A nonlinear adaptive wavelet-based method for spiky deconvolution

In this paper we present a now approach for computing sparse solutions to discrete Fredholm integral equations of the first kind corrupted by additive noise. Thus, considering the joint detection-estimation character that spiky deconvolution problems have, we introduce a novel algorithm which involves two principal steps at each iteration: 1) detecting the nonzero values to the solution using an adaptive nonlinear projection selection operator based on the estimation of the pointwise Lipschitz regularity; 2) estimating the amplitudes of this values by obtaining a regularized solution of the original equation using the a priori knowledge and the above approximation. This technique achieves good performance with a low computational cost and offers the possibility of including any kind of constraint, linear and nonlinear.