Development of myelin, a fatty sheath that insulates nerve fibers, is critical for brain function. Myelination during infancy has been studied with histology, but postmortem data cannot evaluate the longitudinal trajectory of white matter development. Here, we obtained longitudinal diffusion MRI and quantitative MRI measures of longitudinal relaxation rate (R1) of white matter in 0, 3 and 6 months-old human infants, and developed an automated method to identify white matter bundles and quantify their properties in each infant's brain. We find that R1 increases from newborns to 6-months-olds in all bundles. R1 development is nonuniform: there is faster development in white matter that is less mature in newborns, and development rate increases along inferior-to-superior as well as anterior-to-posterior spatial gradients. As R1 is linearly related to myelin fraction in white matter bundles, these findings open new avenues to elucidate typical and atypical white matter myelination in early infancy.