  In this effort, we propose a convex optimization approach based on weighted
$\ell_1$-regularization for reconstructing objects of interest, such as signals
or images, that are sparse or compressible in a wavelet basis. We recover the
wavelet coefficients associated to the functional representation of the object
of interest by solving our proposed optimization problem. We give a specific
choice of weights and show numerically that the chosen weights admit efficient
recovery of objects of interest from either a set of sub-samples or a noisy
version. Our method not only exploits sparsity but also helps promote a
particular kind of structured sparsity often exhibited by many signals and
images. Furthermore, we illustrate the effectiveness of the proposed convex
optimization problem by providing numerical examples using both orthonormal
wavelets and a frame of wavelets. We also provide an adaptive choice of weights
which is a modification of the iteratively reweighted $\ell_1$-minimization
method.
