  We provide a new algorithm for the treatment of inverse problems which
combines the traditional SVD inversion with an appropriate thresholding
technique in a well chosen new basis. Our goal is to devise an inversion
procedure which has the advantages of localization and multiscale analysis of
wavelet representations without losing the stability and computability of the
SVD decompositions. To this end we utilize the construction of localized frames
(termed "needlets") built upon the SVD bases. We consider two different
situations: the "wavelet" scenario, where the needlets are assumed to behave
similarly to true wavelets, and the "Jacobi-type" scenario, where we assume
that the properties of the frame truly depend on the SVD basis at hand (hence
on the operator). To illustrate each situation, we apply the estimation
algorithm respectively to the deconvolution problem and to the Wicksell
problem. In the latter case, where the SVD basis is a Jacobi polynomial basis,
we show that our scheme is capable of achieving rates of convergence which are
optimal in the $L_2$ case, we obtain interesting rates of convergence for other
$L_p$ norms which are new (to the best of our knowledge) in the literature, and
we also give a simulation study showing that the NEED-D estimator outperforms
other standard algorithms in almost all situations.
