Wavelet domain signal deconvolution with singularity-preserving regularization

In this paper, we consider a wavelet based singularity-preserving regularization scheme for use in signal deconvolution problems. The inverse problem of finding solutions with singularities to discrete Fredholm integral equations of the first kind arises in many applied fields, e.g. in Geophysics. This equation is usually an ill-posed problem when it is considered in a Hilbert space framework, requiring regularization techniques to control arbitrary error amplifications and to get adequate solutions. Thus, considering the joint detection-estimation character this kind of signal deconvolution problems have, we introduce two novel algorithms which involve two principal steps at each iteration: (a) detecting the positions of the singularities by a nonlinear projection selection operator based on the estimation of Lipschitz regularity using the discrete dyadic wavelet transform; and (b) estimating the amplitudes of these singularities by obtaining a regularized solution of the original equation using the a priori knowledge and the above approximation. Some simulation examples serve to appreciate the high performance of the proposed techniques in this kind of problems.