Summary: | Background. 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 the
known operator (functional) from the object. 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 problem 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 Gato (or Freshe) derivatives of
nonlinear operators. Results and conclusions. An approximate method for reconstructing a
signal from its autocorrelation function and calculating the phase of its spectrum from the
reconstructed 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.
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