Geodesic
Solving inverse problems of achieving super-resolution using neural networks
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 18 (2025) no. 1, pp. 104-117

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The actual problem of obtaining approximate numerical solutions of inverse problems in the form of Fredholm integral equations of the first kind for radio and sonar systems and remote sensing is considered. The obtained solutions make it possible to significantly increase the accuracy of measurements, as well as to bring the angular resolution to values exceeding the Rayleigh criterion. This allows you to: – receive detailed radio images of various objects and probed areas; – determine the number of individual small-sized objects in the composition of complex targets that were not recorded separately without the presented signal processing; – to obtain the coordinates of such small-sized objects with high accuracy; – to increase the probability of obtaining correct solutions to problems of recognition and identification of objects. The method is applicable for modern multi-element measuring systems. It is based on extrapolation of signals received by all elements outside the system itself. The problem of creating the necessary neural network and its training has been solved. As a result, a new virtual measuring system of a much larger size is synthesized, which makes it possible to dramatically increase the angular resolution and thereby improve the quality of approximate solutions to the inverse problems under consideration. Examples demonstrate the effectiveness of the method, assess the adequacy and stability of the solutions obtained. The degree of excess of the Rayleigh criterion by the virtual goniometer system depending on the signal-to-noise ratio is investigated.
Keywords: Rayleigh criterion, Fredholm integral equation, neural network.
Mots-clés : extrapolation 
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     title = {Solving inverse problems of achieving super-resolution using neural networks},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
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B. A. Lagovsky; E. Y. Rubinovich; I. A. Yurchenkov. Solving inverse problems of achieving super-resolution using neural networks. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 18 (2025) no. 1, pp. 104-117. http://geodesic.mathdoc.fr/item/VYURU_2025_18_1_a7/