On the application of the Fourier method to solve the problem of correction of thermographic images

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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.

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1. 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

About the authors

Obaida Baaj

Peoples’ Friendship University of Russia (RUDN University)

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

postgraduate student of Nikolskiy Mathematical Institute

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


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