  Sparsity is one of the key concepts that allows the recovery of signals that
are subsampled at a rate significantly lower than required by the
Nyquist-Shannon sampling theorem. Our proposed framework uses arbitrary
multiscale transforms, such as those build upon wavelets or shearlets, as a
sparsity promoting prior which allow to decompose the image into different
scales such that image features can be optimally extracted. In order to further
exploit the sparsity of the recovered signal we combine the method of
reweighted $\ell^1$, introduced by Cand\`es et al., with iteratively updated
weights accounting for the multilevel structure of the signal. This is done by
directly incorporating this approach into a split Bregman based algorithmic
framework. Furthermore, we add total generalized variation (TGV) as a second
regularizer into the split Bregman algorithm. The resulting algorithm is then
applied to a classical and widely considered task in signal- and image
processing which is the reconstruction of images from their Fourier
measurements. Our numerical experiments show a highly improved performance at
relatively low computational costs compared to many other well established
methods and strongly suggest that sparsity is better exploited by our method.
