4$ independent coefficients each in
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
$\mathcal{I}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ corresponding to
$(\tilde{\mathcal{R}})_{11}$, $(\tilde{\mathcal{R}})_{22}$,
$(\tilde{\mathcal{R}})_{33}$,
$(\tilde{\mathcal{R}})_{13}=(\tilde{\mathcal{R}})_{31}$ at each frequency with
similar for $\mathcal{I}$. The model with chamfers, which lacks this symmetry,
has $6$ independent coefficients each in $\tilde{\mathcal{R}}$ and
$\mathcal{I}$ (at each frequency), which means that all coefficients of the
tensor are independent of each other.

We show a comparison of the MPT spectra using the principal invariants
$I_{i}$, $i=1,2,3$ for $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and ${\mathcal{I}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ that have been obtained using the PODP approach
in Figure 28. In this figure, we have restricted consideration to frequencies
such that $10^{2}\leq\omega\leq 10^{8}\text{ rad/s}$ in order to allow
comparisons with the earlier key and coin results. In practice, the eddy
current model brakes down at a frequency $\omega_{limit}<2\times 10^{6}\text{
rad/s}$ for all the TT-33 models considered and so, in practice, higher
frequencies are not relevant. While the $I_{i}({\mathcal{I}})$, $i=1,2,3$,
invariants for the TT-33 models are similar to that for the keys and coins,
the behaviour of $I_{i}(\tilde{\mathcal{R}})$, $i=1,2,3$ are quite different
due to $\mu_{r}\neq 1$. For $I_{1}(\tilde{\mathcal{R}})$, we see the curves
are monotonically decreasing with $\log\omega$, but do not asymptote to $0$
for small $\omega$. For $I_{2}(\tilde{\mathcal{R}})$ and
$I_{3}(\tilde{\mathcal{R}})$ we see the curves are neither monotonically
increasing or decreasing with $\log\omega$ and the curves do not asymptote to
$0$ for small $\omega$ motivating that discrimination between the object is
possible. The L-shape exhibits significant differences to the TT-33 models
with a different location of resonant peak in $I_{i}({\mathcal{I}})$,
$i=1,2,3$ and a significantly different behaviour for
$I_{i}(\tilde{\mathcal{R}})$, $i=1,2,3$. The results for the TT–33 with a
trigger hole and TT-33 with no internals models are similar with further
differences for the TT-33 without chamfers and TT-33 with chamfers. However,
all these latter four cases exhibit a resonance peak of around
$\omega=10^{4}\text{rad/s}$ for $I_{i}({\mathcal{I}})$, $i=1,2,3$. The
magnitude of the resonance peak for $I_{3}({\mathcal{I}})$ decreases in
sequence of the associated volume of the different TT-33 models.

The corresponding results obtained for the invariants $J_{i}$, $i=2,3$ for
$\tilde{\mathcal{R}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ and
${\mathcal{I}}[\alpha B,\omega,\sigma_{*},\mu_{r}]$ that have been obtained
using the PODP approach are shown in Figure 29. These results again illustrate
the significant difference between the L-shape and the other models. With the
exception of the L-shape, the results for $J_{2}(\tilde{\mathcal{R}})$ show a
monotonic increase with $\log\omega$, those for $J_{3}(\tilde{\mathcal{R}})$
show a monotonic decrease, $J_{2}({\mathcal{I}})$ and $J_{3}({\mathcal{I}})$
show a single local maximum. The results for TT-33 with and without chamfers
are similar with greater differences exhibited between the TT-33 with no
internals and TT-33 with trigger hole. The results shown in Figures 28 and 29
indicate the significant difference between the spectral signatures of the
L-shape and the more realistic models, which adds the credibility that the MPT
spectral signature makes it possible to distinguish between a carpenters
metallic set-square (which closely resembles an L-shape) and the receiver of a
pistol, for example.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig28RealInvariants_1.pdf}&\includegraphics[scale={0.5}]{Fig28ImagInvariants_1.pdf}\\\
\text{(a) }I_{1}(\tilde{\mathcal{R}})&\text{(b) }I_{1}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig28RealInvariants_2.pdf}&\includegraphics[scale={0.5}]{Fig28ImagInvariants_2.pdf}\\\
\text{(c) }I_{2}(\tilde{\mathcal{R}})&\text{(d) }I_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig28RealInvariants_3.pdf}&\includegraphics[scale={0.5}]{Fig28ImagInvariants_3.pdf}\\\
\text{(e) }I_{3}(\tilde{\mathcal{R}})&\text{(f)
}I_{3}(\mathcal{I})\end{array}$

Figure 28: Set of receiver models for TT-33 pistol: Comparison of tensor
invariants. (a) $I_{1}(\tilde{\mathcal{R}})$, (b) $I_{1}(\mathcal{I})$ (c)
$I_{2}(\tilde{\mathcal{R}})$, (d) $I_{2}(\mathcal{I})$, (e)
$I_{3}(\tilde{\mathcal{R}})$ and (f) $I_{3}(\mathcal{I})$.

