A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets

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Abstract
A two-step shape reconstruction method for electromagnetic (EM) tomography is presented which uses adjoint fields and level sets. The inhomogeneous background permittivity distribution and the values of the permittivities in some penetrable obstacles are assumed to be known, and the number, sizes, shapes, and locations of these obstacles have to be reconstructed given noisy limited-view EM data. The main application we address in the paper is the imaging and monitoring of pollutant plumes in environmental cleanup sites based on cross-borehole EM data. The first step of the reconstruction scheme makes use of an inverse scattering solver which recovers equivalent scattering sources for a number of experiments, and then calculates from these an approximation for the permittivity distribution in the medium. The second step uses this result as an initial guess for solving the shape reconstruction problem. A key point in this second step is the fusion of the `level set technique' for representing the shapes of the reconstructed obstacles, and an `adjoint field technique' for solving the nonlinear inverse problem. In each step, a forward and an adjoint Helmholtz problem are solved based on the permittivity distribution which corresponds to the latest best guess for the representing level set function. A correction for this level set function is then calculated directly by combining the results of these two runs. Numerical experiments are presented which show that the derived method is able to recover one or more objects with nontrivial shapes given noisy cross-borehole EM data.