Properties of Generalised Magnetic Polarizability Tensors

TL;DR

This paper develops higher-order generalized magnetic polarizability tensors (GMPTs) for better characterizing conducting permeable objects in low-frequency magnetic fields, with explicit formulas, symmetry properties, and potential for object classification.

Contribution

It introduces a comprehensive framework for higher-order GMPTs, including explicit formulas, symmetry analysis, and invariant features for object identification.

Findings

Derived explicit formulas for GMPT coefficients.

Identified symmetry properties of GMPTs.

Proposed invariant features for object classification.

Abstract

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 characterisation to higher order tensor descriptions by introducing generalised 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 behaviour 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.