An explicit reconstruction method for magnetic resonance electrical property tomography based on the generalized Cauchy formula

This paper presents an explicit reconstruction formula for magnetic resonance electrical property tomography (MREPT). We derive a Dbar problem from the time-harmonic Maxwell equations under the assumptions that $H_z=0$, $\partial H^{+}\ne 0$, and ${\partial }_z{H}^{+}=0$, where the z-axis is parallel to the body axis, $H^{+}\equiv \frac{1}{2}(H_x+i{H}_y)$ is the measured magnetic field, and $\partial \equiv \frac{1}{2}({\partial }_x-i{\partial }_y)$. Then, by using the generalized Cauchy formula, the electrical conductivity and permittivity are explicitly expressed in terms of their boundary values and the measured magnetic field. We also propose an iterative algorithm based on the explicit reconstruction formula without the assumption that ${\partial }_z{H}^{+}=0$. Numerical simulations show that the proposed methods can reconstruct the electrical properties even with noisy data. (paper)