On reconstruction of astronomical images in observations through turbulent atmosphere B. Zhilyaev 1 , V. Reshetnyk 2 , V. Petukhov 1 1 Main Astronomical Observatory of National Academy of Sciences of Ukraine, Kiev 2 Kyiv National Taras Shevchenko University, Kyiv zhilyaev@mao.kiev.ua (Submitted on 20.02.2018. Accepted on 01.05.2018) Abstract. We demonstrate the technique for reconstruction of astronomical images, obtained during observations through a turbulent atmosphere. We use combined method, which includes: averaging of centered images; the filtration of their Fourier transforms by the inverse optical transfer function for the Kolmogorov model of atmospheric turbulence, and subsequent inverse Fourier transform. We use observational data from the high-speed CMOS camera on the Zeiss-600 telescope of Andrushevska Observatory with an exposure time of 0.01 sec and a shooting frequency of 30 frames per second. We show that the angular resolution can be improved up to the diffraction limit for observations made on small telescopes. Key words: atmospheric turbulence - image reconstruction - stars: individual: 61 Cyg A Introduction Atmospheric turbulence distorts the wavefront of the light wave. This re- sults in degradation of the image, loss of angular resolution, which can be from one to several arc seconds. Fried (1967) established some important properties of atmospheric tur- bulence: the coherence radius r 0 , the coherence time, and the isoplanatic angle. The coherence radius is defined as the length of a wavefront area over which the rms phase variations are equal to 1 rad. At best sites r 0 ranges from 10-30 cm, the coherence time is 10-50 milliseconds and the isoplanatic angle is 2-10 arc seconds. Another important property related to turbulence is seeing. It is defined as the full width of the star image at half maximum (FWHM). At best sites seeing disk is 1/2 to 1/3 arc second. Image degradation has the main two types: image motion and image blurring. The motion of the image is due to the distortion of the slopes of the wavefront. The blurring looks like a spotted pattern. The way to reduce the turbulence effect is to reduce the exposure time, so that it freezes the atmospheric turbulence. The exposure time is usually chosen to be equal to the coherence time. The degradation of the image is usually eliminated by methods of adap- tive optics. At the same time, it is possible to achieve a diffraction quality of the image in a small area of the focal plane (area of isoplanatism). It requires, however, a complex and expensive technique (Labeyrie, 2013). We can specify one of the image recovery methods. This is the so-called ”lucky imaging”. The randomness that prevails in a turbulent field rarely produces ”lucky” images of high quality. A special technique (stacking, shift-and-add) allows to combine these images together, achieving nearly diffraction quality. A number of large telescopes in the world are equipped with systems for observing ”lucky imaging”. Here should be mentioned the 5 m telescope at Bulgarian Astronomical Journal 29, 2018