Restoration of a blurred photographic image of a moving object obtained at the resolution limit

Abstract

Objectives. When processing images of the Earth’s surface obtained from satellites, the problem of restoring a blurry image of a moving object is of great practical importance. The aim of this work is to study the possibility of improving the quality of restoration of blurry images obtained at the limit of the resolution of the camera.Methods. Digital signal processing methods informed by the theory of incorrect and ill-conditioned problems were used.Results. The proposed method for restoring a blurred photographic image of a moving object differs from traditional approaches in that the discrete convolution equation, to which the problem of restoring a blurred image is reduced, is obtained by approximating the corresponding integral equation based on the Kotelnikov interpolation series rather than on the traditional basis of the quadrature formula. In the work, formulas are obtained for calculating the kernel of the convolution obtained using the Kotelnikov interpolation series. The discrete convolution inversion problem, which belongs to the class of ill-posed problems, requires regularization. Results of traditional approaches to restoring blurred images using the quadrature formula with Tikhonov regularization and the proposed method based on the Kotelnikov interpolation series are compared. Although the quality of the blurred image restoration is almost the same in both cases, in the quadrature formula the blur value is expressed as an integer number of pixels, while, when using the Kotelnikov series, this value can also be specified in fractions of a pixel.Conclusions. The expediency of discretizing the convolution describing the image distortion of the blur type on the basis of the Kotelnikov interpolation series when processing a blurred image obtained at the limit of the resolution of the camera is demonstrated. In this case, the amount of blur can be expressed in fractions of a pixel. This situation typically arises when processing satellite photography of the Earth’s surface.

Keywords

• blurred image of a moving object
• resolution
• image restoration
• tikhonov regularization
• inverse problem
• kotelnikov interpolation series
• edge effect