Analysis of global and local extrema of regularizing functionals in the problem of the reconstruction of images observed through randomly inhomogeneous media

An approach to the reconstruction problem of undistorted images of objects observed through randomly inhomogeneous media is considered. The approach is based on the solution to the integral equation of the first kind (convolution equation) with an unknown kernel and inaccurately defined right-hand side. It is demonstrated that the procedure for the construction of an inverse operator needed for the solution is determined by the minimization of the nonlinear regularizing multiextremum functional, which takes into account the a priori data regarding the positive intensity distribution in the image of an object. The analysis of the structure of the multiextremum functional is presented. The estimates of the reconstructed images that correspond to the global and local extrema are studied. The results of a numerical experiment on the reconstruction of the optical image of an object observed through the turbulent atmosphere are presented.