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Spheroidal First Order Polarization Tensor (SFOPT) Toolkit
Abstract
The polarization tensor (PT) has been utilized in numerous applications involving electric and electromagnetic fields, such as metal detection, landmine detection, and electrical imaging. In these applications, the PT is implemented in the process of identifying object, where the object can be represented in the form of the first order PT. Thus, it is crucial to have an application that provides easy access to calculate the first order PT for the object. However, the existing application in the literature has some limitations, focusing solely on computing the first order PT for a prolate spheroid with semi axes and an oblate spheroid with semi axes . Therefore, the purpose of this study is to invent a graphical user interface (GUI) for the Spheroidal First Order Polarization Tensor (SFOPT) Toolkit which facilitates efficient computations and visualizations related to spheroids. The SFOPT Toolkit will be developed by using App Designer in MATLAB. The SFOPT interface integrates four essential functions: computation of the first order PT for a spheroid, classification of spheroidal types, three-dimensional visualization of spheroids, and determination of semi axes from the computed first order PT. Through illustrative examples, we demonstrate the effectiveness and versatility of the SFOPT framework, offering insights into its practical utility and potential applications in diverse fields. The reliability of the toolkit is also presented, and the findings shows the error of computations are small. The toolkit is a user-friendly application since the users can easily access it by downloading the application instead of running the coding themselves. Moreover, this toolkit can be a reference for researchers to compute the first order PT for a spheroid and determine the semi axes (size) of the spheroid from the given first order PT.
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The magnetic polarizability tensor (MPT) is an economical characterisation of a conducting magnetic object, which can assist with identifying hidden targets in metal detection. The MPT’s coefficients depend on multiple parameters of interest including the object shape, size, electrical conductivity, magnetic permeability, and the frequency of excitation. The computation of the coefficients follow from post-processing an eddy current transmission problem solved numerically using high-order finite elements. To reduce the computational cost of constructing these characterisations for multiple different parameters, we compare three methods by which the MPT can be efficiently calculated for two-dimensional parameter sets, with different levels of code invasiveness. We compare, with numerical examples, a neural network regression of MPT eigenvalues with a projection-based reduced order model (ROM) and a neural network enhanced ROM (POD–NN) for predicting MPT coefficients.
We present new alternative complete asymptotic expansions for the time harmonic low‐frequency magnetic field perturbation caused by the presence of a conducting permeable object as its size tends to zero for the eddy current approximation of the Maxwell system. Our new alternative formulations enable a natural extension of the well‐known rank 2 magnetic polarizability tensor (MPT) object characterization to higher order tensor descriptions by introducing generalized MPTs (GMPTs) using multi‐indices. In particular, we identify the magnetostatic contribution, provide new results on the symmetries of GMPTs, derive explicit formulae for the real and imaginary parts of GMPT coefficients and also describe the spectral behavior of GMPT coefficients. We also introduce the concept of harmonic GMPTs (HGMPTs) that have fewer coefficients than other GMPT descriptions of the same order. We describe the scaling, translation and rotational properties of HGMPTs and describe an approach for obtaining those HGMPT coefficients that are invariant under the action of a symmetry group. Such an approach is one candidate for selecting features in object classification for hidden object identification using HGMPTs.
In order to enhance identification of objects in electrical imaging or metal detection, the polarization tensor is used to characterize the perturbation in electric or electromagnetic field due to the presence of the conducting objects. This is similar as describing the uniform fluid flow that is disturbed after a solid is immersed in the fluid during the study of fluid mechanics. Moreover, in some applications, it is beneficial to determine a spheroid based on the first order polarization tensor in order to understand what is actually represented by the tensor. The spheroid could share similar physical properties with the actual object represented by the polarization tensor. The purpose of this paper is to present how scaling on the matrix for the first order polarization tensor will affect the original spheroid represented by that first order polarization tensor. In the beginning, we revise the mathematical property regarding how scaling the semi axes of a conducting spheroid has an effect to its first order polarization tensor. After that, we give theoretical results with proofs on how scaling the matrix for the first order polarization tensor affects the volume and semi axes of the spheroid. Following that, some numerical examples are provided to further justify the theory. Here, different scalar factors will be used on the given first order polarization tensor before the new volume and semi axes of the spheroid are computed. In addition, we also investigate how the size of the scale on the first order polarization tensor influence the accuracy of computing the related volume and semi axes. In this case, it is found that a large error could occur to the volume and the semi axes when finding them by solving the first order PT with that has being scaled by a very large scaling factor or a too small scaling factor. A suggestion is then given on how to reduce the errors.
