Abstract: Blind deconvolution problems arise in image analysis when both the extent of image blur, and the true image, are unknown. If a model is available for at least one of these quantities then, in theory, the problem is solvable. It is generally not solvable if neither the image nor the point-spread function, which controls the extent of blur, is known parametrically. In this paper we develop methods for solution when a model is known for the point-spread function, but the image is assessed only nonparametrically. We assume that the image includes sharp edges -- mathematically speaking, lines of discontinuity of image brightness. However, the locations, shapes and other properties of these lines are not needed for our algorithm. Our technique involves mathematically ``focussing'' the restored image until the edges are as sharp as possible, with sharpness being measured using a difference-based approach. We pay special attention to the Gaussian point-spread function. Although this context is notorious in statistical deconvolution problems on account of the difficulty of finding a usable solution, it is arguably the central, and the most important, setting for restoring blurred images. Numerical simulation, application to an image, and theoretical analysis demonstrate the effectiveness of our approach.
Key words and phrases: Adaptive estimation, blur, deblurring, Gaussian blur, ill-posed problem, image restoration, inverse problem, noise, nonparametric regression, point-spread function, regularisation, test pattern.