Eddy current tomography using a binary Markov model

Abstract

The non-destructive evaluation (NDE) problem we treat is the testing of a globally homogeneous conductive medium for anomalies such as cracks and notches. The medium is illuminated with a monochromatic electric field; the anomalies induce eddy currents and they modify the total field which can be measured. The tomographic approach, aimed to draw up an image of the medium, is recent in this area. It corresponds to an extremely difficult ill-posed inverse problem and its resolution needs the use of pertinent prior information. The considered anomalies can be represented using images whose pixels can only take the values 0 and 1. Our main contribution lies in the regularization of a large-support ill-posed observation operator using a locally constant binary image Markov random field. The resulting high-dimensional combinatorial optimization problem is tedious: neither exact resolution nor simulated annealing are feasible. Instead, we establish an equivalent continuous-valued optimization problem. A nearly optimal solution is then calculated using a graduated non-convexity algorithm adapted for this purpose. The proposed inversion technique surpasses the particular NDE problem and can be applied whenever a binary image is observed using a linear system and corrupted by Gaussian noise.