Deconvolution with wavelet footprints for ill-posed inverse problems

In recent years, wavelet based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool, because it manages to efficiently represent sharp discontinuities. Indeed, discontinuities carry most of the signal information and, so, they represent the most critical part to analyse. We have recently introduced the notion of footprints, which form an overcomplete basis built on the wavelet transform. With footprints, one can exactly model the dependency across scales of the wavelet coefficients generated by a discontinuity and this allows to further improve wavelet based algorithms. In this paper we present a footprint based algorithm for signal deconvolution. The algorithm is fast and works for blind deconvolution too. With footprints we manage to deconvolve efficiently the irregular part of the signal. Thanks to the property of footprints of exactly modeling discontinuities, the deconvolved signal does not present artifacts around discontinuities. Moreover, we show that the residual, that is, the difference between the deconvolved signal with footprints and the observed signal, is regular. Thus, this residual can be further deconvolved with any other traditional method. We show that our system outperforms other deconvolution methods.