Discrete Wavelet Transforms in Walsh Analysis
A review of discrete wavelet transforms defined through Walsh functions and used for image processing, compression of fractal signals, analysis of financial time series, and analysis of geophysical data is presented. Relationships of the discrete transforms considered with wavelet bases recently con...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 257; no. 1; pp. 127 - 137 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Springer US
04.08.2021
Springer Springer Nature B.V |
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| Abstract | A review of discrete wavelet transforms defined through Walsh functions and used for image processing, compression of fractal signals, analysis of financial time series, and analysis of geophysical data is presented. Relationships of the discrete transforms considered with wavelet bases recently constructed and frames on the Cantor and Vilenkin groups are noted. |
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| AbstractList | A review of discrete wavelet transforms defined through Walsh functions and used for image processing, compression of fractal signals, analysis of financial time series, and analysis of geophysical data is presented. Relationships of the discrete transforms considered with wavelet bases recently constructed and frames on the Cantor and Vilenkin groups are noted. Keywords and phrases: Walsh function, Haar system, Weierstrass function, wavelet, frame, zero-dimensional group, discrete transform, image processing, signal coding, analysis of geophysical data. AMS Subject Classification: 42C40, 65T60 A review of discrete wavelet transforms defined through Walsh functions and used for image processing, compression of fractal signals, analysis of financial time series, and analysis of geophysical data is presented. Relationships of the discrete transforms considered with wavelet bases recently constructed and frames on the Cantor and Vilenkin groups are noted. |
| Audience | Academic |
| Author | Farkov, Yu. A. |
| Author_xml | – sequence: 1 givenname: Yu. A. surname: Farkov fullname: Farkov, Yu. A. email: farkov@list.ru, farkov-ya@ranepa.ru organization: Russian Presidential Academy of National Economy and Public Administration |
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| Cites_doi | 10.1016/j.jmaa.2014.11.061 10.1137/1.9781611970104 10.1142/8431 10.1109/78.735301 10.18500/1816-9791-2016-16-2-217-225 10.1109/SAMPTA.2017.8024368 10.4213/sm1126 10.1090/mmono/239 10.4236/ajcm.2012.22011 10.1016/j.jmaa.2016.08.062 10.46753/pjaa.2015.v02i02.004 10.1016/j.jat.2008.10.003 10.1134/S2070046611040030 10.4213/mzm8704 10.1142/S0219691311004195 10.1134/S0371968514020125 10.46753/pjaa.2015.v02i02.002 10.1134/S2070046611030022 10.1007/BF01198002 10.1142/S0219691315500368 10.1142/S0219691315500022 10.1109/78.668788 10.1090/S0002-9947-1949-0032833-2 10.1155/S0161171298000428 |
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| Keywords | image processing discrete transform 42C40 Walsh function Weierstrass function wavelet zerodimensional group signal coding Haar system analysis of geophysical data 65T60 frame |
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| References | DaubechiesITen Lectures on Wavelets1992PhiladelphiaSIAM10.1137/1.9781611970104 Bl. Sendov, “Adapted multiresolution analysis,” in: Functions, Series, Operators, Memorial Conf. in Honor of the 100th Anniversary of the Birth of Prof. G. Alexits (1899–1978), Budapest, Hungary, August 9–13, 1999(L. Leindler et al., eds.), János Bolyai Math. Soc., Budapest (2002), pp. 23–38. InFKimSAn Introduction to Wavelet Theory in Finance: A Wavelet Multiscale Approach2012SingaporeWorld Scientific10.1142/8431 LangWCFractal multiwavelets related to the Cantor dyadic groupInt. J. Math. Math. Sci.199821307317160973910.1155/S0161171298000428 TchobanouMKMultidimensional Multi-Speed Signal Processing Systems2009MoscowTekhnosfera M. Vetterli and J. Kovačević, Wavelets and