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Detection threshold for non-parametric estimation

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Abstract

A new threshold is presented for better estimating a signal by sparse transform and soft thresholding. This threshold derives from a non-parametric statistical approach dedicated to the detection of a signal with unknown distribution and unknown probability of presence in independent and additive white Gaussian noise. This threshold is called the detection threshold and is particularly appropriate for selecting the few observations, provided by the sparse transform, whose amplitudes are sufficiently large to consider that they contain information about the signal. An upper bound for the risk of the soft thresholding estimation is computed when the detection threshold is used. For a wide class of signals, it is shown that, when the number of observations is large, this upper bound is from about twice to four times smaller than the standard upper bounds given for the universal and the minimax thresholds. Many real-world signals belong to this class, as illustrated by several experimental results.

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References

  1. Abramowitz M. and Stegun I. (1972). Handbook of Mathematical Functions. Dover Publications Inc., New York

  2. Berman, S.M.: Sojourns and Extremes Of Stochastic Processes. Wadsworth and Brooks/Cole (1992)

  3. Bruce, A.G., Gao, H.Y.: Understanding waveshrink: Variance and bias estimation. Research Report 36, StatSci (1996)

  4. Coifman, R.R., Donoho, D.L.: Translation invariant de-noising, pp. 125–150. Lecture Notes in Statistics, vol. 103 (1995)

  5. Donoho D.L. (1993). Unconditional bases are optimal bases for data compression and for statistical estimation. Appl. Comput. Harmon. Anal. 1(1): 100–115

  6. Donoho D.L. (1995). De-noising by soft-thresholding. IEEE Trans Inform. Theory 41(3): 613–627

  7. Donoho D.L. and Johnstone I.M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3): 425–455

  8. Donoho D.L. and Johnstone I.M. (1996). Neo-classical minimax problems, thresholding and adaptive function estimation. Bernoulli 2(1): 39–62

  9. Donoho D.L. and Johnstone I.M. (1999). Asymptotic minimaxity of wavelet estimators with sampled data. Statistica Sinica 9(1): 1–32

  10. Johnstone I.M. (1999). Wavelets and the theory of non-parametric function estimation. J. R. Statist. Soc. A 357: 2475–2493

  11. Krim H., Tucker D., Mallat S. and Donoho D.L. (1999). On denoising and best signal representation. IEEE Trans. Inform. Theory 45(7): 2225

  12. Luisier F., Blu T. and Unser M. (2007). A new sure approach to image denoising: interscale orthonormal wavelet thresholding. IEEE Trans. Image Process. 16(3): 593–606

  13. Mallat S. (1999). A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, New York

  14. Pastor, D.: A theoretical result for processing signals that have unknown distributions and priors in white gaussian noise. To appear in Computational Statistics & Data Analysis, CSDA, http://dx.doi.org/10.1016/j.csda.2007.10.011

  15. Pastor, D., Amehraye, A.: Algorithms and applications for estimating the standard deviation of awgn when observations are not signal-free. J. Comput. 2(7) (2007)

  16. Pastor D., Gay R. and Gronenboom A. (2002). A sharp upper bound for the probability of error of likelihood ratio test for detecting signals in white gaussian noise. IEEE Trans. Inform. Theory 48(1): 228–238

  17. Poor H.V. (1994). An Introduction to Signal Detection and Estimation, 2nd edn. Springer, New York

  18. Portilla J., Strela V., Wainwright M.J. and Simoncelli E.P. (2003). Image denoising using scale mixtures of gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11): 1338–1351

  19. Serfling R.J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York

  20. Wald A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Am. Math. Soc. 49(3): 426–482

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Correspondence to Abdourrahmane M. Atto.

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Atto, A.M., Pastor, D. & Mercier, G. Detection threshold for non-parametric estimation. SIViP 2, 207–223 (2008). https://doi.org/10.1007/s11760-008-0051-x

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Keywords

  • Non-parametric estimation
  • Soft thresholding
  • Sparse transform
  • Wavelet transform
  • Non-parametric detection