  Compressed sensing (CS) methods in magnetic resonance imaging (MRI) offer
rapid acquisition and improved image quality but require iterative
reconstruction schemes with regularization to enforce sparsity. Regardless of
the difficulty in obtaining a fast numerical solution, the total variation (TV)
regularization is a preferred choice due to its edge-preserving and structure
recovery capabilities. While many approaches have been proposed to overcome the
non-differentiability of the TV cost term, an iterative shrinkage based
formulation allows recovering an image through recursive application of linear
filtering and soft thresholding. However, providing an optimal setting for the
regularization parameter is critical due to its direct impact on the rate of
convergence as well as steady state error. In this paper, a regularizer
adaptively varying in the derivative space is proposed, that follows the
generalized discrepancy principle (GDP). The implementation proceeds by
adaptively reducing the discrepancy level expressed as the absolute difference
between TV norms of the consistency error and the sparse approximation error. A
criterion based on the absolute difference between TV norms of consistency and
sparse approximation errors is used to update the threshold. Application of the
adaptive shrinkage TV regularizer to CS recovery of parallel MRI (pMRI) and
temporal gradient adaptation in dynamic MRI are shown to result in improved
image quality with accelerated convergence. In addition, the adaptive TV-based
iterative shrinkage (ATVIS) provides a significant speed advantage over the
fast iterative shrinkage-thresholding algorithm (FISTA).
