Summary: The goal of image deconvolution is to restore an image within a given area, from a blurred and noisy specimen. It is well known that the convolution operator integrates not only the image in the field of view of the given specimen, but also part of the scenery in the area bordering it. The result of a deconvolution algorithm which ignores the non-local properties of the convolution operator will be a restored image corrupted by distortion artefacts. These artefacts, which tend to be more pronounced near the boundary, can propagate to the entire image.
We propose two different ways to compensate for boundary artefacts, both of a statistical nature. The first one is based on the restoration of an extended image, on whose exterior boundary we impose statistics-based boundary conditions. In the second one, the contribution to the convolution integral coming from the area outside the field of view is treated as noise. In both cases, the methodological tools come from Bayesian statistical inversion and the problems are reduced to the case where the signal to estimate and the noise are mutually independent Gaussian white noise random variables. Computed examples illustrate the performance of the two approaches.