A multilevel based reweighting algorithm with joint regularizers for sparse recovery

We propose an algorithmic framework based on ADMM/split Bregman that combines a multilevel adapted, iterative reweighting strategy and a second total generalized variation regularizer. The level adapted reweighting strategy is a combination of reweighted `-minimization and additional compensation factors for a uniform treatment of the sparsity structure across all levels. Classical multilscale transforms that are very well suited for this algorithm are, for instance, the wavelet transform and the shearlet transform. The proposed algorithm is tested for the reconstruction of images from their Fourier measurements and Radon measurements, respectively. The numerical experiments show a highly improved performance at relatively low additional computational costs compared to many other well established methods.