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Scanning (electrical) impedance imaging (SII) is a novel high-resolution imaging modality that has the potential of imaging the electrical properties of thin biological tissues. In this paper, we apply the reciprocity principle to the modeling of the SII system and develop a fast nonlinear inverse method for image reconstruction. The method is fast because it uses convolution to eliminate the requirement of a numerical solver for the 3-D electrostatic field in the SII system. Numerical results show that our approach can accurately reveal the exact conductivity distribution from the measured current map for different 2-D simulation phantoms. Experiments were also performed using our SII system for a piece of butterfly wing and breast cancer cells. Two-dimensional current images were measured and corresponding quantitative conductivity images were restored using our approach. The reconstructed images are quantitative and reveal details not present in the measured images.