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Gaussian DCT Coefficient Models

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Abstract

It has been known that the distribution of the discrete cosine transform (DCT) coefficients of most natural images follow a Laplace distribution. However, recent work has shown that the Laplace distribution may not be a good fit for certain type of images and that the Gaussian distribution will be a realistic model in such cases. Assuming this alternative model, we derive a comprehensive collection of formulas for the distribution of the actual DCT coefficient. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. Finally, the superior performance of the derived distributions over the Gaussian model is illustrated. It is expected that this work could serve as a useful reference and lead to improved modeling with respect to image analysis and image coding.

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References

  1. Ahmed, N., Natarajan, T., Rao, K.R.: Discrete cosine transform. IEEE Trans. Comput. 23, 90–93 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Wang, Z., Hunt, B.: The discrete W transform. Appl. Math. Comput. 16, 19–48 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Jain, A.K.: Fundamentals of Digital Image Processing. Prentice Hall, New York (1989)

    MATH  Google Scholar 

  4. Rao, K.R., Yip, P.: Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic Press, Boston (1990)

    MATH  Google Scholar 

  5. Duhamel, P., Vetterli, M.: Fast Fourier transforms: a tutorial review and a state of the art. Signal Process. 19, 259–299 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Reininger, R.C., Gibson, J.D.: Distribution of the two–dimensional DCT coefficients for images. IEEE Trans. Commun. 31, 835–839 (1983)

    Article  Google Scholar 

  7. Lam, E.Y.: Analysis of the DCT coefficient distributions for document coding. IEEE Signal Process. Lett. 11, 97–100 (2004)

    Article  Google Scholar 

  8. Lam, E.Y., Goodman, J.W.: A mathematical analysis of the DCT coefficient distributions for images. IEEE Trans. Image Process. 9, 1661–1666 (2000)

    Article  MATH  Google Scholar 

  9. Lam, E.Y.: Statistical modelling of the wavelet coefficients with different bases and decomposition levels. IEE Proc. Vis. Image Signal Process. 151, 203–206 (2004)

    Article  Google Scholar 

  10. Teichroew, D.: The mixture of normal distributions with different variances. Ann. Math. Stat. 28, 510–512 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nadarajah, S., Kotz, S.: On the DCT coefficient distributions. IEEE Signal Process. Lett. 13, 601–603 (2006)

    Article  MathSciNet  Google Scholar 

  12. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 1. Gordon and Breach Science, Amsterdam (1986)

    Google Scholar 

  13. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000)

    MATH  Google Scholar 

  14. van Dorp, J.R., Kotz, S.: A novel extension of the triangular distribution and its parameter estimation. J. R. Stat. Soc. Ser. D—Stat. 51, 63–79 (2002)

    Google Scholar 

  15. van Dorp, J.R., Kotz, S.: The standard two-sided power distribution and its properties: with applications in financial engineering. Am. Stat. 56, 90–99 (2002)

    Article  Google Scholar 

  16. Tukey, J.W.: Exploratory Data Analysis. Addison-Wesley, Reading (1977)

    MATH  Google Scholar 

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  1. University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

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  1. Saralees Nadarajah
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Correspondence to Saralees Nadarajah.

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Nadarajah, S. Gaussian DCT Coefficient Models. Acta Appl Math 106, 455–472 (2009). https://doi.org/10.1007/s10440-008-9307-2

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  • DOI: https://doi.org/10.1007/s10440-008-9307-2

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