On an Approximate Method for Recovering a Function from Its Autocorrelation Function

When solving many physical and technical problems, a situation arises when only operators (functionals) from the objects under study (signals, images, etc.) are available for observations (measurements). It is required to restore the object from its known operator (functional). In many cases, the correlation (autocorrelation) function acts as an operator. A large number of papers have been devoted to the study of the existence of a solution to the problem of signal reconstruction from its autocorrelation function and the uniqueness of this solution. Since the solution to the problem of restoring a function from its autocorrelation function is not known in an analytical form, the problem of developing approximate methods arises. This is relevant not only in the problems of signal and image recovery, but also in solving the phase problem. From the above, the relevance of the problem of restoring a function (images) from the autocorrelation function follows. The article is devoted to approximate methods for solving this problem. Materials and methods. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations, based on the theory of stability of solutions to ordinary differential equation systems. The method is stable under perturbations of the parameters of the mathematical model and, when solving nonlinear operator equations, does not require the reversibility of the Gateaux (or Freshet) derivatives of nonlinear operators. Results. In this work, an approximate method of signal reconstruction from its autocorrelation function and calculation of the phase of its spectrum from the reconstructed signal is constructed and substantiated. Conclusions. An approximate method for reconstructing a signal from its autocorrelation function and calculating the phase of its spectrum from the recovered signal is constructed and substantiated. The method does not require additional information about the signal under study. The results of the work can be used in solving a number of problems in optics, crystallography, and biology.