A quantitative algorithm for parameter estimation in magnetic induction tomography

Abstract

Magnetic induction tomography aims at producing images of the spatial distribution of electric conductivity and/or magnetic permeability in a given region of space. The associated inverse problem is ill posed and strongly nonlinear, making it very hard to deal with. However, if ‘a priori’ knowledge is incorporated into the solution, the problem may be simplified and an inversion procedure that produces quantitative results for some situations may be obtained. This paper describes an approach that follows this rationale. The spatial distribution in the region of interest is restricted to a single object of cylindrical shape and the physical property to be imaged to the magnetic permeability. Solving the problem will mean in this sense estimating only four parameters: two spatial coordinates for the position of the cylinder's centre, its radius and its magnetic permeability. The geometry of data collection is simple enough for the direct problem to be solved analytically and allows one to concentrate on the main characteristics of the inversion. It is shown that the problem is invertible, although in the limit, when the radius tends to zero, this is no longer true. A Newton–Raphson- type algorithm was developed for inversion purposes. It converges in a small number of iterations and produces exact values of the parameters for a predefined range of magnetic permeability and cylinder radius, if no noise is present. The inversion procedure is studied and its performance evaluated in several situations, related to the amount of S/N in the measurement vector, the degree of overdetermination in the system of equations and the choice of the sampling points.