Generalised Polarisation Tensors for Maxwell's Equations

Lead Research Organisation: Swansea University
Department Name: College of Engineering


The detection of hidden targets from measurements of electromagnetic fields has many important applications, which include improving food safety by locating foreign objects, using medical imaging to find cancerous tumours, detecting landmines and weapons as well as in the application of geological surveys to find new aquifers. Depending on the application, the measured electromagnetic fields might relate to low frequency magnetic induction, medium frequency radio wave propagation, microwave radiation or higher frequency visible light, ultraviolet and x-rays. To determine the shape, location and material properties of hidden target from the measured electromagnetic fields requires the solution of the "Maxwell inverse problem". This process is mathematically and computationally challenging.

In this project, we will assemble a team of international scientists, mathematicians and engineers who will develop new mathematical results that describe how electromagnetic fields behave at different distances from the target across a range of frequencies. This, in turn, will lead to the individual classification of targets using a mathematical construction called a "generalised polarisation tensor". These results will open up possibilities for the next generation of inverse solution algorithms, which are capable of the rapid and accurate detection of hidden targets from electromagnetic field measurements.

Planned Impact

Our proposed feasibility study has, in the longer term, the possibility to bring benefits to a range of beneficiaries in the UK and beyond.

In the short term, our project will bring academic benefits to applied mathematicians, physicists and engineers, working in the field of computational and theoretical electromagnetism and inverse problems, in the form of new theoretical developments. These theoretical developments will lead to possibilities for new forms of low-cost imaging that can be exploited by engineers, medical physicists and geophysicists involved in the design and development of new and improved forms of metal detectors, medical imaging equipment and geophysical survey equipment. We will ensure that the academic beneficiaries benefit from our research by the publication of papers and paying any publication costs to make these freely available to all Universities and other Research Organisations.

Our feasibility study will also bring rapid societal and economic benefits. It will bring immediate benefits to the "Find a Better Way" Charity, which is devoted to accelerating the detection and safe removal of landmines. This will be achieved by providing a more complete theoretical basis to the practical studies they are funding as part of the Scanning Electromagnetic Mine Imaging System (SEMIS) being developed by Professor Petyon at The University of Manchester. It will bring immediate benefits to companies specialising in the development of scanning equipment, for ensuring safety and security at public events and airports, companies developing metal detectors for food safety applications, organisations involved in the development of novel medical imaging modularities as well as those developing geophysical survey equipment. We will ensure they benefit from our research by engaging representatives from the groups in future proposals that will build on the success of this feasibility study.

In the medium term, this feasibility study, together with our proposed future research projects, will help to inform members of the public sector, such as policy makers, making decision over investments in new technologies for security and anti-terrorism measures at public events and airports, and health care managers, making decisions over investments in new medical technologies for medical technologies. It will also assist managers of companies involved in food production and preparation making decisions over new technologies for improving food safety and managers of water companies making decisions over new technologies for the prospecting of ground water aquifers.

In the longer term, the general public will benefit from the research initiated in this feasibility study. Examples of potential benefits include new techniques for the accelerated detection and safe removal of land mines, improvements to screening techniques that lead to greater safety and security from acts of violence and terrorism at public events and airports, improvements to medical diagnosis, through new technological developments, improvements to food safety, by reducing the risk of foreign objects through better detection techniques, and reductions in water shortages, through the improved prospecting of ground water aquifers.
Description The key achievements of this project to date include:

1) The development of asymptotic formulae that describe the scattering from dielectric and conducting objects at distances that are large compared to the object's size. The leading order terms we have obtained are expressed in terms of the Polya-Szego polarization tensor, which include information about the shape and material properties of the inclusion.

2) Providing a rigorous mathematical justification for the engineering prediction in magnetic induction that the perturbed magnetic field due to the presence of a general conducting object placed in a low frequency background field can be expressed in terms of a rank 2 complex symmetric tensor, which contains information about the shape and material properties of the object. To show this, a recently derived asymptotic formula for the perturbed magnetic field due to the presence of a conducting object, which is expressed in terms of a new class of rank 4 polarization tensors (H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov Target detection and characterization from electromagnetic induction data, Journal de Mathe ´matiques Pures et Applique ´es (2014), 101:54-75.) was applied. We have been able to show that the rank 4 tensor, in their leading order term, does in fact reduce to a new complex symmetric rank 2 tensor, thus providing a solid theoretical foundation for the engineering prediction. Moreover, the reduction in rank means that only 6 complex independent coefficients are required, which has important consequences for engineers wishing to characterize conducting objects in applications such as landmine detection, event security and food safety. For objects with rotational and mirror symmetries we show that the number of coefficients is still smaller.

3) We have developed new results that describe the low and high frequency response of the polarization tensor coefficients for magnetic induction. We have found that the topology of the object plays an important role in the tensor coefficients at the upper of these extremes. Whereas at the former the form of the tensor implies to the Poyla-Szego tensor. Such results are useful as they can be used to distinguish between permeable and conducting objects and to distinguish information about an object shape (e.g. whether it is a coin or a ring).

4) We have established links with world leaders in development of asymptotic expansions involving polarization tensors including Professors Kang (Department of Mathematics, Inha University, South Korea) Ammari (Department of Mathematics and its Applications, Ecole Normale Superieure, France) and Volkov (Worcester Polytechnic Institute, USA). Receiving invitations to present our work in high profile international conferences, such as the Applied Inverse Problems Conference in South Korea in 2013, British Applied Mathematics Colloquium in 2015 and the IMA Conference on Numerical Simulation in 2015..
Exploitation Route We have already made significant efforts to communicate our findings to the wider academic community. Specifically, we submitted two papers to highly regarded journals in the applied mathematics and electrical engineering communities. Two articles have already appeared in the IMA Journal of Applied Mathematics and a further article has been accepted by IEEE Transactions on Magnetics to widen the impact of the research. We have presented the results of our research to the applied mathematics and engineering communities both nationally and internationally at the following conferences MAFELAP 2013, Brunel University London, Applied Inverse Problems 2013, Daejeon South Korea, British Applied Mathematics Colloquium, Cardiff University 2014, British Applied Mathematics Colloquium 2015, The University of Cambridge, IMA Conference on Numerical Simulation, The University of Oxford, 2015. We have engaged with the engineering group of Professor Peyton and comparison of our computed polarisation tensors with his field measurements are underway. We have been successful in gaining further EPSRC support to further the research started in this feasibility study and in the new project starting in 2018 we will engage with a range of industrial representations in order to promote and widen the impact of our research.
Sectors Aerospace, Defence and Marine,Healthcare,Security and Diplomacy

Description In this small mathematical sciences project, we have obtained new mathematical results that provide an explicit expression for the magnetic polarizability tensor and obtained results that describe the behaviour of the tensor coefficients under low and high frequencies. These results helped provide the basis for the larger 3 year EPSRC funded project Reducing the threat to public safety: Improved object characterisation, location and detection project, which has gone to provide a new provide a new mathematical theory for metal detection and develop new algorithms and software for identifying hidden threat objects for security applications. The mathematical results developed have also benefited industry and practitioners, in particular through the software that has been subsequently developed in the latter project, EPSRC funded project Reducing the threat to public safety: Improved object characterisation, location and detection, which builds on the work in this project, to compute the new characterisations for realistic object geometries. This software is open source and has also led to open data sets. The software has been promoted to project partners and to the wider industry through an impact event, documentation and training videos.
First Year Of Impact 2015
Sector Aerospace, Defence and Marine,Manufacturing, including Industrial Biotechology,Security and Diplomacy
Impact Types Societal,Economic

Description Responsive Mode
Amount £648,740 (GBP)
Funding ID EP/R002134/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 01/2018 
End 12/2020
Title hpfem-mpt 
Description This is a MATLAB program for numerically computing the rank 2 magnetic polarizability tensor (MPT) using the hp-finite element method (hpfem) for different shaped objects. It is designed as an educational and research tool for engineers, mathematicians and physicists working both academia and industry and it is hoped those interested in characterising conducting permeable objects will find it useful. 
Type Of Technology Software 
Year Produced 2017 
Open Source License? Yes  
Impact This program was used to numerically generate MPTs for a range of different objects and the results have been presented in Journal articles in the IMA Journal of Applied Mathematics and IEEE Transactions on Magnetics in 2015 and 2016, respectively. The results have also been presented in a range of conferences.