(46-1) 08 * << * >> * Russian * English * Content * All Issues

Increasing the spatial resolution of signals in optical systems
V.I. Guzhov 1, I.O. Marchenko 1, E.E. Trubilina 1

1 Novosibirsk State Technical University,
630073, Novosibirsk, Novosibirsk, Russia, Nemirovicha-Danchenko 136

 PDF, 991 kB

DOI: 10.18287/2412-6179-CO-924

Pages: 65-70.

Full text of article: Russian language.

Abstract:

Reconstruction of the signal in the intervals between discrete values is of great importance in solving the problem of spatial superresolution in optical microscopy and digital holography. The article deals with the issue of restoring high-resolution image elements from a certain number of raster images displaced by a sub-pixel shift. The numerical values of the image samples are obtained by spatial integration over some finite area of regular rasters. The spatial resolution is increased using an analytical expression for the spectrum of discrete signals obtained using the apparatus of generalized functions. Unlike ideal sampling, the spectrum of the function is supplemented by a multiplier, whose form depends on the type of aperture. To obtain high-resolution image elements, it is necessary to divide the Fourier spectrum of the sampled image by a factor depending on the selected aperture. The spectrum of the aperture is usually used, therefore, if the spectrum of the image obtained by averaging over a certain aperture is known, then the spectrum of the original image can also be obtained.
     Apertures of various shapes are used, for example, elliptical, diamond-shaped, hexagonal, but most often rectangular apertures. The simulation results are presented for a rectangular aperture but in the case of its substitution with, for example, a set of regular apertures in the form of a circle, the expression will be true for regular circular rasters.
     The analytical expression for the spectrum of the image obtained by averaging over a certain aperture can be used to reconstruct the spectrum of the original image. Having received the inverse Fourier transform from it, it is possible to obtain the original image.
     With an increase in the spatial resolution it becomes possible to carry out studies by methods of digital holography of volumetric diffuse objects while retaining the quality of analog holography (when recording in photographic media) and create optical superresolution systems based on optical microscopes.

Keywords:
sampling, sampling rate, spatial frequencies, generalized functions, Kotelnikov's theorem, Fourier transform, spectrum, superresolution.

Citation:
Guzhov VI, Marchenko IO, Trubilina EE. Increasing the spatial resolution of signals in optical systems. Computer Optics 2022; 46(1): 65-70. DOI: 10.18287/2412-6179-CO-924.

References:
  1. Kotel'nikov VA. On the transmission capacity of "ether" and wire in telecommunications. Physics-Uspekhi 2006; 49(7): 736-744.
  2. Nyquist H. Certain topics in telegraph transmission theory. Trans Am Inst Electr Eng 1928; 47: 617-644.
  3. Wittaker ET. On the function which are represented by the expansion of interpolating theory. Proc R Soc Edinb 1915; 35: 181-194.
  4. Shannon CE. Communication in the presence of noise. Proceedings of Institute of Radio Engineers 1949; 37(1): 10-21.
  5. Guzhov VI, Trubilina EE, Marchenko IO. Signal recovery with a limited number of ideal discrete samples [In Russian]. Scientific Bulletin of NSTU 2020; 1(78): 147-156. DOI: 10.17212/1814-1196-2020-1-147-156.
  6. Vaskov ST, Efimov VM, Reznik AL. Fast digital reconstruction of signals and images by the criterion of minimum energy [In Russian]. Avtometriya 2003; 39(4): 13-20.
  7. Belov AM, Denisova AY. Spectral and spatial super-resolution method for Earth remote sensing image fusion. Computer Optics 2018; 42(5): 855-863. DOI: 10.18287/2412-6179-2018-42-5-855-863.
  8. Popov MA, Stankuvich SA, Shklyar SV. Algorithm for increasing the resolution of subpixel-shifted images [In Russian]. Mathematical Machines and Systems 2015; 1: 29-36.
  9. Guzhov VI, Marchenko IO, Pozdnyakov GA, Il'inykh SP. Super resolution in digital holography [In Russian]. Engineering Journal of Don: Electronic Scientific Journal 2018; 3.
  10. Wagner O, Schwarz A, Shemer A, Ferreira C, Garcia J, Zalevsky Z. Superresolved imaging based on wavelength multiplexing of projected unknown speckle patterns. Appl Opt 2015; 54(13): D51-D60.
  11. Blazhevich SV, Selyutina ES. Sub-pixel scanning to produce super-resolution digital images [In Russian]. Scientific Statements. Series: Mathematics. Physics 2014; 5(176:34): 186-190.
  12. Guzhov VI, Il'inykh SP, Marchenko IO. Method of increasing spatial resolution in digital holographic microscopy [In Russian]. Avtometriya 2018; 54(3): 104-110. DOI: 10.15372/AUT20180313.
  13. Guzhov VI, Il'Inykh SP, Marchenko IO. Method of increasing the spatial resolution in digital holographic microscopy. Optoelectronics, Instrumentation and Data Processing 2018; 54(3): 301-306. DOI: 10.3103/S8756699018030135.
  14. Kokoshkin AV, Korotkov VA, Korotkov KV, Novichikhin EP. Estimation of super-resolution imaging errors based on the use of multiple frames. Computer Optics 2017; 41(5): 701-711. DOI: 10.18287/2412-6179-2017-41-5-701-711.
  15. Pavelyeva EA. Image processing and analysis based on the use of phase information. Computer Optics 2018; 42(6): 1022-1034. DOI: 10.18287/2412-6179-2018-42-6-1022-1034.
  16. Guzhov VI, Marchenko IO, Trubilina EE, Trubilin AA. Sampling of signals using a finite set of apertures [In Russian]. Omsk Scientific Bulletin 2021; 1.
  17. Schwartz L. Théorie des distributions. Paris: Hermann; 1950-1951.
  18. Schwartz L. Mathematical methods for physical sciences [In Russian]. Moscow: "Mir" Publisher; 2005.
  19. Gelfand IM, Shilov GE. Generalized functions and actions on them [In Russian]. Moscow: "Gosudarstvennoe Izdatel'stvo Fiziko-Matematicheskoi Literatury" Publisher; 1959.
  20. Vladimirov VS. Generalized functions in mathematical physics [In Russian]. Moscow: Science" Publisher; 1979.
  21. Sobolev SL. Some applications of functional analysis in mathematical physics [In Russian]. Moscow: "Nauka" Publisher; 1988.
  22. Kech V, Teodorescu P. Introduction to the theory of generalized functions with applications in technology [In Russian]. Moscow: "Mir" Publisher; 1978.

© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: journal@computeroptics.ru ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20