Skip to main content
SpringerLink
Log in

3D Image Reconstructions and the Nyquist–Shannon Theorem

  • 3DR Express
  • Published:
3D Research

Abstract

Fracture surfaces are occasionally modelled by Fourier’s two-dimensional series that can be converted into digital 3D reliefs mapping the morphology of solid surfaces. Such digital replicas may suffer from various artefacts when processed inconveniently. Spatial aliasing is one of those artefacts that may devalue Fourier’s replicas. According to the Nyquist–Shannon sampling theorem the spatial aliasing occurs when Fourier’s frequencies exceed the Nyquist critical frequency. In the present paper it is shown that the Nyquist frequency is not the only critical limit determining aliasing artefacts but there are some other frequencies that intensify aliasing phenomena and form an infinite set of points at which numerical results abruptly and dramatically change their values. This unusual type of spatial aliasing is explored and some consequences for 3D computer reconstructions are presented.

Graphical Abstract

Fourier’s replicas of scanned surfaces correctly reproduce all morphological features only if the number N of harmonic terms in the Fourier sum does not exceed a critical value derived from the Nyquist frequency of the scanned original. Incorporating harmonic terms of higher frequencies in the Fourier sum, an abrupt increase of aliasing effects occurs whenever the frequencies in the Fourier sum reach even multiples of the Nyquist frequency. Illustration N = 50 represents a correct reproduction whereas illustrations N = 228 and N = 440 show critical abrupt increases of aliasing artifacts.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Gottlieb, D., & Shu, C. W. (1997). On the Gibbs phenomenon and its resolution. SIAM Review, 39, 644–668.

    Article  MATH  MathSciNet  Google Scholar 

  2. Gottlieb, D., Shu, C. W., Solomonoff, A., & Vandeven, H. (1992). On the Fourier partial sum of a nonperiodic analytic function. Journal of Applied Mathematics and Computing, 43(1–2), 91–98.

    MathSciNet  Google Scholar 

  3. Gelb, A., & Tanner, J. (2006). Robust reprojection methods for the resolution of the Gibbs phenomenon. Applied and Computational Harmonic Analysis, 20(1), 3–25.

    Article  MATH  MathSciNet  Google Scholar 

  4. Eckhoff, K. S. (1993). Accurate and efficient reconstruction of discontinuous functions from truncated series expansions. Mathematics of Computation, 61(204), 745–763.

    Article  MATH  MathSciNet  Google Scholar 

  5. Eckhoff, K. S. (1995). Accurate reconstruction of functions of finite regularity from truncated Fourier series expansions. Mathematics of Computation, 64(210), 671–690.

    Article  MATH  MathSciNet  Google Scholar 

  6. Eckhoff, K. S. (1998). On a high order numerical method for functions with singularities. Mathematics of Computation, 67(223), 1063–1087.

    Article  MATH  MathSciNet  Google Scholar 

  7. Driscoll, T. A., & Fornberg, B. (2001). A Pade-based algorithm for overcoming the Gibbs phenomenon. Numerical Algorithms, 26(1), 77–92.

    Article  MATH  MathSciNet  Google Scholar 

  8. Boyd, J. P. (2002). A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. Journal of Computational Physics, 178(1), 118–160.

    Article  MATH  MathSciNet  Google Scholar 

  9. Huybrechs, D. (2010). On the Fourier extension of nonperiodic functions. SIAM Journal on Numerical Analysis, 47(6), 4326–4355.

    Article  MATH  MathSciNet  Google Scholar 

  10. Jung, J.-H., & Shizgal, B. D. (2004). Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon. Journal of Computational and Applied Mathematics, 172(1), 131–151.

    Article  MATH  MathSciNet  Google Scholar 

  11. Mertz, P., & Gray, F. (1934). A theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television. Bell System Technical Journal, 13(1), 464–515.

    Article  Google Scholar 

  12. Whittaker, J. M. (1935). Interpolatory function theory. Cambridge: Cambridge University Press.

    Google Scholar 

  13. Shannon, C. E. (1949). Communication in the presence of noise. Proceedings of the Institute of Radio Engineers, 37, 10–21.

    MathSciNet  Google Scholar 

  14. James, J. F. (1995). A student’s guide to Fourier transforms. Cambridge: Cambridge University Press.

    Google Scholar 

  15. Crow, F. C. (1977). Aliasing problem in computer-generated shaded images. Communications of the ACM, 20(11), 799–805.

    Article  Google Scholar 

  16. Shekarforoush, H., Berthold, M., & Zerubia, J. (1995). 3D super-resolution using generalized sampling expansion. In: International conference on image processing, vol. 2 (pp. 300–303, ISBN 0-8186-7310-9).

  17. Cary, P. W. (1999). Genraized sampling and “beyond Nyquist” imaging. CREWES Research Report, 11, 1–23.

    Google Scholar 

  18. Vaidyanathan, P. P. (2001). Generalizations of the sampling theorem: Seven decades after Nyquist. IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, 48(9), 1094–1109.

