Skip to main content
SpringerLink
Log in

Generalized Sampling Approximation for Multivariate Discontinuous Signals and Applications to Image Processing

  • Chapter
  • First Online:
New Perspectives on Approximation and Sampling Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

The aim of the paper is to extend some results concerning univariate generalized sampling approximation to the multivariate frame. We give estimates of the approximation error of the multivariate generalized sampling series for not necessarily continuous functions in \(L^{p}(\mathbb{R}^{n})\)-norm, using the averaged modulus of smoothness of Sendov and Popov type. Finally, we study some concrete examples of sampling operators and give applications to image processing dealing, in particular, with biomedical images.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In one dimension, where the variable t is often identified with time, we also speak of a time-limited kernel.

References

  1. Andreev, A., Popov, V.A., Sendov, B.: Jackson-type theorems for the best one-sided approximations by trigonometric polynomials and splines. Mat. Zametki 26(5), 791–804, 816 (1979) [English translation in Math. Notes 26(5–6), 889–896 (1980)]

    Google Scholar 

  2. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals. Analysis (Munich) 23(4), 299–340 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 316(1), 269–306 (2006). doi:10.1016/j.jmaa.2005.04.042

    Article  MATH  MathSciNet  Google Scholar 

  4. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Trans. Inform. Theory 56(1), 614–633 (2010). doi:10.1109/TIT.2009.2034793

    Article  MathSciNet  Google Scholar 

  5. Bevilacqua, R., Bini, D., Capovani, M., Menchi, O.: Metodi Numerici [Numerical Methods]. Zanichelli, Bologna (1992)

    Google Scholar 

  6. de Boor, C., DeVore, R.: Approximation by smooth multivariate splines. Trans. Am. Math. Soc. 276(2), 775–788 (1983). doi:10.2307/1999083

    Article  MATH  Google Scholar 

  7. de Boor, C., Höllig, K.: Recurrence relations for multivariate B-splines. Proc. Am. Math. Soc. 85(3), 397–400 (1982). doi:10.2307/2043855

    Article  MATH  Google Scholar 

  8. de Boor, C., Höllig, K.: B-splines from parallelepipeds. J. Anal. Math. 42, 99–115 (1982/1983). doi:10.1007/BF02786872

    Google Scholar 

  9. Butzer, P.L., Splettstößer, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahr. Deut. Math. Ver. 90(1), 1–70 (1988)

    MATH  Google Scholar 

  10. Butzer, P.L., Engels, W., Ries, S., Stens, R.L.: The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines. SIAM J. Appl. Math. 46(2), 299–323 (1986). doi:10.1137/0146020

    Article  MATH  MathSciNet  Google Scholar 

  11. Butzer, P.L., Fischer, A., Stens, R.L.: Generalized sampling approximation of multivariate signals; theory and some applications. Note Mat. 10(Suppl 1), 173–191 (1990)

    MathSciNet  Google Scholar 

  12. Butzer, P.L., Fischer, A., Stens, R.L.: Generalized sampling approximation of multivariate signals; general theory. Atti Sem. Mat. Fis. Univ. Modena 41(1), 17–37 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Butzer, P.L., Higgins, J.R., Stens, R.L.: Sampling theory of signal analysis. In: Pier, J.-P. (ed.) Development of Mathematics 1950–2000, pp. 193–234. Birkhäuser, Basel (2000)

    Google Scholar 

  14. Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Nonuniform Sampling: Theory and Practice, pp. 17–121. Kluwer/Plenum, New York (2001). doi:10.1007/978-1-4615-1229-5

    Chapter  Google Scholar 

  15. Chui, C.K.: Multivariate splines. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54. Society of Industrial and Applied Mathematics (SIAM), Philadelphia (1988). doi:10.1137/1.9781611970173

    Google Scholar 

  16. Costarelli, D., Vinti, G.: Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Boll. Unione Mat. Ital. (9) 4(3), 445–468 (2011)

    Google Scholar 

  17. Costarelli, D., Vinti, G.: Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numer. Funct. Anal. Optim. 34(6), 1–26 (2013)

    MathSciNet  Google Scholar 

  18. Engels, W., Stark, E.L., Vogt, L.: Optimal kernels for a general sampling theorem. J. Approx. Theory 50(1), 69–83 (1987). doi:10.1016/0021-9045(87)90067-0

    Article  MATH  MathSciNet  Google Scholar 

  19. Fischer, A., Stens, R.L.: Generalized sampling approximation of multivariate signals; inverse approximation theorems. In: Szabados, J., Tandori, K. (eds.) Approximation Theory (Proc. Conf. Kecskemét, Hungary, 6–11 Aug 1990). Colloq. Math. Soc. János Bolyai, vol. 58, pp. 275–286. North- Holland Publishing Co., Amsterdam (1991)

