Dealing with boundary artifacts in MCMC-based deconvolution

Many numerical methods for deconvolution problems are designed to take advantage of the computational efficiency of spectral methods, but classical approaches to spectral techniques require particular conditions be applied uniformly across all boundaries of the signal. These boundary conditions – traditionally periodic, Dirichlet, Neumann, or related – are essentially methods for generating data values outside the domain of the signal, but they often lack physical motivation and can result in artifacts in the reconstruction near the boundary. In this work we present a data-driven technique for computing boundary values by solving a regularized and well-posed form of the deconvolution problem on an extended domain. Further, a Bayesian framework is constructed for the deconvolution, and we present a Markov chain Monte Carlo method for sampling from the posterior distribution. There are several advantages to this approach, including that it still takes advantage of the efficiency of spectral methods, that it allows the boundaries of the signal to be treated in a non-uniform manner – thereby reducing artifacts – and that the sampling scheme gives a natural method for quantifying uncertainties in the reconstruction.