Quantitative magnetic resonance imaging (MRI) techniques are powerful tools for the study of human tissue, but, in practice, their utility has been limited by lengthy acquisition times. Here, we introduce the Constrained, Adaptive, Low-dimensional, Intrinsically Precise Reconstruction (CALIPR) framework in the context of myelin water imaging (MWI); a quantitative MRI technique generally regarded as the most rigorous approach for noninvasive, in vivo measurement of myelin content. The CALIPR framework exploits data redundancy to recover high-quality images from a small fraction of an imaging dataset, which allowed MWI to be acquired with a previously unattainable sequence (fully sampled acquisition 2 hours:57 min:20 s) in 7 min:26 s (4.2% of the dataset, acceleration factor 23.9). CALIPR quantitative metrics had excellent precision (myelin water fraction mean coefficient of variation 3.2% for the brain and 3.0% for the spinal cord) and markedly increased sensitivity to demyelinating disease pathology compared to a current, widely used technique. The CALIPR framework facilitates drastically improved MWI and could be similarly transformative for other quantitative MRI applications.