On a modification of the Hamming method for summing discrete Fourier series and its application to solve the problem of correction of thermographic images

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Abstract

The paper considers mathematical methods of correction of thermographic images (thermograms) in the form of temperature distribution on the surface of the object under study, obtained using a thermal imager. The thermogram reproduces the image of the heat-generating structures located inside the object under study. This image is transmitted with distortions, since the sources are usually removed from its surface and the temperature distribution on the surface of the object transmits the image as blurred due to the processes of thermal conductivity and heat exchange. In this paper, the continuation of the temperature function as a harmonic function from the surface deep into the object under study in order to obtain a temperature distribution function near sources is considered as a correction principle. This distribution is considered as an adjusted thermogram. The continuation is carried out on the basis of solving the Cauchy problem for the Laplace equation - an ill-posed problem. The solution is constructed using the Tikhonov regularization method. The main part of the constructed approximate solution is presented as a Fourier series by the eigenfunctions of the Laplace operator. Discretization of the problem leads to discrete Fourier series. A modification of the Hamming method for summing Fourier series and calculating their coefficients is proposed.

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1. Introduction Thermal imaging methods are widely used in medicine as a means of early diagnostics [1-4]. Visualization (thermogram) of the temperature distribution on the surface of the patient’s body contains information about sources of heat inside the body associated with the functioning of internal organs. In particular, it contains information about temperature anomalies associated with pathologies of internal organs. The image on the thermogram, as a rule, is distorted due to the process of thermal conductivity, heat exchange and the relative remoteness of heat sources from the surface of the body. Within the framework of the chosen mathematical model, it is possible to correct the image on the thermogram in order to increase the effectiveness of diagnostics. Since the evolution of the temperature distribution in the patient’s body is relatively slow, it makes it possible to use stationary models, in particular, models of harmonic temperature distribution. As an adjusted thermogram, we will consider the temperature distribution near the sources obtained by the continuation of the harmonic function from the boundary (similar to the continuation of gravitational fields in geophysics problems [5]). In [6], based on the method [7], one of the possible solutions to such a problem is proposed. The problem, as ill-posed, is solved using the Tikhonov regularization method [8]. When forming computational algorithms, discrete Fourier series [9, 10] are used, the coefficients of which are calculated from functions depending on the coefficient number [11]. To sum up such series, a modification of the Hamming method [9] is proposed here. 2. Mathematical model and inverse problem As a mathematical model, we consider a homogeneous heat-conducting body in the form of a rectangular cylinder
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About the authors

Evgeniy B. Laneev

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: elaneev@yandex.ru
ORCID iD: 0000-0002-4255-9393

Doctor of Physical and Mathematical Sciences, Professor of Mathematical Department

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Obaida Baaj

Peoples’ Friendship University of Russia (RUDN University)

Email: 1042175025@rudn.ru
ORCID iD: 0000-0003-4813-7981

Post-Graduate Student of Mathematical Department

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

References

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Copyright (c) 2022 Laneev E.B., Baaj O.

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