Polarization tensor (PT) has been widely used in engineering application, particularly in electric and magnetic field areas. In this case, suitable method must be employed in the evaluation of PT in order to make sure that the tensor obtained is higher in its accuracy. Our aim in this paper is to provide simple and easy implemented method to compute first order PT which is based on Gaussian quadrature numerical integration involving linear interpolation. This study provides the comparison between two different orders of Gaussian quadrature, which is order one and order three. The numerical technique of higher order Gaussian quadrature as PT being calculated offer more accurate and higher in its convergence. Relative error for PT is calculated by using provided analytical solution and higher order Gaussian quadrature gives high accuracy and convergence compared to lower order Gaussian quadrature. In this study, the validation of the results obtained by using our proposed method is provided by comparing it with the analytical solution derived from previous researcher. We illustrate the behavior of a tensor of a sphere and ellipsoid with graphical representation.
The detection of small metallic objects buried in mineralized soil poses a challenge for metal detectors, especially when the response from the metallic objects is orders of magnitude below the response from the soil. This article describes a new, handheld, detector system based on magnetic induction spectroscopy (MIS), which can be used to detect buried metallic objects, even in challenging soil conditions. Experimental results consisting of 1669 passes across either buried objects or empty soil are presented. Fourteen objects were buried at three different depths in three types of soil including nonmineralized and mineralized soils. A novel processing algorithm is proposed to demonstrate how spectroscopy can be used to detect metallic objects in mineralized soils. The algorithm is robust across all types of soil, objects, and depths used in this experiment and achieves a true positive rate over 99% at a false-positive rate of less than 5%, based on just a single pass over the object. It has also been shown that the algorithm does not have to be trained separately for each soil type. The data gathered in the experiment are also published to enable more research on the processing algorithms for MIS-based detectors.
The magnetic polarizability tensor (MPT) has attracted considerable interest due to the possibility it offers for characterizing conducting objects and assisting with the identification and location of hidden targets in metal detection. An explicit formula for its calculation for arbitrary-shaped objects is missing in the electrical engineering literature. Furthermore, the circumstances for the validity of the magnetic dipole approximation of the perturbed field, induced by the presence of the object, are not fully understood. On the other hand, in the applied mathematics community, an asymptotic expansion of the perturbed magnetic field has been derived for small objects and a rigorous formula for the calculation of the MPT has been obtained. The purpose of this paper is to relate the results of the two communities, to provide a rigorous justification for the MPT, and to explain the situations in which the approximation is valid.
This article present the manual to determine ellipsoids that has the same first order polarization tensor to any conducting objects included in electrical field. Given the first order polarization tensor for an object at specified conductivity, the analytical formula of the first order polarization tensor for ellipsoid in the integral form is firstly expressed as system of nonlinear equation by the trapezium rule. We will then discuss how the derived equations are simultaneously solved by appropriated numerical method to uniquely compute all semi principal axes of the ellipsoid. Few examples to use the proposed technique in this study are also provided in three different situations. In each case, the first order polarization tensor for the obtained ellipsoid can be calculated back from the analytical formula to examine the effectiveness of the method.
Weakly electric fish generate electric current and use hundreds of voltage sensors on the surface of their body to navigate and locate food. Experiments (von der Emde and Fetz 2007 J. Exp. Biol. 210 3082–95) show that they can discriminate between differently shaped conducting or insulating objects by using electrosensing. One approach to electrically identify and characterize the object with a lower computational cost rather than full shape reconstruction is to use the first order polarization tensor (PT) of the object. In this paper, by considering experimental work on Peters' elephantnose fish Gnathonemus petersii, we investigate the possible role of the first order PT in the ability of the fish to discriminate between objects of different shapes. We also suggest some experiments that might be performed to further investigate the role of the first order PT in electrosensing fish. Finally, we speculate on the possibility of electrical cloaking or camouflage in prey of electrosensing fish and what might be learnt from the fish in human remote sensing.
