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ISSN: 1023-5086


ISSN: 1023-5086

Scientific and technical

Opticheskii Zhurnal

A full-text English translation of the journal is published by Optica Publishing Group under the title “Journal of Optical Technology”

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DOI: 10.17586/1023-5086-2020-87-02-56-63

УДК: 621.397.3

Restoration of nonuniformly smeared images

For Russian citation (Opticheskii Zhurnal):

Сизиков В.С., Довгань А.Н., Цепелева А.Д. Восстановление изображений, смазанных неравномерно // Оптический журнал. 2020. Т. 87. № 2. С. 56–63.


Sizikov V.S., Dovgan A.N., Tsepeleva A.D. Restoration of nonuniformly smeared images [in Russian] // Opticheskii Zhurnal. 2020. V. 87. № 2. P. 56–63.

For citation (Journal of Optical Technology):

V. S. Sizikov, A. N. Dovgan, and A. D. Tsepeleva, "Restoration of nonuniformly smeared images," Journal of Optical Technology. 87(2), 110-116 (2020).


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.


smeared image, uniform and nonuniform smear, integral equations, Tikhonov regularization method, truncation and diffusion of edges, piecewise continuously nonuniform smears, MatLab


The research was supported by Megafaculty of Computer Technology and Management of ITMO University (617026, 619296).

OCIS codes: 100.0100


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