Function spaces and optimal currents in impedance tomography

Abstract

Abstract The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j, which contain the most information about the difference between the unknown physical conductivity σ* and a given approximation σ 0. The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.