ABSTRACT
Chapter 5 introduces two wavelet-based recovery schemes using wavelet sparsity and the underlying structure of the wavelet coefficients. The initial part of the chapter is devoted to the method based on structured sparsity that includes the tree and block-based recovery schemes. In the latter half of the chapter, the use of wavelet sparsity for MR image recovery is discussed in depth. Wavelet sparsity–based formulations are found to be highly influenced by the threshold to be used in each iteration. The threshold has a direct impact on the speed of convergence and the steady-state reconstruction error. For simultaneous improvements in both speed and steady-state errors, the most feasible technique is to use an iteration-dependent threshold in which one uses a large value of the threshold at the start of the iterations that is adaptively decreased in the succeeding iterations until the target regularization is achieved. A data-driven continuation scheme that uses the updated quantities in each iteration can be more appropriate in this case. Several variants of this scheme with applications to pMRI are discussed with numerical examples.
