We introduce a new type of object characterisation, which is capable of accurately describing small isolated inclusions for potential field inverse problems such as in electrostatics, magnetostatics and related low frequency Maxwell problems. Relevant applications include characterising ferrous unexploded ordnance (UXO) from magnetostatic field measurements in magnetometry, describing small conducting inclusions for medical imaging using electrical impedance tomography (EIT), performing geological ground surveys using electrical resistivity imaging (ERT), characterising objects by electrosensing fish to navigate and identify food as well as describing the effective properties of dilute composites. Our object characterisation builds on the generalised polarizability tensor (GPT) object characterisation concept and provides an alternative to the compacted GPT (CGPT). We call the new characterisations harmonic GPTs (HGPTs) as their coefficients correspond to products of harmonic polynomials. Then, we show that the number of independent coefficients of HGPTs needed to characterise objects can be significantly reduced by considering the symmetry group of the object and propose a systematic approach for determining the subspace of symmetric harmonic polynomials that is fixed by the group and its dimension. This enable us to determine the independent HGPT coefficients for different symmetry groups.
翻译:我们引入了一种新的物体特征,能够准确描述对潜在场面反问题,如电阻、磁阻和低频马克斯韦尔问题等潜在场面问题进行小规模孤立的包容,相关的应用包括磁度测量磁性场测量产生的有色未爆弹药特征,描述利用电阻阻断断断层成像进行地质地面测量的小型医疗成像整合,利用电阻断层成像进行地质地面测量,通过电传鱼类对物体进行定位,以导航和识别食物,以及描述稀释复合物的有效特性。我们的物体特征建立在通用极化振标(GPT)对象特征概念的基础上,为压缩的GPT提供了替代物。我们称新的色化GPTs(HGPTs)为其与调制聚度产品相匹配的系数。然后,我们表明,考虑到该物体的对称组,可以大幅减少HGPTs为特性所需的独立系数数量,并提议一种系统的方法,用以确定其亚空基面的GPT值。