On the statistical properties of risk estimate in the problem of inverting the Radon transform with a random volume of projection data

When reconstructing tomographic images, it is necessary to solve the problem of suppressing the noise arising from registration of projection data. Methods for solving this problem based on wavelet algorithms and threshold processing procedures have several advantages, including computational efficiency and ability to adapt to local features of images. An analysis of the errors of these methods is an important practical task, since it makes it possible to evaluate the quality of both the methods themselves and the equipment used. When using threshold processing procedures, it is usually assumed that the number of decomposition coefficients is fixed and the noise distribution is Gaussian. This model has been well studied in the literature, and the optimal threshold values have been calculated for different classes of functions. However, in some situations, the sample size is not fixed in advance and must be modeled with some random variable. This paper considers a model with a random number of observations and investigates the asymptotic properties of the mean-square risk estimate. It is proved that the limiting distribution of this estimate belongs to the class of shift-scale mixtures of normal laws.