Electroanatomical mapping, a keystone diagnostic tool in cardiac electrophysiology studies, can provide high-density maps of the local electric properties of the tissue. It is therefore tempting to use such data to better individualize current patient-specific models of the heart through a data assimilation procedure and to extract potentially insightful information such as conduction properties. Parameter identification for state-of-the-art cardiac models is however a challenging task. In this work, we introduce a novel inverse problem for inferring the anisotropic structure of the conductivity tensor, that is fiber orientation and conduction velocity along and across fibers, of an eikonal model for cardiac activation. The proposed method, named PIEMAP, performed robustly with synthetic data and showed promising results with clinical data. These results suggest that PIEMAP could be a useful supplement in future clinical workflows of personalized therapies.