SeMA Journal

, Volume 76, Issue 1, pp 27–36 | Cite as

Construction of dual wavelet frame pairs and signal recovery

  • Ali Akbar ArefijamaalEmail author
  • Fahimeh Arabyani Neyshaburi
  • Samaneh Matindoost


Signal processing is an enabling technology that helps us to denote any operation which modifies or analyzes the information contained in a signal. In this paper, we first decompose the original signal by a wavelet packet frame and analyze the coefficients. Then, by using dual wavelet frames, we reconstruct the original signal. In this reconstruction, the standard choice for duals which plays a key role is the canonical dual. Our aim is to develop new duals to obtain more accurate results. To this end, we consider wavelet frames which Fourier transform of generators form a partition of unity. Then we introduce several explicit duals for them and compare the advantage of these duals in signal processing. This indicates that we may obtain more reliable estimates by alternate duals.


Wavelet frame Dual wavelet frame Partition of unity 

Mathematics Subject Classification

Primary 42C15 Secondary 42C40 


  1. 1.
    Allabakash, S., Yasodha, P., Venkatramana Reddy, S., Srinivasulu, P.: Wavelet transform-based methods for removal of ground clutter and denoising the radar wind profiler data. Signal Process. IET 9, 440–448 (2015)CrossRefGoogle Scholar
  2. 2.
    Arefijamaal, A., Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35, 535–540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arefijamaal, A., Zekaee, E.: Image processing by alternate dual Gabor frames. Bull. Iran. Math. Soc. 42(6), 1305–1314 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Casazza, P.G., Kutyniok, G., Lammers, M.C.: Duality principles in frame theory. Fourier Anal. Appl. 10(4), 383–408 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Y., Zhang, Y., Hu, H., Ling, H.: A novel gray image watermarking scheme. J. Softw. 6(5), 849–856 (2011)Google Scholar
  6. 6.
    Christensen, O.: Frames and Bases. An Introductory Course. Birkhäuser, Boston (2008)zbMATHGoogle Scholar
  7. 7.
    Christensen, O.: Pairs of dual Gabor frames with compact support and desired frequency localization. Appl. Comput. Harmon. Anal. 20, 403–410 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Christensen, O.: Time-frequency analysis and its applications in denoising. Department of informatics, University of Bergan, Thesis (2002)Google Scholar
  9. 9.
    Christensen, O., Kim, R.Y.: On dual Gabor frame pairs generated by polynomials. Fourier Anal. Appl. 16, 11–16 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chui, C.K., Shi, X.: Orthonormal wavelets and tight frames with arbitrary real dilations. Appl. Comput. Harmon. Anal. 9(3), 243–264 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Daubechies, I.: The wavelet transform, time frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Deng, L., Yu, C.L., Chakrabarti, C., Kim, J., Narayanan, V.: Efficient image reconstruction using partial 2D Fourier transform. In: Proceedings of the IEEE Workshop on Signal Processing Systems, pp. 49–54. SiPS, Washington, DC Metro Area, USA (2008)Google Scholar
  14. 14.
    Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika 81(3), 425–455 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. Am. Stat. Assoc. 90(432), 1200–1224 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Feichtinger, H.G., Strohmer, T.: Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Boston (1997)zbMATHGoogle Scholar
  17. 17.
    Gautier, M., Lienard, J.: Efficient wavelet packet modulation for wireless communication. In: Proceedings of the 3rd Advanced International Conference, pp. 1–19. France Telecom R and D, France (2007)Google Scholar
  18. 18.
    Găianu, M., Onchis, D.M.: Face and marker detection using Gabor frames on GPUs. Signal Process. 96, 90–93 (2014)CrossRefGoogle Scholar
  19. 19.
    He, J., Li, Z., Qian, H.: Cryptography based on spatiotemporal chaos system and multiple maps. J. Softw. 5(4), 421–428 (2010)CrossRefGoogle Scholar
  20. 20.
    Heil, C.E., Walnut, D.F.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projection, 2nd edn. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Huang, F., Qu, X.: Design of image encryption algorithm based on compound two-dimensional maps. J. Softw. 6(10), 1953–1960 (2011)Google Scholar
  23. 23.
    Janssen, A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. Fourier Anal. Appl. 1(4), 403–436 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization. Adv. Comput. Math. 30, 231–248 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mallat, S.G.: A wavelet tour of signal processing. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  26. 26.
    Porter, R., Canagarajah, N.: Robust rotation-invariant texture classification: wavelet, Gabor filter and GMRF based schemes. IEE Proc. Vis. Image Signal Process. 144(3), 180–188 (1997)CrossRefGoogle Scholar
  27. 27.
    Ron, A., Shen, Z.: Weyl–Heisenberg frames and Riesz bases in \(L^2(\mathbb{R}^d )\). Duke Math. J. 89, 237–282 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tong, C.S., Leung, K.T.: Super-resolution reconstruction based on linear interpolation of wavelet coefficients. Multidimens. Syst. Signal Process. 18(2–3), 153–171 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Thamarai, P., Adalarasu, K.: Denoising of EEG, ECG and PPG signals using wavelet transform. J. Pharm. Sci. and Res. 10(1), 156–161 (2018)Google Scholar
  30. 30.
    Verma, R., Mahrishi, R., Srivastava, G. K., Siddavatam, R.: A novel image reconstruction using second generation wavelets. In: IEEE International Conference on Advances in Recent Technologies in Communication and Computing, pp. 509–513. IEEE Press, Kerala (2009)Google Scholar
  31. 31.
    Wexler, J., Raz, S.: Discrete Gabor expansions. Signal Process. 21, 207–221 (1990)CrossRefGoogle Scholar
  32. 32.
    Yang, X., Min, J., Shi, Y.: Parametrisation construction frame of lifting scheme. IET Signal Process. 5, 1–15 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhang, Y., Yuanyuan, W., Chen, Z.: Efficient discrete cosine transform model-based algorithm for photoacoustic image reconstruction. J. Biomed. Opt. 18(6), 066008(1)–066008(9) (2013)CrossRefGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  • Ali Akbar Arefijamaal
    • 1
    Email author
  • Fahimeh Arabyani Neyshaburi
    • 1
  • Samaneh Matindoost
    • 2
  1. 1.Department of Mathematics and Computer SciencesHakim Sabzevari UniversitySabzevarIran
  2. 2.Department of Electrical EngineeringHakim Sabzevari UniversitySabzevarIran

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