Higher degree total variation for 3-D image recovery

We extend the novel higher degree total variation (HDTV) image regularization penalties to 3-D signals. These penalties generalize the popular total variation (TV) penalty to incorporate higher degree image derivatives. We adapt a fast alternating minimization algorithm designed for solving 2-D image recovery problems with HDTV regularization to the 3D setting. Numerical experiments on the compressed sensing recovery of 3-D magnetic resonance images show that HDTV and generalized HDTV improve the image quality significantly compared with TV. We also investigate the relationship between the recently introduced Hessian Schatten-norms and HDTV.