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In magnetic resonance electrical impedance tomography, among several conductivity image reconstruction algorithms, the harmonic Bz algorithm has been successfully applied to Bz data from phantoms and animals. The algorithm is, however, sensitive to measurement noise in Bz data. Especially, in in vivo animal and human experiments where injection current amplitudes are limited within a few milliampere at most, measured Bz data tend to have a low SNR. In addition, magnetic resonance (MR) signal void in outer layers of bones and gas-filled organs, for example, produces salt-pepper noise in the MR phase and, consequently, Bz images. The Bz images typically present areas of sloped transitions, which can be assimilated to ramps. Conductivity contrasts change ramp slopes in Bz images and it is critical to preserve positions of those ramps to correctly recover edges in conductivity images. In this paper, we propose a ramp-preserving denoising method utilizing a structure tensor. Using an eigenvalue analysis, we identified local regions of salt-pepper noise. Outside the identified local regions, we applied an anisotropic smoothing to reduce noise while preserving their ramp structures. Inside the local regions of salt-pepper noise, we used an isotropic smoothing. After validating the proposed denoising method through numerical simulations, we applied it to in vivo animal imaging experiments. Both numerical simulation and experimental results show significant improvements in the quality of reconstructed conductivity images.