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A conductivity inclusion, inserted in a homogeneous background, induces aperturbation in the background potential. This perturbation admits a multipoleexpansion whose coefficients are the so-called generalized polarization tensors(GPTs). GPTs can be obtained from multistatic measurements. As a modificationof GPTs, the Faber polynomial polarization tensors (FPTs) were recentlyintroduced in two dimensions. In this study, we design two novel analyticalnon-iterative methods for recovering the shape of a simply connected inclusionfrom GPTs by employing the concept of FPTs. First, we derive an explicitexpression for the coefficients of the exterior conformal mapping associatedwith an inclusion in a simple form in terms of GPTs, which allows us toaccurately reconstruct the shape of an inclusion with extreme or near-extremeconductivity. Secondly, we provide an explicit asymptotic formula in terms ofGPTs for the shape of an inclusion with arbitrary conductivity by consideringthe inclusion as a perturbation of its equivalent ellipse. With this formula,one can non-iteratively approximate an inclusion of general shape witharbitrary conductivity, including a straight or asymmetric shape. Numericalexperiments demonstrate the validity of the proposed analytical approaches.

Analytical shape recovery of a conductivity inclusion based on Faber polynomials