$\begin{array}[]{cc}\includegraphics[scale={0.5}]{Fig29RealDeviatoric_2.pdf}&\includegraphics[scale={0.5}]{Fig29ImaginaryDeviatoric_2.pdf}\\\
text{(a)}J_{2}(\tilde{\mathcal{R}})&\text{(b) }J_{2}(\mathcal{I})\\\
\includegraphics[scale={0.5}]{Fig29RealDeviatoric_3.pdf}&\includegraphics[scale={0.5}]{Fig29ImaginaryDeviatoric_3.pdf}\\\
\text{(c) }J_{3}(\tilde{\mathcal{R}})&\text{(d)
}J_{3}(\mathcal{I})\end{array}$

Figure 29: Set of receiver models for TT-33 pistol: Comparison of tensor
invariants. (a) $J_{2}(\tilde{\mathcal{R}})$, (b) $J_{2}(\mathcal{I})$, (c)
$J_{3}(\tilde{\mathcal{R}})$ and (d) $J_{3}(\mathcal{I})$.

For the TT-33 models, each of the associated MPT frequency spectra have
independent coefficients that are associated with both on and off diagonal
entries of the tensor. The behaviour of $\sqrt{I_{2}({\mathcal{Z}}[\alpha
B,\omega,\sigma_{*},\mu_{r}])}$ for the different models is shown in Figure
30.

Figure 30: Set of receiver models for TT-33 pistol: Comparison of the tensor
invariant $\sqrt{I_{2}({\mathcal{Z}})}$.

Finally, in Figure 31 we show the contours of $|\text{Re}({\bm{J}}^{e})|$ and
field lines for $\text{Re}({\bm{J}}^{e})$ on the plane spanned by $\bm{e}_{1}$
and $\bm{e}_{3}$ with $\xi_{2}=0$ and the TT-33 with chamfers model for the
situations where
${\bm{J}}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{1}^{(1)}$,
${\bm{J}}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{2}^{(1)}$,
${\bm{J}}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}$ are the eddy
currents corresponding to $\omega=10^{3}\text{ rad/s}$.

$\begin{array}[]{cc}\includegraphics[scale={0.15}]{TT-33_om_1e3_E1_real_colour_around_E2_axis_E1Vec.jpeg}&\includegraphics[scale={0.15}]{TT-33_om_1e3_E2_real_colour_around_E2_axis_E2Vec.jpeg}\\\
\text{(a)
}\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{1}^{(1)}&\text{(b)
}\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{2}^{(1)}\end{array}$
$\begin{array}[]{c}\includegraphics[scale={0.15}]{TT-33_om_1e3_E3_real_colour_around_E2_axis_E3Vec.jpeg}\\\
\text{(c)
}\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}\end{array}$

Figure 31: Set of receiver models for TT-33 pistol: Contours of
$|\text{Re}({\bm{J}}^{e})|$ and streamlines for $\text{Re}({\bm{J}}^{e})$ on
the plane spanned by $\bm{e}_{1}$ and $\bm{e}_{3}$ with $\xi_{2}=0$ and the
TT-33 with chamfers model, showing (a)
$\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{1}^{(1)}$, (b)
$\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{2}^{(1)}$ and (c)
$\bm{J}^{e}=\mathrm{i}\omega\sigma_{*}\bm{\theta}_{3}^{(1)}$

###  5.4 Threat items: Knives

Knife crime in the U.K. is a persistent issue with 47 000 offences involving a
knife or sharp instrument in England and Wales from April 2018 - March 2019,
285 of which currently recorded as homicide [17] with trend being a
significant increase in the last 8-9 years. The early recognition of threat
objects through metal detection may help to reduce the number of offences
involving a sharp instrument. In this section, we present MPT spectral
signature characterisations for exemplar knife models.

