Methods for reconstructing tomographic images based on the inversion of the Radon transform are used in problems arising in medicine, biology, astronomy, and many other fields. In the presence of noise in the projection data, as a rule, it is necessary to apply regularization methods. Recently, methods of the threshold processing of wavelet expansion coefficients have become popular. The analysis of errors of these methods is an important practical task, since it makes it possible to assess the quality of both the methods themselves and the equipment used. When using threshold processing, it is usually assumed that the number of expansion coefficients is fixed, and the noise distribution is Gaussian. This model is well studied in the literature, and the optimal values of the threshold processing parameters are calculated for different classes of functions. However, in some situations, the amount of data is not known in advance and has to be modeled with a certain random variable. In this paper, we consider a model with a random amount of data containing Gaussian noise, and estimate the order of the mean-square risk with an increasing number of decomposition coefficients.
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References
D. Donoho, “Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition,” Appl. Comput. Harmon. Anal., 2, 101–126 (1995).
M. Jansen, Noise Reduction by Wavelet Thresholding, Springer Verlag (2001).
N. Lee, Wavelet-vaguelette decompositions and homogenous equations, Ph.D. Thesis, Purdue University, West Lafayette (1997).
F. Abramovich and B. W. Silverman, “Wavelet decomposition approaches to statistical inverse problems,” Biometrika, 85, No. 1, 115–129 (1998).
A.V. Markin and O. V. Shestakov, “Asymptotics of the risk estimate for the threshold processing of wavelet-vaguelette coefficients in a tomography problem,” Inform. Appl., 4, No. 2, 36–45 (2010).
O. V. Shestakov, “Mean-square thresholding risk with a random sample size,” Inform. Appl., 12, No. 3, 14–17 (2018).
O. V. Shestakov, “Averaged probability of the error in calculating wavelet coefficients for the random sample size,” J. Math. Sci., 237, No. 6, 826–830 (2019).
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, New York (1999).
A. A. Eroshenko and O. V Shestakov, “Asymptotic properties of risk estimate in the problem of reconstructing images with correlated noise by inverting the Radon transform,” Inform. Appl., 8, No. 4, 32–40 (2014).
D. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: Asymptopia?” J. Roy. Stat. Soc. Ser. B, 57, No. 2, 301–369 (1995).
T. Cai and L. Brown, “Wavelet estimation for samples with random uniform design,” Stat. Prob. Lett., 42, 313–321 (1999).
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Proceedings of the XXXV International Seminar on Stability Problems for Stochastic Models, Perm, Russia, September 24–28, 2018. Part II.
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Shestakov, O.V. Mean-Square Risk of the Threshold Processing in the Problem of Inverting the Radon Transform with a Random Sample Size. J Math Sci 248, 46–50 (2020). https://doi.org/10.1007/s10958-020-04854-6
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DOI: https://doi.org/10.1007/s10958-020-04854-6