Restoration of nonuniformly smeared images

Abstract

This paper considers the problem of mathematically eliminating a nonuniform rectilinear smear in an image, for example, a picture taken by a motionless camera of runners on a track, running at different speeds. The problem is described by a set of one-dimensional integral equations of a general type (not a convolution type) with a two-dimensional point scattering function or one two-dimensional integral equation with a four-dimensional point scattering function. The integral equations are solved by Tikhonov regularization and quadrature/cubature. It is shown that, in the case of a nonuniform smear, the use of a set of one-dimensional integral equations is preferable to one two-dimensional integral equation. In the direct problem, the image smear is supplemented by truncating it, evading the boundary conditions, and diffusing its edges to suppress the Gibbs effect in the inverse problem. Cases of piecewise uniform and continuously (linearly) nonuniform smear are considered. Illustrative results are presented.