This paper presents an inversion procedure to estimate the location and magnetic polarizability tensor of metal targets from broadband electromagnetic induction (EMI) data. The solution of this inversion produces a spectral target signature, which may be used in identifying metal targets in landmines from harmless clutter. In this process, the response of the metal target is modelled with dipole moment and fitted to planar EMI data by solving a minimization least squares problem. A computer simulation platform has been developed using a modelled EMI sensor to produce synthetic data for inversion. The reconstructed tensor is compared with an assumed true solution estimated using a modelled tri-axial Helmholtz coil array. Using some test examples including a sphere which has a known analytical solution, results show the inversion routine produces accurate tensors to within 12% error of the true tensor. A good convergence rate is also demonstrated even when the target location is mis-estimated by a few centimeters. Having verified the inversion routine using finite element modelling, a swept frequency EMI experimental setup is used to compute tensors for a set of test samples representing examples of metallic landmine components and clutter for a broadband range of frequencies (kHz to tens of kHz). Results show the reconstructed spectral target signatures are very distinctive and hence potentially offer an efficient physical approach for landmine identification. The accuracy of the evaluated spectra is similarly verified using a uniform field forming sensor.
Walk-through metal detectors are used at check points for preventing personnel and passengers from carrying threatening metallic objects, such as knives and guns, into a secure area. These systems are capable of detecting small metallic items, such as handcuff keys and blades, but are unable to distinguish accurately between threatening objects and innocuous items. This paper studies the extent to which a K-nearest-neighbour classifier can distinguish various kinds of metallic objects, such as knives, shoe shanks, belts and containers. The classifier uses features extracted from the magnetic polarizability tensor, which represents the electromagnetic properties of the object. The tests include distinguishing threatening objects from innocuous ones, classifying a set of objects into 13 classes, and distinguishing between several similar objects within an object class. A walk-through metal detection system is used as source for the test data, which consist of 835 scans and 67 objects. The results presented show a typical success rate of over 95% for recognizing threats, and over 85% for correct classification. In addition, we have shown that the system is capable of distinguishing between similar objects reliably. Overall, the method shows promise for the field of security screening and suggests the need for further research.
This paper describes the development of a large-scale multi-coil arrangement capable of characterizing the Magnetic Polarizability Tensor (MPT) of large threat objects such as firearms. The system has been applied to the measurement of a comprehensive range of weapons made available by the National Firearms Centre of the UK. For comparison, a number of non-threat items such as metallic belt buckles, keys and coins have also been characterized. Clear differences in the magnitude and spectroscopic response of the MPT data for different firearm types and non-threat items are presented. The application of unsupervised Machine Learning (ML) algorithms to MPT data of threat and non-threat objects enable a better understanding of target object classification. The presented results are encouraging as they demonstrate the ability of the MPT used in combination with the adopted classification algorithms to robustly discriminate between threat and non-threat objects.
This paper studies the use of electromagnetic induction in localization of wireless capsule endoscopes (WECs). There is still currently a need for an accurate localization system to enable localizing possible findings in the gastrointestinal tract, and to develop an active steering system for the capsule. Developing an optimal localization system requires the sensitivity of the system to be analyzed. In this paper, three different coil geometries are modelled with a computer simulation platform, and their sensitivities and target responses are compared. In order to do that, a formulation for the sensitivity based on the dipole model approximation is presented. The first coil array is based on literature and is used as a reference. The second array presents how having more transmit-receive channels in the array effects the sensitivity. The third coil array simulates the effect of increasing the field excitation intensity in different directions by using a three-axial Helmholtz array. In addition, both proposed coil arrays utilize larger coils than the reference. As a result, it seems that both increasing the coil size and the number of field projections interrogating the target increase the overall sensitivity in the region of interest and the target response. The findings suggest that an optimal coil array could utilize both large coils and multiple transmit-receive channels to increase the number of independent fields incident onto the target.
