Abstract

Let us consider the deconvolution problem, i.e. to recover a latent source x(center dot) from the observations y=[y(1),..., y(N)] of a convolution process y = x star h + eta, where eta is an additive noise, the observations in y might have missing parts with respect to y, and the filter h could be unknown. We propose a novel strategy to address this task when x is a continuoustime signal: we adopt a Gaussian process prior on the source x, which allows for closed-form Bayesian non-parametric deconvolution. We first analyse the direct model to establish the conditions under which the model is well-defined. Then, we turn to the inverse problem, where we study (i) some necessary conditions under which Bayesian deconvolution is feasible and (ii) to which extent the filter h can be learned from data or approximated for the blind deconvolution case. The proposed approach, termed Gaussian process deconvolution, is compared to other deconvolution methods conceptually, via illustrative examples, and using real-world datasets.