# Development of a mathematical model of the process of transferring landscape images through a noisy communication channel

### Abstract

The necessity of modeling systems for transmitting landscape images through communication channels is substantiated. Algorithms for transforming images for the modeling process have been developed. An algorithm for generating random noise in the communication channel based on the Poisson distribution has also been developed. A model has been developed that can be used to solve a variety of remote sensing data transmission problems.

#### For citations:

Kostrov B.V.,
Grinchenko N.N.,
Khizrieva N.I.,
Fokina N.S.
Development of a mathematical model of the process of transferring landscape images through a noisy communication channel. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2020;(4):101-107.
https://doi.org/10.38013/2542-0542-2020-4-101-107

At the present time, for error protection of transmitted information, methods of noiseless coding are used, aimed at detection and correction of errors occurring in the course of data transmission through communication channels. One of the important stages in designing a data transmission system is selection of an appropriate coding method from a large number of the existing ones. To make the task simpler, a modelling process is often applied, during which a model of data transmission system is used for experimental research and the most suitable method of noiseless coding is determined, with its parameters best befitting a particular situation [1, 2].

In the process of modelling a mathematical model is built, being both the means and the object of experiments. Using the revealed mathematical relationships, it is necessary to simulate progressing through all stages of system operation. Certain initial data are fed to the model input, and parameters requiring correction are set. Obtained at the output is a certain result which may either conform to the expectations or, on the contrary, be just the opposite. Depending on this, the modelling parameters are corrected and new relationships are revealed. In the course of modelling there may as well emerge new interrelationships of parameters and exceptional situations that were not detected at the hypothesis shaping stage.

Selected as an object of modelling is the process of data transmission through a noisy communication channel. When images are transferred through a communication channel, errors will occur [3]. They appear in the form of distorted pixels or areas in the image [4]. Images themselves are transferred row-by-row, therefore they have to be vectorised. Since data are transmitted through communication channel in the binary form, to simulate channel operation, images must be presented in the form of binary sequences [5]. For that purpose an algorithm for image transformation into binary vector and an inverse transform algorithm were developed.

When using the direct transform algorithm, it is necessary to proceed through the initial image matrix row-by-row and alternate transfer of elements to the vector from left to right and right to left, depending on row number. It is presumed that in this case there will be no break in pixel brightness values at vertical boundaries of the image and no ripple of traffic in the network.

The transform formula can be written as:

where i and j – numbers of row and column of the initial image matrix, N – number of rows in the initial image, k – number of bits for image pixel representation, M – initial image matrix, V – resultant vector.

A schematic depiction of the transform is given in Figs. 1 and 2.

**. Direct transform**

Fig. 1

Fig. 1

**. Direct transform – order of pixels in transmission vector**

Fig. 2

Fig. 2

The direct transform algorithm comprises the following steps:

1. Matrix elements of the initial grayImg view are substituted for their binary representations in binMatrix of M×(N×k) size, where k – maximum number of binary symbols required for representation of grayImg matrix elements, and M×N – initial image size.

2. Vector binArray with the length of M×(N×k) is generated, filled with zeros.

3. In the cycle, it is necessary to come through all binMatrix rows. If it is an odd row, matrix row elements are successively, from the first one through (N×k), entered into the vector. If it is an even row, then the elements are entered into the vector starting from the last one in the row, i.e. from (N×k), to the first.

During information transfer to DTN, the communication channel is exposed to various external effects and noise. As a result, signal distortions occur, having a negative impact on the quality of information transmitted through the channel. In many cases noise occurs randomly. Therefore, in the modelling process it is necessary to select a suitable law of random variable distribution (RVD), on the base of which noise in the communication channel model will be generated. According to the modelling conditions, image is transmitted through the communication channel in binary form. Accordingly, noise is generated in binary form as well. It follows, then, that random variable in this case takes a value from {0; 1} array, which means that the given random variable is a discrete one [6]. For modelling of the noise generation process, it was decided to use the Poisson distribution. This distribution is discrete, representing one of the extreme cases of binomial distribution [7]. Using the Poisson distribution it is possible to simulate occurrence of rarer events, when normal distribution cannot be applied for their description. The probability formula in this case has the view:

where k = 0, 1, 2…, λ – parameter being set.

The results obtained from calculation of probabilities according to this distribution are closest to the actual pattern of distortions occurring in the communication channel.

