Issues

 / 

2001

 / 

May

  

Reviews of topical problems


Wavelets and their uses

, ,
Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation

This review paper is intended to give a useful guide for those who want to apply the discrete wavelet transform in practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to the corresponding literature. The multiresolution analysis and fast wavelet transform have become a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for the achievement of a goal. Analysis of various functions with the help of wavelets allows one to reveal fractal structures, singularities etc. The wavelet transform of operator expressions helps solve some equations. In practical applications one often deals with the discretized functions, and the problem of stability of the wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves to a few examples only. The authors would be grateful for any comments which would move us closer to the goal proclaimed in the first phrase of the abstract.

Fulltext is available at IOP
PACS: 02.60.−x, 02.70.−c, 05.40.−a, 87.57.−s (all)
DOI: 10.1070/PU2001v044n05ABEH000918
URL: https://ufn.ru/en/articles/2001/5/a/
Citation: Dremin I M, Ivanov O V, Nechitailo V A "Wavelets and their uses" Phys. Usp. 44 447–478 (2001)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Дремин И М, Иванов О В, Нечитайло В А «Вейвлеты и их использование» УФН 171 465–501 (2001); DOI: 10.3367/UFNr.0171.200105a.0465

References (92) ↓ Cited by (124) Similar articles (20)

