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Recognition of wavefront aberrations types corresponding to single Zernike functions from the pattern of the point spread function in the focal plane using neural networks
I.A. Rodin 1, S.N. Khonina 1,2, P.G. Serafimovich 2, S.B. Popov 1,2

Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34,
IPSI RAS – Branch of the FSRC "Crystallography and Photonics" RAS,
443001, Samara, Russia, Molodogvardeyskaya 151

 PDF, 1363 kB

DOI: 10.18287/2412-6179-CO-810

Pages: 923-930.

Full text of article: Russian language.

In this work, we carried out training and recognition of the types of aberrations corresponding to single Zernike functions, based on the intensity pattern of the point spread function (PSF) using convolutional neural networks. PSF intensity patterns in the focal plane were modeled using a fast Fourier transform algorithm. When training a neural network, the learning coefficient and the number of epochs for a dataset of a given size were selected empirically. The average prediction errors of the neural network for each type of aberration were obtained for a set of 15 Zernike functions from a data set of 15 thousand PSF pictures. As a result of training, for most types of aberrations, averaged absolute errors were obtained in the range of 0.012 – 0.015. However, determining the aberration coefficient (magnitude) requires additional research and data, for example, calculating the PSF in the extrafocal plane.

wavefront aberrations, point spread function, focal plane, fast Fourier transform, neural networks.

Rodin IA, Khonina SN, Serafimovich PG, Popov SB. Recognition of wavefront aberrations types corresponding to single Zernike functions from the pattern of the point spread function in the focal plane using neural networks. Computer Optics 2020; 44(6): 923-930. DOI: 10.18287/2412-6179-CO-810.

The study was carried out with the financial support of the RFBR in the framework of the scientific project No. 19-29-09054 in terms of machine learning and neural networks, as well as the Ministry of Science and Higher Education of the Russian Federation in within the framework of work under the State task of the Federal Research Center "Crystallography and Photonics" RAS (agreement No. 007-GZ / Ch3363 / 26) parts of aberrated wavefront modeling and PSF calculation.


