Circular resistor networks for electrical impedance tomography with partial boundary measurements

We introduce an inversion algorithm for electrical impedance tomography (EIT) with partial boundary measurements in two dimensions. It gives stable and fast reconstructions using sparse parameterizations of the unknown conductivity on optimal grids that are computed as part of the inversion. We follow the approach in Borcea et al (2008 Inverse Problems 24 035013) and Vasquez (2006 PhD thesis Rice University, Houston, TX, USA) that connects inverse discrete problems for resistor networks to continuum EIT problems, using optimal grids. The algorithm in Borcea et al (2008 Inverse Problems 24 035013) and Vasquez (2006 PhD Thesis Rice University, Houston, TX, USA) is based on circular resistor networks, and solves the EIT problem with full boundary measurements. It is extended in Borcea et al (2010 Inverse Problems 26 045010) to EIT with partial boundary measurements, using extremal quasi-conformal mappings that transform the problem to one with full boundary measurements. Here we introduce a different class of optimal grids, based on resistor networks with pyramidal topology, that is better suited for the partial measurements setup. We prove the unique solvability of the discrete inverse problem for these networks and develop an algorithm for finding them from the measurements of the Dirichlet to Neumann map. Then, we show how to use the networks to define the optimal grids and to approximate the unknown conductivity. We assess the performance of our approach with numerical simulations and compare the results with those in Borcea et al (2010).