Properties of the SURE Estimates When Using Continuous Thresholding Functions for Wavelet Shrinkage

Wavelet analysis algorithms in combination with thresholding procedures are widely used in nonparametric regression problems when estimating a signal function from noisy data. The advantages of these methods lie in their computational efficiency and the ability to adapt to the local features of the estimated function. It is usually assumed that the signal function belongs to some special class. For example, it can be piecewise continuous or piecewise differentiable and have a compact support. These assumptions, as a rule, allow the signal function to be economically represented on some specially selected basis in such a way that the useful signal is concentrated in a relatively small number of large absolute value expansion coefficients. Then, thresholding is performed to remove the noise coefficients. Typically, the noise distribution is assumed to be additive and Gaussian. This model is well studied in the literature, and various types of thresholding and parameter selection strategies adapted for specific applications have been proposed. The risk analysis of thresholding methods is an important practical task, since it makes it possible to assess the quality of both the methods themselves and the equipment used for processing. Most of the studies in this area investigate the asymptotic order of the theoretical risk. In practical situations, the theoretical risk cannot be calculated because it depends explicitly on the unobserved, noise-free signal. However, a statistical risk estimate constructed on the basis of the observed data can also be used to assess the quality of noise reduction methods. In this paper, a model of a signal contaminated with additive Gaussian noise is considered, and the general formulation of the thresholding problem with threshold functions belonging to a special class is discussed. Lower bounds are obtained for the threshold values that minimize the unbiased risk estimate. Conditions are also given under which this risk estimate is asymptotically normal and strongly consistent. The results of these studies can provide the basis for further research in the field of constructing confidence intervals and obtaining estimates of the convergence rate, which, in turn, will make it possible to obtain specific values of errors in signal processing for a wide range of thresholding methods.