The strong law of large numbers for the risk estimate in the problem of tomographic image reconstruction from projections with a correlated noise

Methods of wavelet analysis based on thresholding of coefficients of the projection decomposition are widely used for solving the problems of tomographic image reconstruction in medicine, biology, astronomy, and other areas. These methods are easily implemented through fast algorithms; so, they are very appealing in practical situations. Besides, they allow the reconstruction of local parts of the images using incomplete projection data, which is essential, for example, for medical applications, where it is not desirable to expose the patient to the redundant radiation dose. Wavelet thresholding risk analysis is an important practical task, because it allows determining the quality of the techniques themselves and of the equipment which is being used. The present paper considers the problem of estimating the function by inverting the Radon transform in the model of data with correlated noise. The paper considers the wavelet-vaguelette decomposition method of reconstructing tomographic images in the model with a correlated noise. It is proven that the unbiased mean squared error risk estimate for thresholding wavelet-vaguelette coefficients of the image function satisfies the strong law of large numbers, i. e., it is a strongly consistent estimate.