Results in Mathematics

, 73:49 | Cite as

Parametric Multi-Wavelets on a Hexagonal Sampling Lattice

  • Hemant Dattatray Bhate
  • Rupali Sadashiv Deshpande


An orthogonal basis is a dictionary of minimum size that can yield a sparse representation if designed to concentrate the signal energy over a set of few vectors. In this paper we construct such dictionary for a two dimensional hexagonal sampling lattice with the 1-parametric family of wavelets associated with Haar scaling function for scale factor 3. We also provide reconstruction formulae at various scales associated with these parametric multi wavelets.


Multiwavelets hexagonal lattice matricial filters 

Mathematics Subject Classification

Primary 42C15 Secondary 42C40 47B20 


  1. 1.
    Aiazzi, B., Baronti, S., Capanni, A., Santurri, L., Vitulli, R.: Advantages of hexagonal sampling grids and hexagonal shape detector elements in remote sensing imagers. In: Signal Processing Conference, 2002 11th European, pp. 1–4. IEEE (2002)Google Scholar
  2. 2.
    Ashino, R., Kametani, M.: A lemma on matrices and a construction of multi-wavelets. Math. Jpn. 45, 267–288 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blanchard, J., Krishtal, I.: Matricial filters and crystallographic composite dilation wavelets. Math. Comput. 81(278), 905–922 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blanchard, J.D., Steffen, K.R.: Crystallographic haar-type composite dilation wavelets. In: Wavelets and Multiscale Analysis, pp. 83–108. Springer (2011)Google Scholar
  5. 5.
    Cabrelli, C.A., Heil, C., Molter, U.M.: Self-Similarity and Multiwavelets in Higher Dimensions, vol. 170. American Mathematical Society, Providence (2004)zbMATHGoogle Scholar
  6. 6.
    Camps, A., Bará, J., Sanahuja, I.C., Torres, F.: The processing of hexagonally sampled signals with standard rectangular techniques: application to 2-d large aperture synthesis interferometric radiometers. IEEE Trans. Geosci. Remote Sens. 35(1), 183–190 (1997)CrossRefGoogle Scholar
  7. 7.
    Chui, C.K., Jian-ao, L.: Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale= 3. Appl. Comput. Harmon. Anal. 2(1), 21–51 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chui, C.K., Lian, J.: A study of orthonormal multi-wavelets. Appl. Numer. Math. 20(3), 273–298 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cohen, A., Daubechies, I., Plonka, G.: Regularity of refinable function vectors. J. Fourier Anal. Appl. 3(3), 295–324 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Donovan, G.C., Geronimo, J.S., Hardin, D.P., Massopust, P.R.: Construction of orthogonal wavelets using fractal interpolation functions. SIAM J. Math. Anal. 27(4), 1158–1192 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Geronimo, J.S., Hardin, D.P., Massopust, P.R.: Fractal functions and wavelet expansions based on several scaling functions. J. Approx. Theory 78(3), 373–401 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    He, X., Jia, W.: Hexagonal structure for intelligent vision. In: Information and Communication Technologies, 2005. ICICT 2005. First International Conference on, pp. 52–64. IEEE (2005)Google Scholar
  13. 13.
    Krommweh, J., Plonka, G.: Directional haar wavelet frames on triangles. Appl. Comput. Harmon. Anal. 27(2), 215–234 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lian, J., Chui, C.K.: Balanced multiwavelets with short filters. IEEE Signal Process. Lett. 11(2), 75–78 (2004)CrossRefGoogle Scholar
  15. 15.
    Mersereau, R.M.: The processing of hexagonally sampled two-dimensional signals. Proc. IEEE 67(6), 930–949 (1979)CrossRefGoogle Scholar
  16. 16.
    Middleton, L., Sivaswamy, J.: Hexagonal Image Processing: A Practical Approach. Springer, New York (2006)zbMATHGoogle Scholar
  17. 17.
    Oschler, K.L., Gray, R.M., Cosman, P.C.: Digital Images and Human Vision, pp. 35–52. MIT Press, Cambridge (1993)Google Scholar
  18. 18.
    Plonka, G., Strela, V.: From wavelets to multiwavelets. FB Mathematik University, Hagen (1997)zbMATHGoogle Scholar
  19. 19.
    Plonka, G., Strela, V.: Construction of multiscaling functions with approximation and symmetry. SIAM J. Math. Anal. 29(2), 481–510 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Puschel, M., Rotteler, M.: Algebraic signal processing theory: 2-D spatial hexagonal lattice. IEEE Trans. Image Process. 16(6), 1506–1521 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Resnikoff, H.L., Raymond Jr., O., et al.: Wavelet Analysis: The Scalable Structure of Information. Springer, New York (2012)zbMATHGoogle Scholar
  22. 22.
    Selesnick, I.W.: Balanced multiwavelet bases based on symmetric fir filters. IEEE Trans. Signal Process. 48(1), 184–191 (2000)CrossRefzbMATHGoogle Scholar
  23. 23.
    Shen, L., Tan, H.H., Tham, J.Y.: Symmetric-antisymmetric orthonormal multiwavelets and related scalar wavelets. Appl. Comput. Harmon. Anal. 8(3), 258–279 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Strang, G., Strela, V.: Orthogonal multiwavelets with vanishing moments. In: SPIE’s International Symposium on Optical Engineering and Photonics in Aerospace Sensing, pp. 2–9. International Society for Optics and Photonics (1994)Google Scholar
  25. 25.
    Strang, G., Strela, V.: Short wavelets and matrix dilation equations. IEEE Trans. Signal Process. 43(1), 108–115 (1995)CrossRefGoogle Scholar
  26. 26.
    Strela, V.: Multiwavelets: regularity, orthogonality, and symmetry via two-scale similarity transform. Stud. Appl. Math. 98(4), 335–354 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Strela, V., Heller, P.N., Strang, G., Topiwala, P., Heil, C.: The application of multiwavelet filterbanks to image processing. IEEE Trans. Image Process. 8(4), 548–563 (1999)CrossRefGoogle Scholar
  28. 28.
    Van De Ville, D., Blu, T., Unser, M., Philips, W., Lemahieu, I., Van de Walle, R.: Hex-splines: a novel spline family for hexagonal lattices. IEEE Trans. Image Process. 13(6), 758–772 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Weidmann, C., Lebrun, J., Vetterli, M.: Significance tree image coding using balanced multiwavelets. In: Image Processing, 1998. ICIP 98. Proceedings. 1998 International Conference on, vol. 1, pp. 97–101. IEEE (1998)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering, Sciences and HumanitiesVishwakarma Institute of TechnologyPuneIndia
  2. 2.Savitribai Phule Pune UniversityPuneIndia
  3. 3.Department of MathematicsSavitribai Phule Pune UniversityPuneIndia

Personalised recommendations