Subband Coding, Prentice Hall, New Jersey (1995). Yu. A. Farkov and E. A. Rodionov, “Algorithms for wavelet construction on Vilenkin groups,” p-Adic Numb. Ultr. Anal. Appl., 3, No. 1, 181–195 (2011). KrivosheinAProtasovVSkopinaMMultivariate Wavelet Frames2016SingaporeSpringer1366.42001 MallatSA Wavelet Tour of Signal Processing1999San DiegoAcademic Press0998.94510 KholshchevnikovaNSkvortsovVAOn U- and M-sets for series with respect to characters of compact zero-dimensional groupsJ. Math. Anal. Appl.20174461383394355473410.1016/j.jmaa.2016.08.062 Yu. A. Farkov, “Orthogonal wavelets in Walsh analysis,” in: Generalized Integrals and Harmonic Analysis (T. P. Lukashenko and A. P. Solodov, eds.), Izd. Mosk. Univ., Moscow (2016), pp. 62–75. RodionovEAOn applications of wavelets to the digital signal processingIzv. Saratov. Univ. Nov. Ser. Mat. Mekh. Inform.2016162217225352291610.18500/1816-9791-2016-16-2-217-225 FarkovYADiscrete wavelets and the Vilenkin–Chrestenson transformMat. Zametki2011896914928290814710.4213/mzm8704 N. Ya. Vilenkin, “On a class of complete orthogonal systems,” Izv. Akad. Nauk SSSR. Ser. Mat., 11, No. 4, 363–400 (1947). ChuiCKMhaskarHNOn trigonometric waveletsConstr. Approx.19939167190121576810.1007/BF01198002 FarkovYAPeriodic wavelets in Walsh analysisCommun. Math. Appl.2012332232422824037 LyubushinAAFarkovYASynchronous components of financial time seriesKomp. Issled. Model.201794639655 G. N. Agaev, N. Ya. Vilenkin, G. M. Jafarli, and A. I. Rubinstein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups [in Russian], Elm, Baku (1981). Z. Cvetkovi´c and M. Vetterli, “Oversampled filter banks,” IEEE Trans. Signal. Proc., 46, No. 5, 1245–1255 (1998). SchippFWadeWRSimonPWalsh Series: An Introduction to Dyadic Harmonic Analysis1990New YorkAdam Hilger0727.42017 E. V. Burnaev and N. N. Olenev, “Proximity measures based on wavelet coefficients for comparing statistical and calculated time series,” in: Collected Scientific and Methodical Papers [in Russian], 10, Izd. Vyatsk. Gos. Univ., Kirov (2006), pp. 41–51. M. Skopina, “p-Adic wavelets,” Poincar´e J. Anal. Appl., 2, 53–63 (2015). FarkovYAConstructions of MRA-based wavelets and frames in Walsh analysisPoincaré J. Anal. Appl.20152133634399391354.42059 FarkovYAOn wavelets related to the Walsh seriesJ. Approx. Theory.20091611259279255815510.1016/j.jat.2008.10.003 Yu. A. Farkov, “Nonstationary multiresolution analysis for Vilenkin groups,” in: Int. Conf. on Sampling Theory and Applications, Tallinn, Estonia, 3-7 July 2017, Tallinn (2017), pp. 595–598. Yu. A. Farkov and E. A. Rodionov, “On biorthogonal discrete wavelet bases,” Int. J. Wavelets Multires. Inf. Process., 13, No. 1, 1550002 (2015). B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Walsh Series and Transformations: Theory and Applications [in Russian], Moscow (2008). FarkovYAMaksimovAYStroganovSAOn biorthogonal wavelets related to the Walsh functionsInt. J. Wavelets Multires. Inform. Process.20119485499280472310.1142/S0219691311004195 BölcskeiHHlawatschFFeichtingerHGFrame-theoretic analysis of oversampled filter banksIEEE Trans. Signal. Proc.199846123256326810.1109/78.735301 FarkovYARodionovEANonstationary wavelets related to the Walsh functionsAm. J. Comput. Math.20122828710.4236/ajcm.2012.22011 KozyrevSVKhrennikovAYShelkovichVMp-Adic wavelets and their applicationsTr. Mat. Inst. Steklova201428516620634799931308.42031 FineNJOn the Walsh functionsTrans. Am. Math. Soc.1949653724143283310.1090/S0002-9947-1949-0032833-2 I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory, American Mathematical Society, Providence, Rhode Island (2011). EvdokimovSSkopinaMOn orthogonal p-adic wavelet basesJ. Math. Anal. Appl.2015422952965329271010.1016/j.jmaa.2014.11.061 FarkovYABiorthogonal wavelets on Vilenkin groupsTr. Mat. Inst. Steklova200926511012425995461178.42037 LyubushinAAYakovlevPVRodionovEAMultivariate analysis of fluctuation parameters of GPS signals before and after the mega-earthquake in Japan March 11, 