Application of the Minimum Principle of a Tikhonov Smoothing Functional in the Problem of Processing Thermographic Data

Abstract

The paper considers a method for correcting thermographic images. Mathematical processing of thermograms is based on the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study to the heat sources. The continuation is performed by solving an ill posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. The cylindrical area is bounded by an arbitrary surface and plane. The Cauchy conditions are set on the surface-the boundary values of the desired function and its normal derivative. Inhomogeneous conditions of the first kind are set on the side faces of the cylinder. The problem is the inverse of the corresponding mixed problem for the Poisson equation. In this paper, an approximate solution of the problem is obtained that is stable with respect to the error in the Cauchy data and inhomogeneity in the boundary conditions. In the course of constructing an approximate solution, the problem is reduced to the Fredholm integral equation of the first kind, which is solved using the minimum smoothing functional principle. The convergence of the approximate solution of the problem is proved when the regularization parameter is matched to the error in the data.