    Article  MATH  MathSciNet  Google Scholar 

  19. Vandewalle, P., Sbaiz, L., Vandewalle, J., & Vetterli, M. (2004). How to take advantage of aliasing in bandlimited signals. In: Proceedings of International Conference on Acoustics, Speech, and Signal, vol. III: Proceedings: Image and Multidimensional Signal Processing Special Sessions (pp. 948–951), Montreal.

  20. Robinson, M. D., Toth, C. A., Lo, J. Y., & Farsiu, S. (2010). Efficient Fourier-wavelet super-resolution. IEEE Transaction on Image Processing, 19(10), 2669–2681.

    Article  MathSciNet  Google Scholar 

  21. Farsiu, S., Robinson, D., Elad, M., & Milanfar, P. (2004). Advances and challenges in super-resolution. International Journal of Imaging System and Technology, 14(2), 47–57.

    Article  Google Scholar 

  22. IEEE Signal Processing Magazine, special issue on super-resolution, May 2003.

  23. Coulange, B., & Moisan, L. (2010). An aliasing detection algorithm based on suspicious colocalization of Fourier coefficients. In: Proceedings of IEEE 17th International Conference on Image Processing, (pp. 2012–2016). Hong Kong.

  24. Ben Hagai, I., Fazi, F. M., & Rafaely, B. (2012). Generalized sampling expansion for functions on the sphere. IEEE Transactions on Signal Processing, 60(11), 5870–5879.

    Article  MathSciNet  Google Scholar 

  25. Ikuma, T., Kunduk, M., & McWhorter, A. J. (2012). Mitigation of temporal aliasing via harmonic modelling of laryngeal waveforms in high-speed videoendoscopy. Journal of the Acoustical Society of America, 132(3), 1636–1645.

    Article  Google Scholar 

  26. Kreymerman, G. (2012). Adjustable active optical low-pass filter. Applied Optics, 51(2), 268–272.

    Article  Google Scholar 

  27. Cho, S. H., Grazioso, R., Zhang, N., Aykac, M., & Schamand, M. (2011). Digital timing: Sampling frequency, anti-aliasing filter and signal interpolation filter dependence on timing resolution. Physics in Medicine & Biology, 56(23), 7569–7583.

    Article  Google Scholar 

  28. Ficker, T., Martišek, D., & Jennings, H. M. (2010). Roughness of fracture surfaces and compressive strength of hydrated cement pastes. Cement and Concrete Research, 40(6), 947–955.

    Article  Google Scholar 

  29. Ficker, T., Martišek, D., & Jennings, H. M. (2011). Surface roughness and porosity of hydrated cement pastes. Acta Polytechnica, 51(3), 7–20.

    Google Scholar 

  30. Ficker, T., & Martišek, D. (2011). Roughness and fractality of fracture surfaces as indicators of mechanical quantities of porous solids. Central European Journal of Physics, 9(6), 1440–1445.

    Google Scholar 

  31. Ficker, T. (2012). Fracture surfaces and compressive strength of hydrated cement pastes. Construction and Building Materials, 27(1), 197–205.

    Article  Google Scholar 

  32. Watt, W. (2000). 3D computer graphics. London: Pearson Education Limited.

    Google Scholar 

  33. Nakamae, E., Ischiyaki, T., Nishita, T., & Takita, S. (1989). Composing 3D images with antialiasing and various shading effects. IEEE Computer Graphics and Applications, 9, 21–29.

    Article  Google Scholar 

  34. Longhurt, P., Debattista, K., Gillibrand, R. & Chalmers, A. (2005). Analytic antialiasing for selective high fidelity rendering. In: Proceedings of the XVIII Brazilian Symposium on Computer Graphics and Image Processing (pp. 359–366).

  35. Boev, A., Bregovic, R., Damyanov, D. & Gotchev, A. (2009). Anti-aliasing filtering of 2D images for multi-view auto-stereoscopic displays. In: International Workshop on Local and Non/local Approsimation in Image Processing (pp. 97–97), Tuusula.

  36. Wilson, T. (Ed.). (1990). Confocal microscopy. London: Academic Press Ltd.

    Google Scholar 

  37. Škrášek, J. (1986). Základy aplikované matematiky (Basics of Applied Mathematics, part II) (pp. 277–278). Prague: SNTL. (in Czech Language).

  38. Bowman, E. T., Soga, K., & Drummond, W. (2001). Particle shape characterization using Fourier descriptor analysis. Géotechnique., 51, 545–554.

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the Grant Agency of the Czech Republic under contract no. 13-03403S. One of us (T.F.) was partly supported on the basis of the grant no. LO 1408 within the program I-LO (NPU I) administered by the Ministry of Education, Youth and Sports of the Czech Republic.

Author information

Authors and Affiliations

  1. Department of Physics, Faculty of Civil Engineering, Brno University of Technology, Veveří 95, 602 00, Brno, Czech Republic

    T. Ficker

  2. Department of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, 616 62, Brno, Czech Republic

    D. Martišek

Authors
  1. T. Ficker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Martišek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Ficker.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ficker, T., Martišek, D. 3D Image Reconstructions and the Nyquist–Shannon Theorem. 3D Res 6, 23 (2015). https://doi.org/10.1007/s13319-015-0057-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13319-015-0057-4

Keywords

Search

Navigation