Mathematical model for reconstructing a damaged bitmap

A. A. Klyachin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2016), pp. 48-56.

Résumé

The paper describes an algorithm for restoring a damaged image, based on the use of maximum and minimum Lipschitz function defined in a flat area. Namely, we will assume that the image is given by the function $u=f(x,y)$, where $x=0,...,M$, $y=0,...,N$, and its value is a brightness level of point $ (x, y) $, which varies in the range of $ u = 0, ..., U $. We consider the current window of the size $ (2n + 1) \times (2n + 1) $ with center at the point $ (x, y) $, where $ n = 1,2, ... $ As the output luminance of the point corresponding to the center of the window, take the value $$ F_{\alpha}^{n}(x,y,z)=\min\{f(i,j)+\alpha\sqrt{(x-i)^2+(y-j)^2+z^2}:|x-i|\leq n, |y-j|\leq n\}, $$ where $x=n,...,M-n$, $y=n,...,N-n$. To suppress the local minima we can use the dual function that looks like this $$ G_{\alpha}^{n}(x,y,z)=\max\{f(i,j)-\alpha\sqrt{(x-i)^2+(y-j)^2+z^2}:|x-i|\leq n, |y-j|\leq n\}. $$ Next, it is necessary to define for each current point $(x,y)$ which of these functions must be applied. To do this, we proceed as follows. In one pass through all the points $(x,y)$ are determined by the image of a local maximum and local minimum points. Repeated passage of this information is taken into account for the determination of the function used. Response $H(x,y,z)$ of our filter is calculated according to the rule $$ H(x,y,z)= \begin{cases} F_{\alpha,n}(x,y,z), if\ (x,y)\ is\ point\ of\ local\ maximum,\\ G_{\alpha,n}(x,y,z), if\ (x,y)\ is\ point\ of\ local\ minimum,\\ f(x,y), otherwise. \end{cases} $$ We show examples of operation of this algorithm for images with varying degrees of damage. We consider images having $20\%$–$75\%$ of the defects. Presented algorithm quite well restores the image with different types of lesions: how random nature with a uniform distribution over the entire image (impulse noise), and concentrated in certain areas.

Détail

Keywords: data recovery, median filter, Lipschitz condition.
Mots-clés : impulse noise, bitmap

Bibliographie

[1] S.\;V. Belim, S.\;A. Seliverstov, “The Analytic Hierarchy Method-Based Algorithm for Restoring Broken Pixels on the Noisy Images”, Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.\;E. Baumana, 2014, no. 11, 521–534

[2] S.\;V. Belim, A.\;O. Mayorov-Zilbernagel, “Algorithm for Searching the Broken Pixels and Eliminating Impulse Noise in Images Using a Method of Association Rules”, Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.\;E. Baumana, 2014, no. 12, 716–737

[3] V.\;Yu. Zinyakov, A.\;E. Gorodetskiy, A.\;Yu. Kuchmin, E.\;I. Zelenev, N.\;V. Alferova, “Reconstructing Two-Dimensional Images with Defects”, Informatsionno-upravlyayushchie sistemy, 2013, no. 3, 8–15

[4] R. Gonsales, R. Vuds, Digital Image Processing, Tekhnosfera Publ., M., 2005, 1071 pp.

[5] A.\;A. Klyachin, V.\;M. Miklyukov, “Traces of Functions with Spacelike Graphs, and the Extension Problem under Restrictions on the Gradient”, Sbornik: Mathematics, 183:7 (1992), 49–64

[6] V.\;S. Sizikov, R.\;A. Ekzemplyarov, “Operating Sequence When Noise Is Being Filtered on Distorted Images”, Journal of Optical Technology, 80:1 (2013), 39–48