    Google Scholar 

  20. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Prentice Hall, Upper Saddle River (2002)

    Google Scholar 

  21. Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis. Clarendon Press, Oxford (1996)

    MATH  Google Scholar 

  22. Hristov, V.H.: A connection between Besov spaces and spaces generated by the averaged modulus of smoothness in R n. C. R. Acad. Bulgare Sci. 38(5), 555–558 (1985)

    MathSciNet  Google Scholar 

  23. Hristov, V.H.: A connection between the usual and the averaged moduli of smoothness of functions of several variables. C. R. Acad. Bulgare Sci. 38(2), 175–178 (1985)

    MATH  MathSciNet  Google Scholar 

  24. Hristov, V.H., Ivanov, K.G.: Operators for onesided approximation of functions. In: Sendov, B., Petrushev, P., Ivanov, K., Rumen, M. (eds.) Constructive Theory of Functions (Proc. Conf., Varna, Bulgaria, 24–31 May 1987), pp. 222–232. Publication House Bulgarian Academy of Sciences, Sofia (1988)

    Google Scholar 

  25. Ivanov, K.G.: On the behaviour of two moduli of functions. II. Serdica 12(2), 196–203 (1986)

    MATH  Google Scholar 

  26. Jerri, A.J.: The Shannon sampling theorem—its various extensions and applications: a tutorial review. Proc. IEEE 65(11), 1565—1596 (1977). doi:10.1109/PROC.1977.10771

    Article  MATH  Google Scholar 

  27. Micchelli, C.A.: On a numerically efficient method for computing multivariate B-splines. In: Schempp, W., Zeller, K. (eds.) Multivariate Approximation Theory (Proc. Conf., Math. Res. Inst., Oberwolfach, Germany, 4–10 Feb 1979), pp. 211–248. Birkhäuser, Basel (1979)

    Google Scholar 

  28. Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001). doi:10.1137/1.9780898718324

    Google Scholar 

  29. Parzen, E.: A simple proof and some extensions of sampling theorems. Technical Report, vol. 7. Stanford University, Stanford (1956)

    Google Scholar 

  30. Petersen, D.P., Middleton, D.: Sampling and reconstruction of wavenumber-limited functions in N-dimensional Euclidean spaces. Inform. Control 5, 279–323 (1962)

    Article  MathSciNet  Google Scholar 

  31. Popov, V.A.: Converse theorem for the onesided trigonometrical approximations. C. R. Acad. Bulgar. Sci. 30(11), 1529–1532 (1977)

    MATH  Google Scholar 

  32. Popov, V.A.: Direct and converse theorems for oneside approximation. In: Butzer, P.L., Nagy, B.Sz. (eds.) Linear Spaces and Approximation (Proc. Internat. Conf., Math. Res. Inst., Oberwolfach, Germany, 20–27 Aug 1977), p. 685. Birkhäuser, Basel (1978)

    Google Scholar 

  33. Popov, V.A.: One-sided approximation of periodic functions of several variables. C. R. Acad. Bulgare Sci. 35(12), 1639–1642 (1982/1983)

    Google Scholar 

  34. Sendov, B., Popov, V.A.: The Averaged Moduli of Smoothness. Wiley, Chichester (1988)

    MATH  Google Scholar 

  35. Splettstößer, W.: Sampling approximation of continuous functions with multidimensional domain. IEEE Trans. Inform. Theory 28(5), 809–814 (1982). doi:10.1109/TIT.1982.1056561

    Article  MATH  MathSciNet  Google Scholar 

  36. Stens, R.L.: Sampling by generalized kernels, Chap. 6. In: Higgins, J.R., Stens, R.L. (eds.) Sampling Theory in Fourier and Signal Analysis. Advanced Topics, vol. 2, pp. 130–157, 273–277. Oxford University Press, Oxford (1999)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Dr. Enrico Cieri and Dr. Giacomo Isernia of the section of Vascular Surgery and Dr. Pietro Pozzilli of the section of Diagnostic and Interventional Radiology of the University of Perugia for their collaboration concerning the applications of our theory to biomedical images.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Universitá degli Studi di Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Carlo Bardaro, Ilaria Mantellini & Gianluca Vinti

  2. Lehrstuhl A für Mathematik, RWTH Aachen, 52056, Aachen, Germany

    Rudolf Stens & Jörg Vautz

Authors
  1. Carlo Bardaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilaria Mantellini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rudolf Stens
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jörg Vautz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Gianluca Vinti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rudolf Stens .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

    Ahmed I. Zayed

  2. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

Additional information

Dedicated to Professor Paul L. Butzer, our teacher and sincere friend, in high esteem

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bardaro, C., Mantellini, I., Stens, R., Vautz, J., Vinti, G. (2014). Generalized Sampling Approximation for Multivariate Discontinuous Signals and Applications to Image Processing. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation