Prof. Dr. Paul Ledger talk
Date: 17.04.19 Time: 16.15 - 17.45 Room: Y27H25
Locating and identifying hidden conducting objects has a range of important applications in metal detection including searching for buried treasure, identifying landmines and in the early detection of concealed terrorist threats. There is a need to distinguish between multiple objects,for example, in benign situations, such as coins and keys accidentally left in a pocket during a security search or a treasure hunter becoming lucky and discovering a hoard of Roman coins, as well as in threat situations, where the risks need to be clearly identified from the background clutter. Furthermore, objects are also often inhomogeneous and made up of several different metals. For instance, the barrel of a gun is invariably steel while the frame could be a lighter alloy, jacketed bullets have a lead shot and a brass jacket and modern coins often consist of a cheaper metal encased in nickel or brass alloy. Thus, in practical metal detection applications, it is important to be able to characterise both multiple objects and inhomogeneous objects. Traditional approaches to the metal detection involve determining the conductivity and permeability distributions in the eddy current approximation of Maxwell's equations and lead to an ill-posed inverse problem. On the other hand, practical engineering solutions in hand held metal detectors use simple thresholding and are not able to discriminate between small objects close to the surface and larger objects buried deeper underground. In this talk, an alternative approach in which prior information about the form of the conducting permeable object has been introduced will be discussed. Ammari, Chen, Chen, Garnier and Volkov  have obtained the leading order term in an asymptotic expansion of the perturbed magnetic field, due to the presence of a homogeneous conducting permeable object, as the object size tends to zero. This expansion separates the object's position from its shape and material description, offering considerable advantages in case of isolated objects. We have shown that this leading order term simplifies for orthonormal coordinates and results in a characterisation of a conducting permeable object by a complex symmetric rank 2 magnetic polarizability tensor (MPT) for the eddy current case . Interestingly, the MPT is different to the symmetric rank 2 Poyla-Szegö polarizability tensor (also known as the Poyla-Szegö polarisation tensor) that is known to characterise small permeable objects in magnetostatics and small conducting objects in electrical impedance tomography . For instance, computing the coefficients of the MPT rely on the solution of vectorial curl-curl transmission problems while the latter on simpler scalar transmission problems. The topology of an object plays an interesting role in the MPT coefficients. Including more terms in the asymptotic expansion increases the accuracy of the representation of the perturbed field. The higher order terms contain higher order tensors, which provide more information about the shape and material parameters of an object, and can help to improve object identification. Complete asymptotic expansions of the perturbed field for the electrical impedance tomography problem have been obtained by Ammari and Kang  and provide a complete characterisation of an object by generalised polarisation tensors. For the eddy current case, we have extended the leading order term obtained in  to a complete expansion of the perturbed magnetic field. The higher order terms in our expansion characterise a conducting permeable object in terms of a new class of generalised magnetic polarizability tensors , of which the rank 2 MPT is the simplest case. Practical metal detection problems contain multiple and inhomogeneous objects. We have also provided an extension of [1, 2] to characterise multiple inhomogeneous objects, including objects that are closely spaced, in terms of MPTs . The talk will review recent work on MPTs and explore the interesting properties exhibited by these tensors for shapes and topologies of objects. The role that a dictionary of MPTs for different shaped objects can play in object classification will also be described. References  H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov, Target detection and characterization from electromagnetic induction data, J. Math Pure Appl. 2014: 101: 54-75.  P.D. Ledger and W.R.B. Lionheart, Characterising the shape and material properties of hidden targets from magnetic induction data, IMA J. Appl. Math. 2015: 80: 1776-1798.  H. Ammari and H. Kang, Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, New-York, Springer, 2007.  P.D. Ledger and W.R.B. Lionheart, Generalised magnetic polarizability tensors, Math. Meth. Appl. Sci, 2018: 41: 3175-3196.  P.D. Ledger, W.R.B. Lionheart and A.A.S. Amad, Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors, Math. Meth. Appl. Sci, 2019:42: 830-860.