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We introduce a new type of object characterisation, which is capable ofaccurately describing small isolated inclusions for potential field inverseproblems such as in electrostatics, magnetostatics and related low frequencyMaxwell problems. Relevant applications include characterising ferrousunexploded ordnance (UXO) from magnetostatic field measurements inmagnetometry, describing small conducting inclusions for medical imaging usingelectrical impedance tomography (EIT), performing geological ground surveysusing electrical resistivity imaging (ERT), characterising objects byelectrosensing fish to navigate and identify food as well as describing theeffective properties of dilute composites. Our object characterisation buildson the generalised polarizability tensor (GPT) object characterisation conceptand provides an alternative to the compacted GPT (CGPT). We call the newcharacterisations harmonic GPTs (HGPTs) as their coefficients correspond toproducts of harmonic polynomials. Then, we show that the number of independentcoefficients of HGPTs needed to characterise objects can be significantlyreduced by considering the symmetry group of the object and propose asystematic approach for determining the subspace of symmetric harmonicpolynomials that is fixed by the group and its dimension. This enable us todetermine the independent HGPT coefficients for different symmetry groups.

Minimal Object Characterisations using Harmonic Generalised Polarizability Tensors and Symmetry Groups