Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3220210.22363/2658-4670-2022-30-3-205-216Research 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 Тихонова[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.]