Image compression using fractal multiwavelet transform


Multiwavelets occur in wavelet theory for more general multiresolution analysis (MRA) using several scaling functions. The muliwavelets feature orthogonality, short support, symmetry, approximation order and regularity at a time which is not possible by using only a single wavelet system. In the present work, theory of MRA and wavelet using several scaling functions is reviewed to define Discrete Multiwavelet Transform and Fractal Multiwavelet Transform in matrix form, using a multifilter. As an application the defined transforms are applied to an image for its compression and the resulting entropy and PSNR values are compared with those occurring with earlier used transforms like Haar and D4 transforms.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Antonini, M., M. Barlaud, P. Mathieu, and I. Daubechies. 1992. Image coding using the wavelet transform. IEEE Trans. Image Process. 1: 205–220.

    Google Scholar 

  2. 2.

    Barnsley, M.F. 1986. Fractal functions and interpolation. Constr. Approx. 2 (4): 303–329.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Cai, W., and J. Wang. 1996. Adaptive wavelet collocation methods for initial boundary value problems of nonlinear PDE’s. SIAM J. Numer. Anal. 33 (3): 937–970.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Chui, C.K., and J. Lian. 1996. A study of orthonormal multi-wavelets. Appl. Numer. Math. 20 (3): 273–298.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Cotronei, M., L.B. Montefusco, and L. Puccio. 1998. Multiwavelet analysis and signal processing. IEEE Trans. Circuits Syst. II 45: 970–987.

    MATH  Google Scholar 

  6. 6.

    Cotronei, M., D. Lazzaro, L.B. Montefusco, and L. Puccio. 2000. Image Compression Through Embedded Multiwavelet Transform Coding. IEEE Trans. Image Process. 9 (2): 1–6.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Daubechies, I. 1988. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41: 909–996.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Donovan, G.C., J.S. Geronimo, D.P. Hardin, and P.R. Massopust. 1996. Construction of orthogonal wavelets using fractal interpolation functions. SIAM J. Math. Anal. 27 (4): 1158–1192.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Geronimo, J.S., D.P. Hardin, and P.R. Massopust. 1994. Fractal functions and wavelet expansions based on several scaling functions. J. Approx. Theory 78: 373–401.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Goodman, T.N.T., and S.L. Lee. 1994. Wavelets of multiplicity r. Trans. Am. Math. Soc. 342 (1): 1–18.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Haar, A. 1910. Zur Theorie der orthogonalen Funktionen-System. Math. Ann. 69: 331–371.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Heil, C., G. Strang, and V. Strela. 1995. Approximation by translates of refinable functions. Numer. Math. 73: 75–94.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Jiang, Q. 1998. Orthogonal multiwavelets with optimum time-frequency resolution. IEEE Trans. Signal Process. 46: 830–844.

    MathSciNet  Google Scholar 

  14. 14.

    Kapoor, G.P., and S.A. Prasad. 2014. Multiresolution analysis based on coalescence hidden variable fractal interpolation functions. Int: J. Comput. Math.

    Google Scholar 

  15. 15.

    Keinert, F. 2005. Wavelets and Multiwavelets, Studies in Advanced Mathematics. Boca Raton: Chapman Hall CRC.

    Google Scholar 

  16. 16.

    Mallat, S. 1989. Multiresolution approximations and wavelet orthonormal bases of \(L^{2}(\mathbb{R})\). Trans. Am. Math. Soc. 315: 69–87.

    MATH  Google Scholar 

  17. 17.

    Massopust, P.R. 1994. Fractal Functions, Fractal Surfaces, and Wavelets. San Diego, CA: Academic Press, Inc.

    Google Scholar 

  18. 18.

    Massopust, P.R. 1998. A multiwavelet based on piecewise \(\cal{C}^{1}\) fractal functions and related applications to differential equations. Bol. Soc. Mat. Mexicana 4 (2): 249–283.

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Meyer, Y. 1992. Wavelets and Operators. Advanced Mathematics. Cambridge: Cambridge University Press.

    Google Scholar 

  20. 20.

    Pan, J., L. Jiao, and Y. Fang. 2001. Construction of orthogonal multiwavelets with short sequence. Signal Process. 81: 2609–2614.

    MATH  Google Scholar 

  21. 21.

    Plonka, G. 1996. Generalized spline wavelets. Constr. Approx. 12: 127–155.

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Ruch, D.K., and P.J. Van Fleet. 2009. Wavelet Theory: An Elementary Approach with Applications. New York: Wiley.

    Google Scholar 

  23. 23.

    Shapiro, J.M. 1993. Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans. Signal Process. 41: 3445–3662.

    MATH  Google Scholar 

  24. 24.

    So, W., and J. Wang. 1997. Estimating the support of a scaling vector. SIAM J. Matrix Anal. Appl. 18 (1): 66–73.

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Strang, G., and V. Strela. 1995. Short wavelets and matrix dilation equations. IEEE Trans. Signal Process. 43: 108–115.

    Google Scholar 

  26. 26.

    Strela, V. 1996. Multiwavelets: Theory and Applications. PhD Thesis, Dept. of Math., Massachusetts Institute of Technology, Cambridge

  27. 27.

    Strela, V., P.N. Heller, G. Strang, P. Topiwala, and C. Heil. 1999. The application of multiwavelet filterbanks to image processing. IEEE Trans. Image Process, 548–563.

  28. 28.

    Strela, V., and G. Strang. 1995. Finite element multiwavelets. In Approximation Theory, Wavelets and Applications, ed. S.P. Singh, 485–496. Norwell, MA: Kluwer Academic.

    Google Scholar 

  29. 29.

    Xia, X.G., and B.W. Suter. 1996. Vector valued wavelets and vector filter bank. IEEE Trans. Signal Process. 44 (3): 508–518.

    Google Scholar 

  30. 30.

    Marasi, H.R., V.N. Mishra, and M. Daneshbastam. 2017. A constructive approach for solving system of fractional differential equations. Waves, Wavelets and Fractals Advanced Analysis 3 (1): 40–47.

    MATH  Article  Google Scholar 

  31. 31.

    Vandana, R., Deepmala Dubey, L.N. Mishra, and V.N. Mishra. 2018. Duality relations for a class of a multiobjective fractional programming problem involving support functions. Am. J. Oper. Res. 8: 294–311.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Md. Nasim Akhtar.

Ethics declarations

Conflict of interest

The authors have declared that no conflict of interest exists.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Akhtar, M.N., Guru Prem Prasad, M. & Kapoor, G.P. Image compression using fractal multiwavelet transform. J Anal 28, 769–789 (2020).

Download citation


  • Fractal interpolation functions
  • Multiresolution analysis
  • Multiwavelets
  • Image compression

Mathematics Subject Classification

  • 28A80
  • 42C40
  • 94A08