350 rub
Journal Nonlinear World №8 for 2013 г.
Article in number:
Simple generator of the fractal images
Keywords: fractal image generator fractional dimension LabVIEW
Authors:
S.A. Ostanin
Abstract:
The generator of stochastic 2-D images of any fractal dimension is described. The generator is built using technology LabVIEW. The mathematical basis of the generator is a 2-D Fourier transform of the noise. The spectrum of 2-D noise is multiplied by a power function, then the inverse Fourier transform. This transformation is equivalent to non-linear filtering of noise. The result is a stochastic fractal 2-D image. Model parameters related to the fractal dimension of the image. The paper presents examples of the generated fractal images and surfaces. Shows a block diagram of a stochastic fractal image generator. The device is simple and can be easily reproduced. Using LabVIEW technology enables the integration of the generator with other programs or devices. The generator can be used, including, for the purposes of mathematical modeling of images and surfaces of different nature.
Pages: 540-544
References

  1. Feder E. Fraktaly'. M.: Mir. 1991.
  2. Mandelbrot B.B. Fractals: Form, Chance and Dimension. San-Francisco: W.H. Freeman and Company. 1977.
  3. Schroder M.R. Fractals, Chaos, Power Laws: Minutes from The Infinite Paradise. New York: Freeman. 1991.
  4. Potapov A.A. Fraktaly' v radiofizike i radiolokaczii: Topologiya vy'borki. M.: Universitetskaya kniga. 2005.
  5. Potapov A.A., Gulyaev Ju.V., Nikitov S.A., Paxomov A.A., German V.A. Novejshie metody' obrabotki izobrazhenij. M.: Fizmatlit. 2008.
  6. Pleshanov V.S., Napryushkin A.A., Kibitkin V.V. Osobennosti primeneniya teorii fraktalov v zadachax analiza izobrazhenij // Avtometriya. 2010. T. 46. № 1. S. 86(96.
  7. Kurdyukov V.I., Ostapchuk A.K., Ovsyannikov V.E., Rogov E.Ju. Razrabotka fraktal'noj matematicheskoj modeli sheroxovatosti poverxnosti // Vestnik KuzGU. 2008. № 5. S. 43-46.
  8. Goroxov D.B., Ogar P.M. Modelirovanie fraktal'ny'x sheroxovaty'x poverxnostej // Matematicheskoe modelirovanie, chislenny'e metody' i kompleksy' programm. 2004. Vy'p. 10. S. 235-240.
  9. Laktyun'kin A.V. Modelirovanie fraktal'ny'x nedifferencziruemy'x poverxnostej i proczessov rasseyaniya imi e'lektromagnitny'x voln // Nelinejny'j mir. 2007. T. 5. № 5. S. 286-287.
  10. Laktyun'kin A.V., Potapov A.A. Zavisimost' proczessov rasseyaniya voln ot statisticheskix parametrov klassicheskix i fraktal'ny'x sheroxovaty'x poverxnostej // Nelinejny'j mir. 2008. T. 6. № 4. S. 231-233.
  11. Maureen E. Cianciolo Cumulus cloud scene simulation modeling using fractals and physics // Proc. SPIE 1819 Digital Image Processing and Visual Communications Technologies in the Earth and Atmospheric Sciences II. 70 (March 26, 1993); doi:10.1117/12.142209
  12. Mandelbrot B. The Fractal Geometry of Nature. San Francisco: Freeman. 1982.
  13. Arakawa K., Krotkov E. Fractal Modeling of Natural Terrain: Analysis and Surface Reconstruction with Range Data // Graphical Models and Image Processing. 1996. V. 58. Is. 5. P. 413-436.
  14. Miller, Gavin S.P. The Definition and Rendering of Terrain Maps. Conference Proceedings: SIGGRAPH 1986 // Computer Graphic. 1986. V. 20. № 4. P. 39(48.
  15. Voss R. Fractals in nature: characterization, measurement, and simulation // Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '87). Anaheim. Calif. USA. July 1987. 
  16. Peitgen H.-O., Hartmut J., Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag. 2004.
  17. Fournier A., Fussell D., Carpenter L. Computer Rendering of Stochastic Models // Communications of the ACM. 1982. V. 25. P. 38.
  18. Zaslavskij G.M., Sharkov E.A. Fraktal'ny'e svojstva zon obrushayushhixsya voln na poverxnosti morya // Dokl. AN SSSR. Mexanika zhidkostej i gazov. 1987. T. 294. C. 1362-1366.
  19. Safonov A.A., Shterenberg A.M. Modelirovanie fraktal'noj poverxnosti polimerov, sformirovanny'x gazorazryadnoj polimerizacziej // Vestn. Sam. gos. texn. un-ta. Ser. Fiz.-mat. nauki. 2010. № 5. T. 21. C. 212-221.
  20. Malinnikov V.A., Nzexa M., Uchaev D.V., Uchaev Dm.V. Metodika geoinformaczionnogo modelirovaniya struktury' drevnix poselenij na osnove fraktal'ny'x metodov // Izvestiya vy'sshix uchebny'x zavedenij. Razdel Geodeziya i ae'rofotos''emka. 2009. № 3. S. 76-79.
  21. Driessen P. A Fractal Model Of The Universe / Examining the Big Bang and Diffuse Background Radiations // International Astronomical Union. Union Astronomique Internationale. 1996. V. 168. P. 467-468.
  22. Beech M. The projection of fractal objects // Astrophysics and Space Science. 1992. V. 192. Is. 1. P. 103-111.
  23. Majumdar A., Bhushan B. Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surfaces // ASME J. of Tribology. 1990. V. 112. P. 205-216.
  24. Shamin R.V. Poverxnostny'e volny' na vode minimal'noj gladkosti // Sovremennaya matematika. Fundamental'ny'e napravleniya. 2010. T. 35. S. 126-140.
  25. Yan W., Komvopoulos K. Contact analysis of elastic-plastic surfaces // J. Appl. Phys., 1998. V. 84. № 7. P. 3617-3624.
  26. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Application of Fractional Differential Equations. North-Holland Mathematics Studies. Amsterdam: Elsevier Science, 2006.
  27. Flandrin P. On the Spectrum of Fractional Brownian Motions // IEEE Trans. on Info. Theory. 1989. V. IT-35. № 1. P. 197(199.
  28. Ostanin S.A. Generator fraktal'nogo shuma proizvol'noj razmernosti // Zhurnal radioe'lektroniki: e'lektron. zhurn. 2012. № 8. URL: http://jre.cplire.ru/jre/aug12/2/text.html (data obrashheniya: 04.05.2013).
  29. Berry M.V. Diffractals // J. Phys. A. 1979. V. 12. № 6. R. 781-797.
  30. Berry M.V., Lewis Z.V. On the Weierstrass-Mandelbrot fractal function // Proc. R. Soc. Lond. 1980. A 370. R. 459-484.
  31. Grudin B.N., Plotnikov V.S., Kislenok E.G., Fishhenko V.K. Analiz, fil'tracziya i dekompozicziya mikroskopicheskix izobrazhenij na osnove ortogonal'ny'x preobrazovanij // Avtometriya. 2007. T. 43. № 1. S. 24-36.