Calderón's Inverse Problem with an Imperfectly Known Boundary and Reconstruction Up to a Conformal Deformation

We consider two-dimensional electrical impedance tomography (EIT), that is, the inverse conductivity problem in the case when the boundary of the domain is not known. In this case, the boundary measurements correspond to the idealization of measuring power needed to maintain different voltage configurations in electrodes attached to the boundary. We give an algorithm that reconstructs the conductivity from the boundary measurement up to a conformal deformation of the domain and estimate quantitatively the distortion in the reconstruction caused by an incorrectly modeled boundary. In our previous results [V. Kolehmainen, M. Lassas, and P. Ola, SIAM J. Appl. Math., 66 (2005), pp. 365–383], it was shown that in the inaccurately modeled domain there is a unique anisotropic conductivity corresponding to the boundary measurements that has the minimal possible anisotropy. This minimally anisotropic conductivity can be found using minimization. In this paper we use the Beltrami equation to isotropize the minimally anisotropic conductivity. The obtained isotropic conductivity coincides with the original isotropic conductivity up to a conformal deformation. The algorithm also makes it possible to give quantitative estimates on the deformation. The proofs are based on Hölder estimates of quasi-conformal maps, and especially on precise bounds given by Mori's theorem.