The dynamics of granular materials, particularly radial and axial segregation in horizontal rotating cylinders containing large and small particles, is studied by Magnetic Resonance Imaging (MRI). Stationary three-dimensional (3D) images and real-time two-dimensional (2D) structural images showing radial segregation, band formation, and band merging are reported. Quantitative local particle concentrations are measured in a noninvasive manner from the different magnetic resonance responses of the seeds throughout segregation. Data are acquired with sufficiently high temporal (300 ms for 2D images) and spatial resolutions (0.94 mm cubic voxels), to give insights into the underlying mechanisms of both radial and axial segregation. In particular, the increasing rate of the local particle concentration during radial segregation is quantified. Particle migration is observed in the bulk material of the 75% and 82% full cylinders during both radial and axial segregation, showing that this region beneath the avalanche layer does not behave as a solid body. We also provide direct experimental evidence to support recent numerical simulations of band merging.