Solving inverse problems of obtaining super-resolution using neural networks

This paper considers the 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 systems using neural networks. The solutions make it possible to significantly increase the accuracy of measurements and to bring the angular resolution to values exceeding the Rayleigh criterion. This allows detailed radio images of various objects and probed areas to be received; the number of individual small-sized objects to be determined in the composition of complex targets that were not recorded separately without the presented signal processing; the coordinates of such small-sized objects to be obtained with high accuracy; and the probability of obtaining correct solutions to problems of recognition and identification of objects to be increased. The method is applicable for multi-element measuring systems. It is based on the extrapolation of signals received by all elements outside the system itself. The problem of creating and training a neural network 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 and assess the adequacy and stability of the solutions. The degree of excess of the Rayleigh criterion by the virtual goniometer system depending on the signal-to-noise ratio is also investigated.