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Mutual induction tomography (MIT) attempts to image the electromagnetic characteristics of an object by measuring the mutual inductances between sets of coils placed around its periphery. The application of MIT for molten metal flow visualization is of interest in this paper, which focuses on computational aspects of the forward and inverse MIT problem. The forward problem has been solved using an edge finite element formulation. The Jacobian matrix is a key to image reconstruction in MIT. The entries of the Jacobian matrix are the sensitivity of the measurement data to the image values, which has been generated by an efficient adjoint formulation. We have implemented a standard regularized Gauss-Newton scheme to solve such a problem. The reconstructed images for a high-contrast conductivity example of steel/argon flow shown in this paper are some of the first nonlinear image reconstruction results for MIT.