2011Geofiz. Issled.20151611423 E. V. Burnaev and N. N. Olenev, “Proximity measure for time series based on wavelet coefficients,” in: Tr. XLVIII Nauch. Konf. MFTI, Dolgoprudny (2005), pp. 108–110. Yu. A. Farkov, “Periodic wavelets on the p-adic Vilenkin group,” p-Adic Numb. Ultr. Anal. Appl., 3, No. 4, 281–287 (2011). FarkovYABorisovMEPeriodic dyadic wavelets and coding of fractal functionsIzv. Vyssh. Ucheb. Zaved.20129546531370481277.42044 ProtasovVYFarkovYADyadic wavelets and scaling functions on the half-lineMat. Sb.200619710129160231011910.4213/sm1126 Wavelet Applications in Economics and Finance (M. Gallegati and W. Semmler, eds.), Springer, Berlin (2014). StroganovSAEstimates of the smoothness of low-frequency microseismic oscillations using dyadic waveletsGeofiz. Issled.20121311722 Yu. A. Farkov, E. A. Lebedeva, and M. A. Skopina, “Wavelet frames on Vilenkin groups and their approximation properties,” Int. J. Wavelets Multires. Inform. Process., 13, No. 5, 1550036 (2015). 5476_CR3 5476_CR1 5476_CR6 SV Kozyrev (5476_CR28) 2014; 285 5476_CR4 WC Lang (5476_CR30) 1998; 21 YA Farkov (5476_CR21) 2012; 2 5476_CR24 5476_CR25 F Schipp (5476_CR37) 1990 5476_CR22 AA Lyubushin (5476_CR31) 2017; 9 I Daubechies (5476_CR7) 1992 YA Farkov (5476_CR19) 2011; 9 5476_CR20 5476_CR42 5476_CR43 H Bölcskei (5476_CR2) 1998; 46 YA Farkov (5476_CR10) 2009; 265 F In (5476_CR26) 2012 VY Protasov (5476_CR35) 2006; 197 YA Farkov (5476_CR17) 2012; 9 YA Farkov (5476_CR9) 2009; 161 MK Tchobanou (5476_CR41) 2009 YA Farkov (5476_CR12) 2011; 89 S Mallat (5476_CR33) 1999 N Kholshchevnikova (5476_CR27) 2017; 446 5476_CR39 5476_CR18 S Evdokimov (5476_CR8) 2015; 422 5476_CR15 EA Rodionov (5476_CR36) 2016; 16 5476_CR16 5476_CR38 AA Lyubushin (5476_CR32) 2015; 16 5476_CR11 5476_CR34 NJ Fine (5476_CR23) 1949; 65 YA Farkov (5476_CR14) 2015; 2 SA Stroganov (5476_CR40) 2012; 13 YA Farkov (5476_CR13) 2012; 3 A Krivoshein (5476_CR29) 2016 CK Chui (5476_CR5) 1993; 9 |
| References_xml | – reference: RodionovEAOn applications of wavelets to the digital signal processingIzv. Saratov. Univ. Nov. Ser. Mat. Mekh. Inform.2016162217225352291610.18500/1816-9791-2016-16-2-217-225 – reference: FarkovYADiscrete wavelets and the Vilenkin–Chrestenson transformMat. Zametki2011896914928290814710.4213/mzm8704 – reference: LyubushinAAFarkovYASynchronous components of financial time seriesKomp. Issled. Model.201794639655 – reference: TchobanouMKMultidimensional Multi-Speed Signal Processing Systems2009MoscowTekhnosfera – reference: FarkovYABorisovMEPeriodic dyadic wavelets and coding of fractal functionsIzv. Vyssh. Ucheb. Zaved.20129546531370481277.42044 – reference: FarkovYAConstructions of MRA-based wavelets and frames in Walsh analysisPoincaré J. Anal. Appl.20152133634399391354.42059 – reference: Yu. A. Farkov, E. A. Lebedeva, and M. A. Skopina, “Wavelet frames on Vilenkin groups and their approximation properties,” Int. J. Wavelets Multires. Inform. Process., 13, No. 5, 1550036 (2015). – reference: KholshchevnikovaNSkvortsovVAOn U- and M-sets for series with respect to characters of compact zero-dimensional groupsJ. Math. Anal. Appl.20174461383394355473410.1016/j.jmaa.2016.08.062 – reference: MallatSA Wavelet Tour of Signal Processing1999San DiegoAcademic Press0998.94510 – reference: BölcskeiHHlawatschFFeichtingerHGFrame-theoretic analysis of oversampled filter banksIEEE Trans. Signal. Proc.199846123256326810.1109/78.735301 – reference: FarkovYABiorthogonal wavelets on Vilenkin groupsTr. Mat. Inst. Steklova200926511012425995461178.42037 – reference: Yu. A. Farkov, “Nonstationary multiresolution analysis for Vilenkin groups,” in: Int. Conf. on Sampling Theory and Applications, Tallinn, Estonia, 3-7 July 2017, Tallinn (2017), pp. 595–598. – reference: E. V. Burnaev and N. N. Olenev, “Proximity measure for time series based on wavelet coefficients,” in: Tr. XLVIII Nauch. Konf. MFTI, Dolgoprudny (2005), pp. 108–110. – reference: FarkovYARodionovEANonstationary wavelets related to the Walsh