We will consider a set of 5 different knife models, which we name as chef,
cutlet, meat cleaver, Santoku and Wusthof. The chef knife is a model of a
cheap chef knife, featuring a 20cm long, 5cm tall blade with a constant
thickness of 1.5mm and a partial tang 555 The tang is the back portion of the
blade, which extends or connects to a handle, a full tang extends the full
length of the handle while a partial tang only extends partially in to the
handle [53]. This model is an example of a stamped knife would normally be
constructed with a plastic handle and only the cutting edge of the knife would
be sharpened to a point. Obviously, we have only modelled the metallic part of
the knife. The cutlet knife is a model of a cheap cutlet, featuring a 11cm
long, 2cm tall blade with a constant thickness of 1.25mm and a partial tang.
This model is an example of a stamped knife and would normally be constructed
with a plastic handle and only the cutting edge of the knife would be
sharpened to a point. The cleaver is a model of a meat cleaver featuring a
20cm long, 9cm tall blade with a thickness of 3mm over the majority of the
blade with a double bevel 6.4 cm from the spin and a full tang with 3 rivets,
which are each 2cm long, have radius 3mm and are spaced 45mm apart centre to
centre. This model could be made using either the method of stamping or
forging. The Santoku features a 17cm long, 4.5cm tall blade with a thickness
of 1.275mm at its spine, which tapers over the height of the blade to a point
at the cutting edge. It features a full tang with 3 rivets, which are each 2cm
long, have radius 3.5mm and are spaced 42.5mm apart centre to centre. This
model would normally be constructed from a single piece of steel with the two
sides of the handle being made with either wood or a plastic material which
are then both stuck and riveted to the steel. Finally, the Wusthof has a 20cm
long, 5cm tall blade with a thickness of 2mm at its spine, which tapers over
the height of the blade to a point at the cutting edge. It features a full
tang with 3 rivets, which are each 2cm long, have radius 3.5mm and are spaced
42.5mm apart centre to centre. In each case, the measurements quoted have been
obtained by approximately measuring the dimensions of common household knives.
The blade of the knives have been assumed to be made of 440 grade stainless
steel, which has a relative permeability $\mu_{r}=62$ [51] and conductivity
$\sigma_{*}=1.6\times 10^{6}$ S/m[41], but modelled instead with a lower
relative permeability $\mu_{r}=5$, and the rivets to be made of copper, which
is non-magnetic having a relative permeability $\mu_{r}=1$ and a conductivity
$\sigma_{*}=5.8\times 10^{7}$ S/m [41]. Note that each of the knives are
simply connected.

A mesh of each of the geometries was generated assuming dimensionless units,
the size parameter $\alpha=0.001$ m and by placing the knife configuration
centrally in a box of dimensions $[-1000,1000]^{3}$ . The resulting meshes
contain $25\,742$, $14\,935$, $55\,226$, $55\,226$ and $79\,945$ unstructured
tetrahedra for the chef, cutlet, meat cleaver, Santoku and Wusthof knives,
respectively, and images of the distribution of elements on the surface of the
object are reproduced in Figure 32. Each of the knives has been orientated so
that the blade is parallel to the ${\bm{e}}_{1}$ direction and lies in the
plane spanned by ${\bm{e}}_{1}$ and ${\bm{e}}_{2}$ with the knife
configuration being symmetrical in the ${\bm{e}}_{3}$ direction. Thus, there
are $4$ independent coefficients each in $\tilde{\mathcal{R}}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ and $\mathcal{I}[\alpha
B,\omega,\sigma_{*},\mu_{r}]$ corresponding to $(\tilde{\mathcal{R}})_{11}$,
$(\tilde{\mathcal{R}})_{22}$, $(\tilde{\mathcal{R}})_{33}$,
$(\tilde{\mathcal{R}})_{13}=(\tilde{\mathcal{R}})_{31}$ at each frequency with
similar for $\mathcal{I}$.

$\begin{array}[]{ccc}\includegraphics[width=138.75967pt,keepaspectratio]{Cheap_Chef_Angle}&\includegraphics[width=138.75967pt,keepaspectratio]{Cheap_Cutlet_Angle}&\includegraphics[width=138.75967pt,keepaspectratio]{Cleaver_Angle}\\\
\text{(a)}&\text{(b)}&\text{(c)}\end{array}$

$\begin{array}[]{cc}\includegraphics[width=138.75967pt,keepaspectratio]{Santoku_Angle}&\includegraphics[width=138.75967pt,keepaspectratio]{Wusthof_Angle}\\\
\text{(d)}&\text{(e)}\end{array}$

Figure 32: Set of knives: surface distribution of elements for (a) chef, (b)
cutlet, (c) meat cleaver, (d) Santoku and (e) Wusthof

The results obtained by using $N=13$ representative full order solution
snapshots at logarithmically spaced frequencies in the range
$10^{1}\leq\omega\leq 10^{10}$ rad/s were found to converge with $p=4$
elements. Then, by applying the PODP algorithm described in Section 2.1 with a
tolerance of $TOL=10^{-6}$, the MPT spectral signature for each of the
receiver models was obtained. A detailed presentation of the results can be
found in [54]. We show a comparison of the MPT spectra using the principal
invariants $I_{i}$, $i=1,2,3$ for $\tilde{\mathcal{R}}[\alpha
B,\omega,\mu_{r},\sigma_{*}]$ and ${\mathcal{I}}[\alpha
B,\omega,\mu_{r},\sigma_{*}]$ that have been obtained using the PODP approach
in Figure 33. In this figure, we have restricted consideration to frequencies
such that $10^{2}\leq\omega\leq 10^{8}\text{rad/s}$ in order to allow
comparisons with the earlier results. In practice, the eddy current model