Learning spectral windowing parameters for regularization using unbiased predictive risk and generalized cross validation techniques for multiple data sets

During the inversion of discrete linear systems noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion, which is a process called regularization. The influence of provided prior information is controlled by non-negative regularization parameter(s). There are a number of methods used to select appropriate regularization parameters, as well as a number of methods used for inversion. New methods of unbiased risk estimation and generalized cross validation are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. It is demonstrated that spectral windowing regularization parameters can be learned from these new estimators applied for multiple data and with multiple windows. The results demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a learning method based on estimating the parameters using true data. The theoretical developments are validated for the case of two dimensional image deblurring. The results verify that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.