Generalized SURE for Exponential Families: Applications to Regularization

Stein´s unbiased risk estimate (SURE) was proposed by Stein for the independent, identically distributed (i.i.d.) Gaussian model in order to derive estimates that dominate least squares (LS). Recently, the SURE criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean-squared error (MSE). However, its use has been limited to the i.i.d. case which precludes many important applications. In this paper we begin by deriving a SURE counterpart for general, not necessarily i.i.d. distributions from the exponential family. This enables extending the SURE design technique to a much broader class of problems. Based on this generalization we suggest a new method for choosing regularization parameters in penalized LS estimators. We then demonstrate its superior performance over the conventional generalized cross validation and discrepancy approaches in the context of image deblurring and deconvolution. The SURE technique can also be used to design estimates without predefining their structure. However, allowing for too many free parameters impairs the estimate´s performance. To address this inherent tradeoff, we propose a regularized SURE objective, and demonstrate its use in the context of wavelet denoising.