  1. Welford WT. Aberrations of optical systems. Bristol, Philadelphia: Adam Hilger Press; 1986. ISBN: 978-0-85274-564-9.
  2. Charman WN. Wavefront aberrations of the eye: A review. Optom Vis Sci 1991; 68(8): 574-583. DOI: 10.1097/00006324-199108000-00002.
  3. Beckers JM. Adaptive optics for astronomy: principles, performance, and applications. Annu Rev Astron Astrophys 1993; 31(1): 13-62. DOI: 10.1146/annurev.aa.31.090193.000305.
  4. Hardy JW. Adaptive optics for astronomical telescopes. Oxford, Oxford University Press; 1998. ISBN: 978-0-19-509019-2.
  5. Booth MJ. Adaptive optics in microscopy. Philos Trans Royal Soc A 2007; 365(1861): 2829-2843. DOI: 10.1098/rsta.2007.0013.
  6. Atchison DA. Wavefront aberrations and their clinical application. Clin Exp Optom 2009; 92(3): 171-172. DOI: 10.1111/j.1444-0938.2009.00380.x.
  7. Lombardo M, Lombardo G. Wave aberration of human eyes and new descriptors of image optical quality and visual performance. J Cataract Refract Surg 2010; 36(2): 313-320. DOI: 10.1016/j.jcrs.2009.09.026.
  8. Khorin PA, Khonina SN, Karsakov AV, Branchevskiy SL. Analysis of corneal aberration of the human eye. Computer Optics 2016; 40(6): 810-817. DOI: 10.18287/0134-2452-2016-40-6-810-8179.
  9. Klebanov IM, Karsakov AV, Khonina SN, Davydov AN, Polyakov KA. Wave front aberration compensation of space telescopes with telescope temperature field adjustment. Computer Optics 2017; 41(1): 30-36. DOI: 10.18287/0134-2452-2017-41-1-30-36.
  10. Buscher DF. Practical optical interferometry. Cambridge: Cambridge University Press; 2015. ISBN: 978-1-107-04217-9.
  11. Malacara D, ed. Optical shop testing. Hoboken, NJ: John Wiley & Sons Inc; 2007. ISBN: 978-0-471-48404-2.
  12. Vasil’ev LA. Schlieren methods. New York, NY, Jerusalem, Israel, London, UK: John Wiley & Sons; 1972.
  13. Hartmann J. Bemerkungen über den bau und die justierung von spektrographen. Zeitschrift für Instrumentenkunde 1900; 20: 17-27, 47-58.
  14. Artzner G. Microlens arrays for Shack-Hartmann wavefront sensors. Opt Eng 1992; 31(6): 1311-1322. DOI: 10.1117/12.56178.
  15. Platt BC, Shack R. History and principles of Shack-Hartmann wavefront sensing. J Refract Surg 2001; 17(5): S573-S577. DOI: 10.3928/1081-597X-20010901-13.
  16. Hongbin Y, Guangya Z, Siong CF, Feiwen L, Shouhua WA. Tunable Shack–Hartmann wavefront sensor based on a liquid-filled microlens array. J Micromech Microeng 2008; 18(10): 105017. DOI: 10.1088/0960-1317/18/10/105017.
  17. Zernike F. How I discovered phase contrast. Science 1955; 121(3141): 345-349. DOI: 10.1126/science.121.3141.345.
  18. Vorontsov MA, Justh EW, Beresnev LA. Advanced phase-contrast techniques for wavefront sensing and adaptive optics. Proc SPIE 2000; 4124: 98-109. DOI: 10.1117/12.407492.
  19. Daria VR, Rodrigo PJ, Sinzinger S, Glückstad J. Phase-only optical decryption in a planar-integrated micro optics system. Opt Eng 2004; 43(10): 2223-2227. DOI: 10.1117/1.1782613.
  20. Sendhil K, Vijayan C, Kothiyal MP. Spatial phase filtering with a porphyrin derivative as phase filter in an optical image processor. Opt Commun 2005; 251(4-6): 292-298. DOI: 10.1016/j.optcom.2005.03.014.
  21. Komorowska K, Miniewicz A, Parka J, Kajzar F. Self-induced nonlinear Zernike filter realized with optically addressed liquid crystal spatial light modulator. J Appl Phys 2002; 92(10), 5635-5641. DOI: 10.1063/1.1515949.
  22. Born M, Wolf E. Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light. 7th ed. Cambridge: Cambridge University Press; 1999. ISBN: 978-0-521-64222-4.
  23. Roddier N. Atmospheric wavefront simulation using Zernike polynomials. Opt Eng 1990; 29(10): 1174-1180. DOI: 10.1117/12.55712.
  24. Neil MAA, Booth MJ, Wilson T. New modal wave-front sensor: a theoretical analysis. J Opt Soc Am A 2000; 17(6): 1098-1107. DOI: 10.1364/JOSAA.17.001098.
  25. Thibos LN, Applegate RA, Schwiegerling JT, Webb R. Standards for reporting the optical aberrations of eyes. J Refract Surg 2002; 18(5): 652-660.
  26. ANSI Z80.28. Methods for reporting optical aberrations of eyes. American National Standards Institute Inc, American National Standards for Ophthalmics; 2004.
  27. Martins AC, Vohnsen B. Measuring ocular aberrations sequentially using a digital micromirror device. Micromachines 2019; 10(2): 117. DOI: 10.3390/mi10020117.
  28. Khonina SN, Kotlyar VV, Soifer VA, Wang Y, Zhao D. Decomposition of a coherent light field using a phase Zernike filter. Proc SPIE 1998; 3573: 550-553. DOI: 10.1117/12.324588.
  29. Sheppard CJR. Zernike expansion of pupil filters: optimization of the signal concentration factor. J Opt Soc Am A 2015; 32(5): 928-933. DOI: 10.1364/JOSAA.32.000928.
  30. Porfirev AP, Khonina SN. Experimental investigation of multi-order diffractive optical elements matched with two types of Zernike functions. Proc SPIE 2016; 9807: 98070E. DOI: 10.1117/12.2231378.
  31. Khonina SN, Karpeev SV, Porfirev AP. Wavefront aberration sensor based on a multichannel diffractive optical element. Sensors 2020; 20(14): 3850. DOI: 10.3390/s20143850.
  32. Khonina SN, Kotlyar VV, Kirsh DV. Zernike phase spatial filter for measuring the aberrations of the optical structures of the eye. J-BPE 2015; 1(2): 146-153. DOI: 10.18287/jbpe-2015-1-2-146.
  33. Gerchberg R, Saxton W. Phase determination for image and diffraction plane pictures in the electron microscope. Optik 1971; 34: 275-284.
  34. Fienup JR. Reconstruction of an object from the modulus of its Fourier transform. Opt Lett 1978; 3(1): 27-29. DOI: 10.1364/OL.3.000027.
  35. Elser V. Phase retrieval by iterated projections. J Opt Soc Am A 2003; 20(1): 40-55. DOI: 10.1364/JOSAA.20.000040.
  36. Marchesini S. A unified evaluation of iterative projection algorithms for phase retrieval. Rev Sci Instrum 2007; 78(1): 011301. DOI: 10.1063/1.2403783.
  37. Zhang C, Wang M, Chen Q, Wang D, Wei S. Two-step phase retrieval algorithm using single-intensity measurement. Int J Opt 2018; 2018: 8643819. DOI: 10.1155/2018/8643819.
  38. Tokovinin A, Heathcote S. DONUT: measuring optical aberrations from a single extrafocal image. Publ Astron Soc Pac 2006; 118(846): 1165-1175. DOI: 10.1086/506972.
  39. Guo H, Korablinova N, Ren Q, Bille J. Wavefront reconstruction with artificial neural networks. Opt Express 2006; 14(14): 6456-6462. DOI: 10.1364/OE.14.006456.
  40. Paine SW, Fienup JR. Machine learning for im-proved image-based wavefront sensing. Opt Lett 2018; 43(6): 1235-1238. DOI: 10.1364/OL.43.001235.
  41. Rivenson Y, Zhang Y, Günaydın H, Teng D, Ozcan A. Phase recovery and holographic image reconstruction using deep learning in neural networks. Light Sci Appl 2018; 7(2): 17141. DOI: 10.1038/lsa.2017.141.
  42. Dzyuba AP. Optical phase retrieval with the image of intensity in the focal plane based on the convolutional neural networks. J Phys Conf Ser 2019; 1368(2): 022055. DOI: 10.1088/1742-6596/1368/2/022055.
  43. Nishizaki Y, Valdivia M, Horisaki R, Kitaguchi K, Saito M, Tanida J, Vera E. Deep learning wavefront sensing. Opt Express 2019; 27(1): 240-251. DOI: 10.1364/OE.27.000240.
  44. Fischer P, Dosovitskiy A, Brox T. Image orientation estimation with convolutional networks. In Book: German Conference on Pattern Recognition. Cham: Springer; 2015: 368-378. DOI: 10.1007/978-3-319-24947-6_30.
  45. Chollet F. Xception: Deep learning with depthwise separable convolutions. Proc IEEE Conf on Comp Vis Pattern Recogn 2017: 1251-1258. DOI: 10.1109/CVPR.2017.195.

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