Spherical harmonics for shape-based inverse problems, as applied to electrical impedance tomography

Electrical Impedance Tomography (EIT) is a badly posed inverse problem. In a 3-D volume too many parameters are required to be able to obtain stable estimates with good spatial resolution and good accuracy. One approach to such problems that has been presented recently in a number of reports, when the relevant constituent parameters can be modeled as isotropic and piecewise continuous or homogeneous, is to use shape-based solutions. In this work, we report on a method, based on a spherical harmonics expansion, that allows us to parameterize the 3-D objects which constitute the conductivity inhomogeneities in the interior; for instance, we could assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape are not. Using this assumption, we have developed a 3-stage optimization algorithm that allows us to iteratively estimate the location of the inhomogeneous objects, to find their external boundaries and to estimate their internal conductivities. The performance of the proposed method is illustrated via simulation in a realistic torso model, as well as via experimental data from a tank phantom.