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Recognition of the types of aberrations corresponding to individual Zernike functions were carried out from the pattern of the intensity of the point spread function (PSF) outside the focal plane using convolutional neural networks. The PSF intensity patterns outside the focal plane are more informative in comparison with the focal plane even for small values/magnitudes of aberrations. The mean prediction errors of the neural network for each type of aberration were obtained for a set of 8 Zernike functions from a dataset of 2 thousand pictures of out-of-focal PSFs. As a result of training, for the considered types of aberrations, the obtained averaged absolute errors do not exceed 0.0053, which corresponds to an almost threefold decrease in the error in comparison with the same result for focal PSFs.

We performed a detailed comparative study of the parametric high degree (cubic, fourth, and fifth) power phase apodization on compensation defocusing and chromatic aberration in the imaging system. The research results showed that increasing the power degree of the apodization function provided better independence (invariance) of the point spread function (PSF) from defocusing while reducing the depth of field (DOF). This reduction could be compensated by increasing the parameter α; however, this led to an increase in the size of the light spot. A nonlinear relationship between the increase in the DOF and spot size was shown (due to a small increase in the size of the light spot, the DOF can be significantly increased). Thus, the search for the best solution was based on a compromise of restrictions on the circle of confusion (CoC) and DOF. The modeling of color image formation under defocusing conditions for the considered apodization functions was performed. The subsequent deconvolution of the resulting color image was demonstrated.

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One of the most important factors for improving the resolution of optical systems is to compensate for the aberrations (distortions) of the wave front. As a rule, whether special measuring devices (wavefront sensors) are used for such compensation or adaptive mirrors that perform iterative correction of the wavefront. However, often (for reasons of compactness or weight reduction), it is not possible to use the special equipment for measuring aberrations. To obtain certain information on the wave front, one can use the measured point spread function (PSF) or the intensity pattern in the focal plane. Methods of processing two PSFs (focal and nonfocal) with the help of neural networks are known. In this paper, we investigate the possibility of recognizing the wave front from a single intensity pattern in the focal plane. The technology of deep machine learning - convolutional neural network is chosen as the way for implementation. The idea of this technology lies in the alternation of convolutional and subsampling layers, for the purpose of efficient image recognition. Such approach will allow to optimize the process of compensation of optical system aberrations and to reduce the amount of required input data.