Discrete and Continuous Models and Applied Computational Science (Oct 2022)
On the application of the Fourier method to solve the problem of correction of thermographic images
Abstract
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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