Magnetic resonance imaging (MRI) reconstruction from sparsely sampled data has
been a difficult problem in medical imaging field. We approach this problem by formulating a cost
functional that includes a constraint term that is imposed by the raw measurement data in k-space
and the L1 norm of a sparse representation of the reconstructed image. The sparse representation
is usually realized by total variational regularization and/or wavelet transform. We have applied
the Bregman iteration to minimize this functional to recover finer scales in our recent work. Here
we propose nonlinear inverse scale space methods in addition to the iterative refinement procedure.
Numerical results from the two methods are presented and it shows that the nonlinear inverse scale
space method is a more efficient algorithm than the iterated refinement method.