  1. Meyer Y Wavelets and Operators (Cambridge: Cambridge Univ. Press, 1992)
  2. Daubechies I Ten Lectures on Wavelets (Philadelphia: SIAM, 1991)
  3. Meyer Y, Coifman R Wavelets, Calderon - Zygmund and Multilinear Operators (Cambridge: Cambridge Univ. Press, 1997)
  4. Meyer Y Wavelets: Algorithms and Applications (Philadelphia: SIAM, 1993)
  5. Progress in Wavelet Analysis and Applications (Eds Y Meyer, S Roques, Gif-sur-Yvette: Editions Frontiers, 1993)
  6. Chui C K An Introduction to Wavelets (San Diego: Academic Press, 1992)
  7. Hernandez E, Weiss G A First Course on Wavelets (Boca Raton: CRC Press, 1997)
  8. Kaiser G A Friendly Guide to Wavelets (Boston: Birkhauser, 1994)
  9. Wavelets: An Elementary Treatment of Theory and Applications (Ed T Koornwinder, Singapore: World Scientific, 1993)
  10. Astaf’eva N M Usp. Fiz. Nauk 166 1145 (1996) [Phys. Usp. 39 1085 (1996)]
  11. Carmona R, Hwang W-L, Torresani B Practical Time-Frequency Analysis (San Diego: Academic Press, 1998)
  12. Grossman A, Morlet J "Decomposition of functions into wavelets of constant shape, and related transforms", in Mathematics+Physics, Lectures on Recent Results Vol. 1 (Ed. L Streit, Singapore: World Scientific, 1985)
  13. Morlet J, Arens G, Fourgeau E, Giard D Geophysics 47 203, 222 (1982)
  14. Wavelets in Physics (Ed. J C Van den Berg, Cambridge: Cambridge Univ. Press, 1999)
  15. Mallat S A Wavelet Tour of Signal Processing (San Diego: Academic Press, 1998)
  16. Erlebacher G, Hussaini M Y, Jameson L M Wavelets: Theory and Applications (New York: Oxford Univ. Press, 1996)
  17. Wavelets in Medicine and Biology (Eds A Aldroubi, M Unser, Boca Raton: CRC Press, 1996)
  18. Haar A Math. Ann. 69 331 (1910)
  19. Sweldens W Appl. Comput. Harmon. Anal. 3 186 (1996)
  20. Goedecker S, Ivanov O V Comput. Phys. 12 548 (1998)
  21. Ausher P Ondelettes Fractales et Applications (Paris: Univ. Paris, Dauphine, 1989)
  22. Beylkin G SIAM J. Numer. Anal. 6 1716 (1992)
  23. Beylkin G, Coifman R, Rokhlin V Commun. Pure Appl. Math. 44 141 (1991)
  24. Jaffard S, Meyer Y Mem. Am. Math. Soc. 123 587 (1996)
  25. Dremin I M Phys. Lett. B 313 209 (1993)
  26. Dremin I M, Hwa R C Phys. Rev. D 49 5805 (1994)
  27. Dremin I M, Gary J W Phys. Rep. 349 301 (2001); hep-ph/0004215
  28. Mallat S, Hwang W L IEEE Trans. Inform. Theory 38 617 (1992)
  29. Mandelbrot B B The Fractal Geometry of Nature (San Francisco: W.H. Freeman, 1982)
  30. Feder J Fractals (New York: Plenum Press, 1988)
  31. Paladin G, Vulpiani A Phys. Rep. 156 147 (1987)
  32. Dremin I M Usp. Fiz. Nauk 160 (8) 105 (1990) [Sov. Phys. Usp. 33 647 (1990)]
  33. Renyi A Probability Theory (Amsterdam: North-Holland, 1970)
  34. Muzy J F, Bacry E, Arneodo A Phys. Rev. Lett. 67 3515 (1991); Int. J. Bifurcat. Chaos 4 245 (1994)
  35. Arneodo A, d’Aubenton-Carafa Y, Thermes C Physica D 96 291 (1996)
  36. De Wolf E A, Dremin I M, Kittel W Phys. Rep. 270 1 (1996)
  37. Takayasu H Fractals in the Physical Sciences (Manchester: Manchester Univ. Press, 1990)
  38. Jaffard S Multifractal Formalism for Functions (Philadelphia: SIAM, 1997)
  39. Donoho D, Johnstone I J. Am. Stat. Assoc. 90 1200 (1995)
  40. Dremin I M, Ivanov O V, Nechitaîlo V A Nauka - Proizvodstvu (6) 13 (2000)
  41. Spiridonov V P Komp’yuterra (8) 38 (1998)
  42. Torresani B, in Self-similar Systems (Eds V B Priezzhev, V P Spiridonov, Dubna: JINR, 1999)
  43. Bowman C, Newell A C Rev. Mod. Phys. 70 289 (1998)
  44. Parr R G, Yang W Density-Functional Theory of Atoms and Molecules (New York: Oxford Univ. Press, 1989)
  45. Goedecker S, Ivanov O V Solid State Commun. 105 665 (1998)
