Many approaches exist to compute the (approximate) inverse of an operator K to recover an approximation to f from a dataset that represents a noise-corrupted version of Kf. Several approaches have been proposed that are adapted to the special case where f has a sparse wavelet expansion, a case that applies to many types of images or other types of signals; an example of the operator K in this context is, e.g., blurring, the convolution with a known function. The talk will present an iterative approach to solve this problem, which can be used with respect to arbitrary orthonormal bases. The algorithm is similar to the Landweber algorithm, except that the prior information incorporated into the variational functional uses a weighted $l^p-norm$ of the wavelet coefficients instead of the $l2-norm$, standard for Landweber methods. This iterative approach converges in norm and is stable; some applications will be shown. This is joint work with Michel Defrise (Vrije Universiteir Brussel) and Christine De Mol (Universite Libre de Bruxelles)