Efficient computation of the magnetic polarizabiltiy tensor spectral signature using proper orthogonal decomposition

Abstract

The identification of hidden conducting permeable objects from measurements of the perturbed magnetic field taken over a range of low frequencies is important in metal detection. Applications include identifying threat items in security screening at transport hubs, location of unexploded ordnance, and antipersonnel landmines in areas of former conflict, searching for items of archeological significance and recycling of valuable metals. The solution of the inverse problem, or more generally locating and classifying objects, has attracted considerable attention recently using polarizability tensors. The magnetic polarizability tensor (MPT) provides a characterization of a conducting permeable object using a small number of coefficients, has an explicit formula for the calculation of their coefficients, and a well understood frequency behavior, which we call its spectral signature. However, to compute such signatures, and build a library of them for object classification, requires the repeated solution of a transmission problem, which is typically accomplished approximately using a finite element discretization. To reduce the computational cost, we propose an efficient reduced order model (ROM) that further reduces the problem using a proper orthogonal decomposition for the rapid computation of MPT spectral signatures. Our ROM benefits from a posteriori error estimates of the accuracy of the predicted MPT coefficients with respect to those obtained with finite element solutions. These estimates can be computed cheaply during the online stage of the ROM allowing the ROM prediction to be certified. To further increase the efficiency of the computation of the MPT spectral signature, we provide scaling results, which enable an immediate calculation of the signature under changes in the object size or conductivity. We illustrate our approach by application to a range of homogenous and inhomogeneous conducting permeable objects.

Identification of metallic objects using spectral magnetic polarizability tensor signatures: Object characterisation and invariants

“…which bound the accuracy of the real and imaginary parts of the PODP MPT prediction with respect to the full-order solution, have been derived 6 and can be computed at low-computational cost during the online stage of the reduced order model for each = m , m = 1, … , 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 any confusion may arise.…”

“…[2][3][4][5] The behavior of the MPT's coefficients as a function of frequency, known as its spectral signature, has been studied theoretically 5 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. 6 These computations employ the established Open Source finite element (FE) package, NGSolve, and the recently derived alternative explicit expressions formulas for the MPT coefficients. 5 The use of NGSolve 7,8 ensures that the solutions to the underlying full-order (eddy current type) transmission problems are accurately computed using high-order H(curl) conforming (high-order edge element) discretizations (see References 9-11 and references therein) and the POD technique ensures their rapid computation over sweeps of frequency.…”

“…The magnetic polarizability tensor (MPT) has been shown to offer an economical characterisation of conducting permeable objects and explicit formulas for the computation of its six 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 2‐5 . The behavior of the MPT's coefficients as a function of frequency, known as its spectral signature, has been studied theoretically 5 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 6 . These computations employ the established Open Source finite element (FE) package, NGSolve, and the recently derived alternative explicit expressions formulas for the MPT coefficients 5 .…”

“…Included in this approach 6 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. The advantages of using this approach to compute MPT coefficients over using commercial software, in terms of accuracy, computational efficiency and applicability to wide set of applications, has previously been discussed and demonstrated in References 3,4,6,12.…”

“…With the exception of References 13,14, all previous studies used measured MPT spectral signature information to build the classifier. As pointed out in Reference 6, 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 general noise (e.g., from amplifiers, parasitic voltages and filtering) 23 .…”