Frequency-difference electrical impedance tomography (fdEIT) has been proposed to deal with technical difficulties of a conventional static EIT imaging method caused by unknown boundary geometry, uncertainty in electrode positions and other systematic measurement artifacts. In fdEIT, we try to produce images showing changes of a complex conductivity distribution with respect to frequency. Simultaneously injecting currents with at least two frequencies, we find differences of measured boundary voltages between those frequencies. In most previous studies, real parts of frequency-difference voltage data were used to reconstruct conductivity changes and imaginary parts to reconstruct permittivity changes. This conventional approach neglects the interplay of conductivity and permittivity upon measured boundary voltage data. In this paper, we propose an improved fdEIT image reconstruction algorithm that properly handles the interaction. It uses weighted frequency differences of complex voltage data and a complex sensitivity matrix to reconstruct frequency-difference images of complex conductivity distributions. We found that there are two major sources of image contrast in fdEIT. The first is a contrast in complex conductivity values between an anomaly and background. The second is a frequency dependence of a complex conductivity distribution to be imaged. We note that even for the case where conductivity and permittivity do not change with frequency, the fdEIT algorithm may show a contrast in frequency-difference images of complex conductivity distributions. On the other hand, even if conductivity and permittivity values significantly change with frequency, there is an example where we cannot find any contrast. The performance of the proposed method is demonstrated by using computer simulations to validate its feasibility in future experimental studies.

With each C 2-domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain and its material parameter. They generalize the concept of Polarization Tensor (PT), which can be seen as the first-order GPT. It is known that given an arbitrary shape, one can find an equivalent ellipse or ellipsoid with the same PT. In this paper we consider the problem of recovering finer details of the shape of a given domain using higher-order polarization tensors. We design an optimization approach which solves the problem by minimizing a weighted discrepancy functional. In order to compute the shape derivative of this functional, we rigorously derive an asymptotic expansion of the perturbations of the GPTs that are due to a small deformation of the boundary of the domain. Our derivations are based on the theory of layer potentials. We perform some numerical experiments to demonstrate the validity and the limitations of the proposed method. The results clearly show that our approach is very promising in recovering fine shape details.

Abstract. We derive high-order terms in the asymptotic expansions of boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with C 2 -boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for C 1 -perturbations and inclusions with extreme conductivities. It extends those already derived for small volume conductivity inclusions and leads us to very effective algorithms for determining lower-order Fourier coefficients of the shape perturbation of the inclusion based on boundary measurements. We perform some numerical experiments using the algorithm to test its effectiveness.