functionsAm. J. Comput. Math.20122828710.4236/ajcm.2012.22011 – reference: M. Vetterli and J. Kovačević, Wavelets and Subband Coding, Prentice Hall, New Jersey (1995). – reference: Yu. A. Farkov and E. A. Rodionov, “Algorithms for wavelet construction on Vilenkin groups,” p-Adic Numb. Ultr. Anal. Appl., 3, No. 1, 181–195 (2011). – reference: LyubushinAAYakovlevPVRodionovEAMultivariate analysis of fluctuation parameters of GPS signals before and after the mega-earthquake in Japan March 11, 2011Geofiz. Issled.20151611423 – reference: Z. Cvetkovi´c and M. Vetterli, “Oversampled filter banks,” IEEE Trans. Signal. Proc., 46, No. 5, 1245–1255 (1998). – reference: FarkovYAPeriodic wavelets in Walsh analysisCommun. Math. Appl.2012332232422824037 – reference: InFKimSAn Introduction to Wavelet Theory in Finance: A Wavelet Multiscale Approach2012SingaporeWorld Scientific10.1142/8431 – reference: B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Walsh Series and Transformations: Theory and Applications [in Russian], Moscow (2008). – reference: I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory, American Mathematical Society, Providence, Rhode Island (2011). – reference: ChuiCKMhaskarHNOn trigonometric waveletsConstr. Approx.19939167190121576810.1007/BF01198002 – reference: KozyrevSVKhrennikovAYShelkovichVMp-Adic wavelets and their applicationsTr. Mat. Inst. Steklova201428516620634799931308.42031 – reference: G. N. Agaev, N. Ya. Vilenkin, G. M. Jafarli, and A. I. Rubinstein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups [in Russian], Elm, Baku (1981). – reference: FarkovYAMaksimovAYStroganovSAOn biorthogonal wavelets related to the Walsh functionsInt. J. Wavelets Multires. Inform. Process.20119485499280472310.1142/S0219691311004195 – reference: Bl. Sendov, “Adapted multiresolution analysis,” in: Functions, Series, Operators, Memorial Conf. in Honor of the 100th Anniversary of the Birth of Prof. G. Alexits (1899–1978), Budapest, Hungary, August 9–13, 1999(L. Leindler et al., eds.), János Bolyai Math. Soc., Budapest (2002), pp. 23–38. – reference: E. V. Burnaev and N. N. Olenev, “Proximity measures based on wavelet coefficients for comparing statistical and calculated time series,” in: Collected Scientific and Methodical Papers [in Russian], 10, Izd. Vyatsk. Gos. Univ., Kirov (2006), pp. 41–51. – reference: DaubechiesITen Lectures on Wavelets1992PhiladelphiaSIAM10.1137/1.9781611970104 – reference: ProtasovVYFarkovYADyadic wavelets and scaling functions on the half-lineMat. Sb.200619710129160231011910.4213/sm1126 – reference: Yu. A. Farkov, “Orthogonal wavelets in Walsh analysis,” in: Generalized Integrals and Harmonic Analysis (T. P. Lukashenko and A. P. Solodov, eds.), Izd. Mosk. Univ., Moscow (2016), pp. 62–75. – reference: Yu. A. Farkov and E. A. Rodionov, “On biorthogonal discrete wavelet bases,” Int. J. Wavelets Multires. Inf. Process., 13, No. 1, 1550002 (2015). – reference: Wavelet Applications in Economics and Finance (M. Gallegati and W. Semmler, eds.), Springer, Berlin (2014). – reference: EvdokimovSSkopinaMOn orthogonal p-adic wavelet basesJ. Math. Anal. Appl.2015422952965329271010.1016/j.jmaa.2014.11.061 – reference: StroganovSAEstimates of the smoothness of low-frequency microseismic oscillations using dyadic waveletsGeofiz. Issled.20121311722 – reference: FarkovYAOn wavelets related to the Walsh seriesJ. Approx. Theory.20091611259279255815510.1016/j.jat.2008.10.003 – reference: FineNJOn the Walsh functionsTrans. Am. Math. Soc.1949653724143283310.1090/S0002-9947-1949-0032833-2 – reference: N. Ya. Vilenkin, “On a class of complete orthogonal systems,” Izv. Akad. Nauk SSSR. Ser. 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| SubjectTerms | Discrete Wavelet Transform Fractal analysis Image compression Image processing Mathematics Mathematics and Statistics Signal processing Wavelet analysis Wavelet transforms |
| Title | Discrete Wavelet Transforms in Walsh Analysis |
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