Use of secondary equivalent sources for solving electrical resistance inverse problems

Abstract

In this paper, we propose a "source-type" solution to the problem of electrical resistance tomography (ERT). The goal of ERT is to develop a map of the electrical properties in a region of space based on observations of voltages collected at the boundary in response to input DC currents also at the boundary. As with many inverse problems, ERT is both nonlinear and poorly posed. Source-type inverse methods have been proposed in the inverse scattering context as a way of quasi-linearizing the problem. Specifically, inhomogeneities in the medium are viewed as secondary sources embedded in a homogeneous medium. One can solve a linear inverse source problem to localize these sources (i.e. determine their geometry); however resolving their spatial contrasts quantitatively is not possible under this method. In a sense, the nonlinearity of the original problem is buried in the amplitude. Our work here is motivated by thefact that use of a source-type formulation has not been considered for ERT to the best of our knowledge. We shall show that the secondary sources for ERT are defined by the inner product of the gradients of true conductivity and electrical potential. Using this equivalence, the inverse problem is easily transformed into a multi-source inversion. Given the ill-posedness of the ERT problem arising from the inherent low sensitivity of the observed data to changes in the internal conductivity of the medium, the proposed transformation provides a better description of the effect of inhomogeneities and therefore leads to more efficient inversion techniques. We introduce and discuss one step as well as iterative methods, especially for piecewise constant media. In the iterative method, we make use of a level set formulation and we replace the update of the steepest descent approach by the correlation coefficient between the residual vector and the response of a specified source. Using the same measure, i.e. the correlation coefficient, we introduce a simple single step imaging method. Results of both methods using simulated data are presented.