Polarization Tensor (PT) has been widely used in some of the applications of electric and electromagnetic such as electrical imaging, metal detection and electrosensing fish. Furthermore, in these applications, polarization tensors can capture significant information such as material, shape and orientation about the related objects (medical images in electrical imaging or metallic target in metal detection). Some physical information about the unknown objects can then be characterized from the given first order polarization tensor that representing the object. Therefore, it is beneficial to determine an ellipsoid based on the given first order polarization tensor due to the possible similarities between the ellipsoid and the unknown object. The main objective of this study is to present a method in order to determine the semi axes of the spheroid, which is an ellipsoid with two identical axes. The method is an extension of the previous method which is only applicable to two types of spheroid. Using the rotation of the first order polarization tensor, we will show that this extended method can be used to determine all semi axes for any types of spheroid, depending on the given first order PT. After that, some numerical examples are provided specifically to compare between the first order PT for the spheroid with the given first order PT for verifying the results obtained. It is expected that the computed first order PT for the spheroid will be almost similar to the given first order PT.
Polarization tensor (PT) is a classical terminology in fluid mechanics and theory of electricity that can describe geometry in a specific boundary domain with different conductivity contrasts. In this regard, the geometry may appear in a different size, and for easy characterizing, the usage of PT to identify particular objects is crucial. Hence, in this paper, the first order polarization tensor for different types of object with a diverse range of sizes are presented. Here, we used three different geometries: sphere, ellipsoid, and cube, with fixed conductivity for each object. The software Matlab and Netgen Mesh Generator are the essential mathematical tools to aid the computation of the polarization tensor. From the analytical results obtained, the first order PT for sphere and ellipsoid depends on the size of both geometries. On the other hand, the numerical investigation is conducted for the first order PT for cube, since there is no analytical solution for the first order PT related to this geometry, to further verify the scaling property of the first order PT due to the scaling on the size of the original related object. Our results agree with the previous theoretical result that the first order polarization tensor of any geometry will be scaled at a fixed scaling factor according to the scaling on the size of the original geometry.
Polarization Tensor (PT) has many applications in science and engineering. It can be used to improve reconstruction of image in medical imaging, improve metal detection for security screening or landmine clearance and also to study electrosensing fish. In some applications, it is useful to determine an ellipsoid that has the same PT with some other objects and this is possible because the explicit formula of the PT for ellipsoid can be set equal to some given PT and then solve to obtain all three axes of the ellipsoid. Therefore, in this study, based on a given first order PT, we will discuss a new method to determine the two distinct semi axes of a spheroid (an ellipsoid with two equal axes) so that the first order PT for the spheroid is similar to the given first order PT. In order to achieve this, we rely on some results about the depolarization factors for ellipsoid established in our previous study. Our new proposed method has suggested that the semi axes of the spheroid can be determined almost analytically from the explicit formula of the semi axes derived from the first order PT for spheroid where, a few examples have also been given here. The numerical results in the examples have also shown that the first order PT for a few spheroids computed based on the values of the semi axes are similar to the original given first order PT.
The magnetic dipole polarizability tensor is an object-specific property possessing information about the size, shape, and material. This information could be used by electromagnetic induction sensors typically used for demining operations to discriminate between buried mines and clutter, reducing false alarm rates, and improving demining throughput and safety. This paper presents a methodology capable of obtaining the spectroscopic tensors of small metallic objects and low metal anti-personnel mine surrogates. The experimental results are validated against simulated and analytical solutions to ensure that the obtained tensor truly represents the absolute object tensor. Absolute tensors for a number of typical clutter items and mine surrogates are presented, with significant variance observed between those of mine and clutter.
This book presents important recent developments in mathematical and computational methods used in impedance imaging and the theory of composite materials. The methods involved come from various areas of pure and applied mathematics, such as potential theory, PDEs, complex analysis, and numerical methods. The unifying thread in this book is the use of generalized polarization and moment tensors.
The main approach is based on modern layer potential techniques. By augmenting the theory with interesting practical examples and numerical illustrations, the exposition brings simplicity to the advanced material. An introductory chapter covers the necessary basics.
With its extensive list of references and open problems, the book should enhance accessibility to specialized literature and stimulate progress in the fields of impedance imaging and composite materials. Graduate students and researchers in applied mathematics will benefit from this book. Researchers in engineering and physics might also find this book helpful.
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