A projection-based level-set approach to enhance conductivity anomaly reconstruction in electrical resistance tomography

In this paper we consider level-set-based methods for anomaly reconstruction for applications involving low-sensitivity data concentrating specifically on the problem of electrical resistance tomography (ERT). Typical descent-based inversion methods suffer from poor reconstruction precision in low-sensitivity regions and typically display quite slow convergence. In this paper, we develop a method for constructing the level-set speed function that is capable of overcoming these problems. In the case of the gradient-descent-based level-set evolution, the speed function is defined in terms of the dot products of residual and sensitivity functions. For Gauss-Newton-type methods, however, the speed function is the solution of the linearized inverse problem at each iteration. Here we propose a projection-based approach where the displacement of the level zero, i.e. estimated anomaly contour, at each point depends on the correlation coefficients between the sensitivity of the data to conductivity perturbations and the residual error. In other words, the proposed velocity field is invariant to the absolute amplitudes of both residual and sensitivity, but rather is a reflection of the angle between these two quantities. We demonstrate that our method is a descent-based reconstruction and we relate the mathematical formulation of the projection-based speed function to that of the gradient-descent method and the Gauss-Newton-type approach. The comparison suggests that the proposed technique can be seen as a diagonal approximation of the Gauss Newton formulation. Using a quasi-linear source-type formulation of the forward problem, we describe an efficient implementation of the projection-based approach for finite-domain imaging problems. The proposed algorithm is simulated with numerical 2D and 3D data and its performance and efficiency are compared to those of the gradient-descent method.