Despite the extra information offered by the quantitative magnetic resonance imaging (qMRI), these methods are not widely used due to their long acquisition times. Our approach for fast qMRI was built on minimized data acquisition with ultra-short echo time imaging, and compressed sensing image reconstruction. To validate the approach, a set of variable flip angle images of equine osteochondral specimens were acquired with the Multi-Band-SWIFT sequence. The 4-D image stack was reconstructed with CS-framework utilizing spatial and contrast regularizations. Results with 81% reduced data showed comparable image quality while maintaining correct contrast modulation for quantitative parameter (T1 relaxation time) estimation.