  46. Wei S, Chou M Y Phys. Rev. Lett. 76 2650 (1997)
  47. Han S, Cho K, Ihm J Phys. Rev. B 60 1437 (1999)
  48. Tymczak C J, Wang X Phys. Rev. Lett. 78 3654 (1997)
  49. Ivanov O V, Antropov V P J. Appl. Phys. 85 4821 (1999)
  50. Ivanov O V Phys. Rev. B (2001, to be published)
  51. Goedecker S, Ivanov O V Phys. Rev. B 59 7270 (1999)
  52. Carruthers P, in Proc. of Hot and Dense Matter, Bodrum, 1993 (Singapore: World Scientific, 1994) p. 65
  53. Lipa P, Greiner M, Carruthers P, in Proc. of Soft Physics and Fluctuations, Crakow, 1993 (Singapore: World Scientific, 1994) p. 105
  54. Greiner M et al. Z. Phys. C 69 305 (1996)
  55. Suzuki N, Biyajima M, Ohsawa A Prog. Theor. Phys. 94 91 (1995)
  56. Huang D Phys. Rev. D 56 3961 (1997)
  57. Sarcevic I, Huang Z, Thews R Phys. Rev. D 54 750 (1996)
  58. Nandi B K et al. (WA98 Collab.), in Proc. of 3rd Int. Conference on Physics and Astrophysics of Quark-Gluon Plasma, Jaipur, 1997 (Singapore: World Scientific, 1998) p. 12
  59. Astafyeva N M, Dremin I M, Kotelnikov K A Mod. Phys. Lett. A 12 1185 (1997)
  60. Apanasenko A V, Dobrotin N A, Dremin I M et al. Pis’ma Zh. Eksp. Teor. Fiz. 30 157 (1979) [JETP Lett. 30 145 (1979)]
  61. Alekseeva K I et al. Izv. Akad. Nauk SSSR 26 572 (1962); J. Phys. Soc. Jpn. 17 409 (1962)
  62. Maslennikova N V et al. Izv. Akad. Nauk SSSR 36 1696 (1972)
  63. Arata N Nuovo Cimento A 43 455 (1978)
  64. Dremin I M, Orlov A M, Tretyakova M I Proc. 17 ICRC 5 149 (1981); Pis’ma Zh. Eksp. Teor. Fiz. 40 320 (1984) [JETP Lett. 40 1115 (1984)]
  65. Marutyan I A et al. Yad. Fiz. 29 1566 (1979)
  66. Adamus M et al. (NA22 Collab.) Phys. Lett. B 185 200 (1987)
  67. Bialas A, Peschanski R Nucl. Phys. B 273 703 (1988)
  68. Dremin I M Pis’ma Zh. Eksp. Teor. Fiz. 30 152 (1979) [JETP Lett. 30 140 (1979)]
  69. Dremin I M Yad. Fiz. 33 1357 (1981)
  70. Dremin I M, Ivanov O V, Kalinin S A et al. Phys. Lett. B 499 97 (2001)
  71. Adamovich M I et al. (EMU01 Collab.) J. Phys. G 19 2035 (1993)
  72. Cherry M L et al. (KLM Collab.) Acta Phys. Pol. B 29 2129 (1998)
  73. Dremin I M, Lasaeva P L, Loktionov A A et al. Yad. Fiz. 52 840 (1990) [Sov. J. Nucl. Phys. 52 840 (1990)]; Mod. Phys. Lett. A 5 1743 (1990)
  74. Agababyan N M et al. Phys. Lett. B 389 397 (1996)
  75. Dremin I M Usp. Fiz. Nauk 170 1235 (2000) [Phys. Usp. 43 1137 (2000)]
  76. Georgantas A A Review of Compressor Aerodynamic Instabilities (Canada: National Aeronautical Establishment, 1994)
  77. Dremin I M, Furletov V I, Ivanov O V et al. Control Eng. Pract. (2001, to be published)
  78. Akay M IEEE Spectrum 34 50 (1997)
  79. Lambrou T, Linney A, Speller R Komp’yuterra (6) 50 (1998)
  80. Thurner S, Feurstein M C, Teich M C Phys. Rev. Lett. 80 1544 (1998)
  81. Amaral L A N, Goldberger A L, Ivanov P C et al. Phys. Rev. Lett. 81 2388 (1998)
  82. Ivanov P C, Amaral L A N, Goldberger A L et al. Nature 399 461 (1999)
  83. Yang F, Liao W IEEE Eng. Med. Biol. 16 17 (1997)
  84. Ivanov P C et al. Nature 383 323 (1996)
  85. Blanco S et al. IEEE Eng. Med. Biol. 16 64 (1997)
  86. Schiff S J et al. Electroen. Clin. Neurophysiol. 91 442 (1994)
  87. Blanco S et al. Phys. Rev. E 54 6661 (1996)
  88. Arneodo A et al. J F Phys. Rev. Lett. 74 3293 (1995)
  89. Arneodo A et al. Physica A 254 24 (1998)
  90. Peng C K et al. Nature 356 168 (1992)
  91. Antonini M, Barlaud M, Mathieu P, Daubechies I IEEE Trans. on Image Process. 1 205 (1992)
  92. Wallace G K, in Compression Standard, Communications of the ACM, April 1991, ftp://ftp.uu.net/graphics/jpeg/wallace.ps.gz

© 1918–2021 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions