The transverse charge and magnetization densities provide insight into the nucleon’s inner structure. In the periphery, the isovector components are clearly dominant, and can be computed in a model-independent way by means of a combination of chiral effective field theory and dispersion analysis. With a novel N/D method, we incorporate the pion electromagnetic form- factor data into the chiral EFT calculation, thus taking into account the pion-rescattering effects and rho-meson pole. As a consequence, we are able to reliably compute the densities down to distances b ∼ 1 fm, therefore achieving a dramatic improvement of the results compared to traditional chiral EFT calculations, while remaining predictive and having controlled uncertainties.