Skip to main content
SpringerLink
Log in

Matrix sweep algorithm for computing multiwavelets of odd order orthogonal to polynomials

  • Modeling in Physical and Technical Research
  • Published:
Optoelectronics, Instrumentation and Data Processing Aims and scope

Abstract

This paper studies polynomial multiwavelets of odd order having a support width equal to two grid spacings and orthogonal to polynomials of the same order on a finite interval. A new approach to calculating a multiwavelet transform is proposed based on an algorithm for solving systems of linear algebraic equations with a block-tridiagonal matrix using the matrix sweep method (block Gauss method). The results of numerical experiments for wavelets of fifth order are presented.

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

Access this article

Log in via an institution

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

References

  1. I. Daubechies, Ten Lectures on Wavelets (Regular and Chaotic Dynamics, Izhevsk, 2001; SIAM, 1992).

    MATH  Google Scholar 

  2. C. K. Chui, An Introduction to Wavelets (Mir, Moscow, 2001; SIAM, Philadelphia, 1992)

    Google Scholar 

  3. S. Malla, A Wavelets Tour for Signal Processing (Mir, Moscow, 2005; Elsevier, 1999).

    Google Scholar 

  4. A. V. Pereberin, “Systematization of Wavelet Transforms,” Vychislit. Metody i Programmirovanie 2 (2), 133–158 (2001).

    Google Scholar 

  5. V. Strela, P. N. Heller, G. Strang, et al., “The Application of Multiwavelet Filterbanks to Image Processing,” IEEE Trans. Signal Process. 8 (4), 548–563 (1999).

    ADS  Google Scholar 

  6. K. Koro, K. Abe, “Non-Orthogonal Spline Wavelets for Boundary Element Analysis,” Eng. Analys. Boundary Elements 25 (3), 149–164 (2001).

    Article  MATH  Google Scholar 

  7. I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory (Fizmatlit, Moscow, 2006) [in Russian].

    Google Scholar 

  8. B. M. Shumilov, “Multiwavelets of Third-Order Hermitian Splines Orthogonal to Cubic Polynomials,” Mat. Model, No. 4, 17–28 (2013).

    MathSciNet  Google Scholar 

  9. B. M. Shumilov and A. Zh. Kuduev, “New Type of Fifth-OrderWavelets Orthogonal to Fifth-Order Polynomials,” Vestn. TGU. Upravlenie, Vychislit. Tekhnika i Informatika, No. 4, 108–116 (2012).

    Google Scholar 

  10. A. A. Samarskii and E. S. Nikolaev, Methods of Solving Grid Equations (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  11. Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  12. E. Stolnits, T. DeRose, and D. Salezin, Wavelets in Computer Graphics (Regular and Chaotic Dynamics, Izhevsk, 2002) [Russian translation].

    Google Scholar 

  13. J. Ortega. Introduction to Parallel and Vector Solution of Linear Systems (Mir, Moscow, 1991; Plenum Press, New York, 1988).

    Google Scholar 

  14. E. A. Vasil’eva, “Sufficient Conditions for the Correctness of the Matrix Sweep Method,” Izv. KGU, No. 20, 2011, http://www.klgtu.ru/science/magazine/201120/14.doc.

  15. F. Arandiga, A. Baeza, and R. Donat, “Discrete Multiresolution Based on Hermitian Interpolation: Computing Derivatives,” Commun. Nonlinear Sci. Numerical Simulation 9 (2), 263–273 (2004).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Yu. E. Voskoboinikov, “Optimization ofWavelet Filtering Algorithms with Multiparameter Threshold Functions,” Avtometriya 50 (6), 69–79 (2014) [Optoelectron., Instrum. Data Process. 50 (6), 589–597 (2014)].

    Google Scholar 

  17. B. M. Shumilov, A. T. Bekmuratov, A. Zh. Kuduev, et al., “Wavelet Transform and Compression of Laser Scanning Data of Automobile Roads,” Vestn. TGASU, No. 4, 228–238 (2011).

    Google Scholar 

  18. B. M. Shumilov, A. N. Baigulov, and Zh. Abdykalyk kyzy, “Algorithm and Program of Wavelet Modeling of Automobile Road Surfaces,” Vestn. TGASU, No. 1, 142–152 (2014).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tomsk State University of Architecture and Building, pl. Solyanaya 2, Tomsk, 634003, Russia

    B. M. Shumilov

Authors
  1. B. M. Shumilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. M. Shumilov.

Additional information

Original Russian Text © B.M. Shumilov, 2015, published in Avtometriya, 2015, Vol. 51, No. 2, pp. 83–92.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shumilov, B.M. Matrix sweep algorithm for computing multiwavelets of odd order orthogonal to polynomials. Optoelectron.Instrument.Proc. 51, 175–183 (2015). https://doi.org/10.3103/S8756699015020119

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S8756699015020119

Keywords

Search

Navigation