Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia3220210.22363/2658-4670-2022-30-3-205-216Research ArticleOn the application of the Fourier method to solve the problem of correction of thermographic imagesBaajObaida<p>postgraduate student of Nikolskiy Mathematical Institute</p>1042175025@rudn.ruhttps://orcid.org/0000-0003-4813-7981Peoples’ Friendship University of Russia (RUDN University)0510202230320521605102022Copyright © 2022, Baaj O.2022<p style="text-align: justify;">The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.</p>thermogramill-posed problemCauchy problem for the Laplace equationintegral equation of the first kindTikhonov regularization methodтермограмманекорректная задачазадача Коши для уравнения Лапласаинтегральное уравнение первого родаметод регуляризации Тихонова1. Introduction Improving the quality and information content of images obtained by thermal imaging methods using a thermal imager that registers thermal electromagnetic radiation from the surface of the object under study in the infrared range by their mathematical (digital) processing is an urgent problem. In particular, in medicine, thermal imaging has become an effective diagnostic tool [1-4]. The image on the thermogram, which is a visualization of the temperature distribution on the surface of the patient’s body, makes it possible © BaajO., 2022 This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by-nc/4.0/legalcode to assess functional anomalies in the state of his internal organs. At the same time, the image on the thermogram in some cases turns out to be somewhat distorted due to the processes of thermal conductivity and heat exchange. The paper proposes a method of image correction on a thermogram within a certain mathematical model. As an adjusted thermogram, the image of the temperature field on the plane near the density of heat sources is considered as more accurately transmitting the image of heat sources. It is proposed to obtain this field as a result of the continuation (similar to the continuation of gravitational fields in geophysics problems [5]) of the temperature distribution from the surface from which the initial thermogram is taken. The problem under consideration is ill-posed, since small errors in the initial data (the initial thermogram) may correspond to significant errors in solving the inverse problem. To construct its stable approximate solution, the Tikhonov regularization method [6] is used. 2. Mathematical model and problem statement Let’s consider a physical and mathematical model, in which we set the task of continuing from the boundary of the stationary temperature distribution. The physical model is a homogeneous heat-conducting body in the form of a rectangular cylinder, bounded by the surface[E. F. J. Ring, “Progress in the measurement of human body temperature,” IEEE Engineering in Medicine and Biology Magazine, vol. 17, no. 4, pp. 19-24, 1998. DOI: 10.1109/51.687959.][E. Y. K. Ng and N. M. Sudarshan, “Numerical computation as a tool to aid thermographic interpretation,” Journal of Medical Engineering and Technology, vol. 25, no. 2, pp. 53-60, 2001. DOI: 10.1080/03091900110043621.][B. F. Jones and P. Plassmann, “Digital infrared thermal imaging of human skin,” IEEE Eng. in Med. Biol. Mag., vol. 21, no. 6, pp. 41-48, 2002. DOI: 10.1109/memb.2002.1175137.][G. R. Ivanitskii, “Thermovision in medicine [Teplovideniye v meditsine],” Vestnik RAN, vol. 76, no. 1, pp. 44-53, 2006, in Russian.][A. N. Tikhonov, V. B. Glasko, O. K. Litvinenko, and V. R. Melihov, “On the continuation of the potential towards disturbing masses based on the regularization method [O prodolzhenii potentsiala v storonu vozmushchayushchih mass na osnove metoda regulyarizatsii],” Izvestiya AN SSSR. Fizika Zemli, no. 1, pp. 30-48, 1968, in Russian.][A. N. Tikhonov and V. J. Arsenin, Methods for solving ill-posed problems [Metody resheniya nekorrektnyh zadach]. Moscow: Nauka, 1979, in Russian.][E. B. Laneev, “Construction of a Carleman function based on the Tikhonov regularization method in an ill-posed problem for the Laplace equation,” Differential Equations, vol. 54, no. 4, pp. 476-485, 2018. DOI: 10.1134/S0012266118040055.][E. B. Laneev, N. Y. Chernikova, and O. Baaj, “Application of the minimum principle of a Tikhonov smoothing functional in the problem of processing thermographic data,” Advances in Systems Science and Applications, vol. 1, pp. 139-149, 2021. DOI: 10.25728/assa.2021.21.1.1055.][E. B. Laneev, M. N. Mouratov, and E. P. Zhidkov, “Discretization and its proof for numerical solution of a Cauchy problem for Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface,” Physics of Particles and Nuclei Letters, vol. 5, no. 3, pp. 164-167, 2002. DOI: 10.1134/S1547477108030059.][R. W. Hamming, Numerical methods for scientists and engineers. New York: McGraw-Hill Book Company, 1962.][H. Pennes, “Analysis of tissue and arterial blood temperature in the resting human forearm,” J. Appl. Physiol., no. 1, pp. 93-122, 1948.][J. P. Agnelli, A. A. Barrea, and C. V. Turner, “Tumor location and parameter estimation by thermography,” Mathematical and Computer Modelling, vol. 53, no. 7-8, pp. 1527-1534, 2011. DOI: 10.1016/j.mcm.2010.04.003.]