2212, 2021


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In the world, scientific research is being conducted to improve the quality level of digital television images, methods for modeling filtration processes and highly efficient control systems in a number of priority areas, including: on the formation of mathematical models of filtration processes, improving the methods of wavelet, Fourier, Haar, Walsh-Hadamard, Karhunen-Loev in increasing the clarity and brightness of images based on linear and nonlinear differential equations; creation of methods for eliminating additive, pulsed and adaptive-Gaussian types of noise in images using additive and adaptive filtering; methods of algorithms and software for introducing intra-frame and inter-frame image transformations; methods of adaptive brightness system control using the Chebyshev matrix series; methods of gradient, static and Laplace methods for image segmentation and dividing it into contours; formation of criteria and conditions for evaluating image quality. Conducting scientific research in the above research areas confirms the relevance of the topic of this article.


Today, in the world in the field of information and communication technologies, close attention is paid to the control system for processing digital television images in video information systems. In the conditions of intensive improvement of modern information and communication systems to increase the volume and information flow, one of the urgent problems is to improve the quality of television images and control the filtration processes from excess information. In this direction, in the field of information and communication technologies in the leading countries of the world, the demand and need for improving filtering methods and increasing the brightness of digital television images are increasing.

Currently, one of the most important issues is the formation of digital television images, based on them, the improvement of the image processing control system, methods of numerical models and algorithms for solving problems of filtering various digital television images using Fourier and wavelet methods. Purposeful scientific research is carried out in this area, including close attention is paid in the following areas: improved method of classification and selection of criteria for observing and evaluating image quality, methods for controlling image clarity at given values of medium-intensity pixels, creating algorithms for modeling the image processing process, methods for controlling the processes of ensuring the level of clarity of a digital image:

Image filtering  by the convolution method with an impulse response  the case of a continuous image is mathematically described as follows [1; 7-8-p.]:

where  brightness distribution in the image after filtering,  integration variables. When implementing this method of filtering digitally, the original image, the image after filtering, as well as the impulse response are represented as arrays of numbers, the elements of which are denoted respectively by  и h(k,n) and the numbers of rows and columns-through  and . In this case, the brightness of the pixels of the filtered image is calculated as follows:


where  and N- the length of the two-dimensional impulse response in both directions. Values  and  they are selected odd in order to avoid shifting the filtered image relative to the original one.

When filtering, the image is scanned by a window (pulse response), the dimensions of which are  pixels. Each window sample represents a weighting factor (the value of the impulse response), by which the image pixel covered by this window sample is multiplied. In this case, the intensity of the pixel of the filtered image, whose coordinates coincide with the coordinates of the center of the window, is found by summing all the products [1].

Impulse response h(k,n)  when developing a digital filter, it is found as follows. First, the frequency transfer function of the analog filter is found . Then, by applying a two-dimensional integral Fourier transform to it, the corresponding impulse response is found h(x,y)


The impulse response found in this way must be converted into a discrete form by means of its spatial sampling, while the step of spatial sampling must be the same as the step of spatial sampling of the filtered image. The next operation to be performed on the sampled impulse response is its truncation, i.e., limiting its size by rows and columns to reasonable limits. The fact is that frequency transfer functions bounded in the frequency space by boundary frequencies  correspond to the impulse characteristics that are unlimited in the coordinate space (x,y) The last and final operation is the normalization of the truncated impulse response, as a result of which the sum of its samples should become equal to one, i.e.

Due to the normalization of the impulse response after its truncation, the correct reproduction of the average brightness in the filtered image is ensured, which would otherwise be disturbed due to the truncation operation. Turning to the problem of truncation of the impulse response, we note that the greater its length, the greater the amount of calculations must be performed when implementing digital filtering by the method under consideration. In addition, the edge effect will appear on most of the image. Simple truncation of the impulse response by multiplying it by the window function  удовлетворяющую условию [3; 2-3-p.]

it leads to the appearance of undesirable “undulation” of the frequency transfer function, as well as to its expansion in the frequency domain. To achieve a compromise between the length of the impulse response in the image space and the frequency transfer function in the frequency space [2], a number of windows of a special shape were developed, among which the most famous are: the triangular Bartlett window, the Blackman window, the Hahn window, the Kaiser window, and the Hamming window satisfying the condition

An important feature of these windows is that when approaching the truncation boundary, the value of  it gradually decreases, due to which the effects of “undulation” and the expansion of the frequency transfer function are weakened. After finding the impulse response  it is necessary to investigate it for separability with respect to variables  and . If it turns out that it is separable, i.e. if

where  one-dimensional impulse characteristics, then the expression (2.1) it should be converted to the form


Calculating values  according to the formula (3) allows you to significantly reduce the number of necessary mathematical operations compared to the number of mathematical operations when using the formula (1). So, for example, if when calculating   according to the formula (2 t o determine the value of one sample of the filtered image, you need to perform (K-1)N multiplication operations and (K-1)(N-1) addition operations, then in the case of calculation   according to the formula (3) the number of necessary multiplication operations is reduced to K+N and the number of addition operations is reduced to K+N-2 If you accept K=7, N=7 what is quite a bit for typical filtering problems, even in this case, the gain in the amount of necessary computational costs provided by using the separability property of the impulse response will be 3.5 times for multiplication operations and 3 times for addition operations. In fact, the gains when using the separability property of the impulse response are significantly greater. It should be noted that a number of impulse characteristics, which often have to deal with in practice, are separable. These include: the impulse response described by the Gaussian law, the impulse response having a constant value inside a rectangular window, and some others. Next, you should pay attention to two more significant circumstances that are important to keep in mind when developing a filter. First, it is necessary to set limiters for the brightness value of the filtered image before its presentation with an eight-digit code, preventing it from going beyond the accepted dynamic range. The appearance of such brightness values is possible if there are outliers on the transition characteristic of the filter, due, for example, to a sharp decline in the frequency transfer function. In this case, the absence of limiters will lead to an overflow of the discharge grid, which will lead to the appearance of black dots and spots on the light areas of the filtered image, and white dots and spots, respectively, on the dark areas. The use of limiters of the dynamic range of the signal from the white side and from the black side allows you to avoid these artifacts, although it introduces the so-called restriction noise into the filtered image [3].

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