The ventricular myocardium is known to exhibit a complex spatial organization, with fiber orientation varying as a function of transmural location. It is now well established that diffusion tensor magnetic resonance imaging (DTMRI) may be used to measure this fiber orientation at high spatial resolution. Cardiac fibers are also known to be organized in sheets with surface orientation varying throughout the ventricles. This article reviews results on use of DTMRI for measuring ventricular fiber orientation, as well as presents new results providing strong evidence that the tertiary eigenvector of the diffusion tensor is aligned locally with the cardiac sheet surface normal. Considered together, these data indicate that DTMRI may be used to reconstruct both ventricular fiber and sheet organization. This article also presents the large deformation diffeomorphic metric mapping (LDDMM) algorithm and shows that this algorithm may be used to bring ensembles of imaged and reconstructed hearts into correspondence (e.g., registration) so that variability of ventricular geometry, fiber, and sheet orientation may be quantified. Ventricular geometry and fiber structure is known to be remodeled in a range of disease processes; however, descriptions of this remodeling have remained subjective and qualitative. We anticipate that use of DTMRI for reconstruction of ventricular anatomy coupled with application of the LDDMM method for image volume registration will enable the detection and quantification of changes in cardiac anatomy that are characteristic of specific disease processes in the heart. Finally, we show that epicardial electrical mapping and DTMRI imaging may be performed in the same hearts. The anatomic data may then be used to simulate electrical conduction in a computational model of the very same heart that was mapped electrically. This facilitates direct comparison and testing of model versus experimental results and opens the door to quantitative measurement, modeling, and analysis of the ways in which remodeling of ventricular microanatomy influences electrical conduction in the heart.