Skip to main content

Advertisement

Springer Nature Link
Log in

Local spectral analysis of images via the wavelet transform based on partial differential equations

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We propose method for the local spectral analysis of images via the two-dimensional continuous wavelet transform with the Morlet wavelet based on its representation as a solution of the partial differential equation. It has been shown that a transformed function uniquely determines an initial value for the equation, i.e. a Cauchy problem is stated. Its solving implies that scale parameter a plays a role of “time variable” and two translation parameters b x , b y are spatial independent variables. Numerical examples are given to illustrate the efficiency of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Antoine J. P., Murenzi R., Vandergheynst P., Ali S.T. (2004) Two-dimensional wavelets and their relatives. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Antoine J. P., Carette P., Murenzi R., Piette B. (1993) Image analysis with two-dimensional continuous wavelet transform. Signal Processing 31: 241–272. doi:10.1016/0165-1684(93)90085-O

    Article  MATH  Google Scholar 

  • Biskri S., Antoine J. P., Inhester B., Mekideche F. (2010) Extraction of solar coronal magnetic loops with the directional 2D morlet wavelet transform. Solar Physics 262: 373–385. doi:10.1007/s11207-010-9533-4

    Article  Google Scholar 

  • Gonnet C., Torresani B. (1994) Local frequency analysis with two-dimensional wavelet transform. Signal Processing 37: 389–404. doi:10.1016/0165-1684(94)90007-8

    Article  MATH  Google Scholar 

  • Haase M. (2000) A family of complex wavelets for characterization of singularities. In: Novak M (eds) Paradigms of complexity. Fractals and structures in the sciences. World Scientific, Singapore, pp 287–288

    Google Scholar 

  • Hundsdorfer W.H., Verwer J.G. (2003) Numerical solution of time-dependent advection– diffusion–reaction equations. Springer, Berlin

    Book  MATH  Google Scholar 

  • Kirby J. F., Swain C. J. (2006) Mapping the mechanical anisotropy of the lithosphere using a 2D wavelet coherence, and its application to Australia. Physics of the Earth and Planetary Interiors 158: 122–138. doi:10.1016/j.pepi.2006.03.02

    Article  Google Scholar 

  • Ning W., Lu C. (2010) Two-dimensional continuous wavelet analysis and its application to meteorological data. Journal of Atmospheric Oceanic Technology 27: 652–666. doi:10.1175/2009JTECHA1338.1

    Article  Google Scholar 

  • Postnikov E. (2006) Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations. Computational Mathamatics and Mathamatics Physics 46: 73–78. doi:1134/S0965542506010088

    Article  MathSciNet  Google Scholar 

  • Postnikov E. B., Loskutov A. Y. (2005) Analysis of small-scale wave structures in the Saturnian A ring based on data from the Cassini interplanetary spacecraft. Journal of Experimental and Theoretical Physics 101: 646–652. doi:10.1134/1.2131932

    Article  Google Scholar 

  • Postnikov E. B., Loskutov A. Y. (2007) Wavelet analysis of fine-scale structures in the Saturnian B and C rings using data from the Cassini spacecraft. Journal of Experimental and Theoretical Physics 104: 417–422. doi:10.1134/S1063776107030065

    Article  Google Scholar 

  • Ruppert-Felsot J., Farge M., Petitjeans P. (2009) Wavelet tools to study intermittency: Application to vortex bursting. Journal of Fluid Mechanics 636: 427–453. doi:10.1017/S0022112009008003

    Article  MATH  MathSciNet  Google Scholar 

  • Shyun H.-C., Sun Y.-S. (2002) Construction of a morlet wavelet power spectrum. Multidimensional System and Signal Processing 13: 101–111. doi:10.1023/A:1013847512432

    Article  Google Scholar 

  • Zhao, Y., & Swamy, M. N. S. (2011). The analysis and design of two-dimensional nearly- orthogonal symmetric wavelet filter banks. Multidimensional System and Signal Processing. doi:10.1007/s11045-011-0165-0 .

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, Kursk State University, Radishcheva st. 33, 305000, Kursk, Russia

    Eugene B. Postnikov

  2. Department of Mathematics, BITS Pilani-K.K. Birla Goa Campus, Zuarinagar, Goa, India

    Vineet K. Singh

Authors
  1. Eugene B. Postnikov
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Vineet K. Singh
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Eugene B. Postnikov.

Additional information

VKS is grateful for financial support of the BOYSCAST Fellowship 2010–2011 program of the Department of Science and Technology of the Government of India.

Electronic Supplementary Material

The Below is the Electronic Supplementary Material.

ESM 1 (PDF 4066 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Postnikov, E.B., Singh, V.K. Local spectral analysis of images via the wavelet transform based on partial differential equations. Multidim Syst Sign Process 25, 145–155 (2014). https://doi.org/10.1007/s11045-012-0196-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-012-0196-1

Keywords