A Factorization Method for Support Recovery in Magnetic Induction Tomography
We develop a factorization framework for magnetic induction tomography in the eddy-current regime. The induced current density is represented by a divergence-free current vector potential, and the external excitation by magnetic dipole densities on an observation sphere. This choice reveals a natural physical factorization of the near-field operator into a data operator, an interior boundary response operator, and an adjoint counterpart. By introducing the Riesz map on the flux space, we recast this factorization in a Hilbert-space setting, which yields a positive-semidefinite operator with a coercive middle factor. Douglas' range theorem then identifies the range of the square root of this dissipative operator with the range of the data operator. For inclusions with a regular boundary, this range identity yields a uniqueness theorem for the support of the conductivity. This characterization leads to a non-iterative spectral indicator based on a regularized Picard quotient computed from the Hermitian imaginary part of the near-field data matrix. The reconstruction stage requires no forward solves. Numerical experiments confirm that the indicator localizes and separates the inclusions accurately, degrades gracefully under increasing noise, and captures non-convex support geometry without any convexity prior.