A non-iterative regularization approach to blind deconvolution

Abstract

Blind deconvolution, where both an original image and a blurring kernel are reconstructed from a blurred and noisy image, is a nonlinear and ill-posed image processing problem. Recently, classical methods for the regularization of non-blind deconvolution have been adapted to this problem. We investigate the behaviour of minimum norm solutions. Under certain applicable conditions, we prove existence as well as uniqueness and derive the explicit form of the minimum norm solution. This constitutes a nonlinear inversion operator for the blind deconvolution problem. The solution depends continuously on the given data provided that the data fulfil a weak smoothness condition. In a sense, blind deconvolution is less ill-posed than non-blind deconvolution. Given noisy data, this smoothness condition is no longer satisfied. We utilize Tikhonov regularization of a Sobolev embedding operator to restore smoothness, so that the inversion operator may be applied. We note that regularization and inversion are two separate tasks. We prove convergence of the regularized solution to the noise-free minimum norm solution and, when the noise-free data fulfil a stronger Sobolev smoothness condition, we give a convergence rate result. Our approach is non-iterative and thus very fast. It conserves mass and symmetry of the kernel and works robustly for a wide range of images and kernels. No knowledge of exact kernel shape and support size is necessary.