Noise vector generation algorithm:

1. Setting parameter λ. For best compliance with reality, it should satisfy the inequation 0.01 ≤ λ ≤ 0.5.

2. Calculating probabilities P1 and P2 for appearance of 0 and 1 under Poisson distribution with the set λ.

3. Calculating ratio P of 0 appearance probability to the sum of probabilities P1 and P2.

4. Generating 0 and 1 in a random way, with 0 appearance probability being equal to P.

5. Writing the obtained elements into a vector of specified length.

Noise vector superposition algorithm:

1. Image vector (imgVector) and noise vector (noiseVector) with length N are supplied to the algorithm input.

2. A resultant vector (noisedVector), equal to the image vector, is created.

3. The resultant vector and the image vector are added element-by-element in the cycle. Elements equal to 2 are replaced by 0.

Given in Fig. 3 is a flowchart of binary noise vector generation algorithm, and in Fig. 4, a flowchart of the algorithm of noise superposition on the image vector.

**. Flowchart of noise vector generation algorithm**

Fig. 3

Fig. 3

**. Flowchart of noise vector superposition algorithm**

Fig. 4

Fig. 4

For inverse transform of the obtained vector to the matrix form, it is necessary to come through the resultant image matrix row-by-row, alternating transfer of elements from vector to matrix from left to right and right to left, depending on row number (Fig. 5). Mathematically, it can be represented as follows:

where i and j – numbers of row and column of the resultant image matrix, N – number of rows in the resultant image, k – number of bits for image pixel representation, M – resultant image matrix, V – vector with superposed noise.

**. Inverse transform**

Fig. 5

Fig. 5

**. Flowchart of inverse transform algorithm**

Fig. 6

Fig. 6

Inverse transform algorithm:

1. Matrix binMatrix of M×(N×k) size is generated, filled with zeros.

2. In the cycle, it is necessary to come through all binMatrix rows. If it is an odd row, elements from the vector, from the first one through (N×k), are successively entered into the matrix row. If it is an even row, elements from the vector are entered into the row starting from the last one in the row, i.e. from (N×k), to the first.

3. The binMatrix elements are written in the decimal form and entered into a matrix of M×N size. This matrix will actually be the image.

Given in Fig. 7 is the initial image, and Figs. 8–10 show the results of implementing these algorithms in the MATLAB program environment [8]. During modelling, noise magnitude is set by means of special parameter λ.

**. Initial image**

Fig. 7

Fig. 7

**. Result for noise with λ = 0.0005**

Fig. 8

Fig. 8

**. Result for noise with λ = 0.01**

Fig. 9

Fig. 9

**. Result for noise with λ = 0.05**

Fig. 10

Fig. 10

This model allows to set different frequencies of error occurrence in the binary communication channel, which makes it possible to evaluate the efficiency of different noiseless coding methods as applies to transmission of Earth remote sensing data. The developed model may as well be used eventually for solving other problems pertaining to efficient transmission of landscape images through communication channels.

## References

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### About the Authors

**B. V. Kostrov**Russian Federation

Kostrov Boris Vasilievich - Dr. Sci. (Engineering), Prof., Head of the Planning and Control Automation Sector, Sigma Design Bureau, Research interests: image processing, artificial intelligence, information technologies.

Ryazan

**N. N. Grinchenko**Russian Federation

Grinchenko Natalya Nikolaevna - Cand. Sci. (Engineering), Assoc. Prof., Leading Software Engineer, Sigma Design Bureau, Research interests: application of information technologies and multimedia systems in image processing.

Ryazan

**N. I. Khizrieva**Russian Federation

Khizrieva Natalya Ilesovna - Software Engineer, Sigma Design Bureau, Research interests: application of information technology and multimedia systems in image processing.

Ryazan

**N. S. Fokina**Russian Federation

Fokina Natalya Sergeevna - Deputy General Director for General Issues, Research interests: application of information technologies and multimedia systems in image processing.

Ryazan### Review

#### For citations:

Kostrov B.V.,
Grinchenko N.N.,
Khizrieva N.I.,
Fokina N.S.
Development of a mathematical model of the process of transferring landscape images through a noisy communication channel. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2020;(4):101-107.
https://doi.org/10.38013/2542-0542-2020-4-101-107