Research algorithm restoring signals in the basis of exponential functions

The paper proposes a method for constructing digital filters for solving inverse problems of recovering signals, time series, and images using an approximation approach. The considered inverse problems belong to the class of incorrectly posed ones and require the use of certain regularizing procedures to synthesize optimal reconstruction algorithms and solve computational problems associated with this. In this regard, a method is proposed for constructing an approximation model of the weight function of the inverse (reconstruction) filter, based on the criterion of the minimum quadratic error of the mismatch of the distorted signal model obtained after the direct (distorting) filter processes the reconstructed (unknown) signal and the existing distorted signal. The weight function of the direct filter is assumed to be known. The statement of the problem of reconstructing signals, time series, and images in the case of a one-dimensional point scattering function is formulated. An algorithm is presented that allows one to reduce the amount of computation when finding the values ​​of the weight function of the inverse filter. The algorithms were tested on model examples when processing real images obtained by remote sensing of the Earth, as well as on specially formed contrast images. To quantify the quality of reconstruction, a relative mean-square measure of the difference between the reference and reconstructed signals (images) was used. The results of testing show that using this approach allows to reduce the error of recovery, which gives an advantage in solving problems of approximation and data recovery.