Risk Estimation Without Using Stein's Lemma -- Application to Image Denoising

We address the problem of image denoising in additive white noise without placing restrictive assumptions on its statistical distribution. In the recent literature, specific noise distributions have been considered and correspondingly, optimal denoising techniques have been developed. One of the successful approaches for denoising relies on the notion of unbiased risk estimation, which enables one to obtain a useful substitute for the mean-square error. For the case of additive white Gaussian noise contamination, the risk estimation procedure relies on Stein's lemma. Sophisticated wavelet-based denoising techniques, which are essentially nonlinear, have been developed with the help of the lemma. We show that, for linear, shift-invariant denoisers, it is possible to obtain unbiased risk estimates of the mean-square error without using Stein's lemma. An interesting consequence of this development is that the unbiased risk estimator becomes agnostic to the statistical distribution of the noise. As a proof of principle, we show how the new methodology can be used to optimize the parameters of a simple Gaussian smoother. By locally adapting the parameters of the Gaussian smoother, we obtain a shift-variant smoother, which has a denoising performance (quantified by the improvement in peak signal-to-noise ratio (PSNR)) that is competitive to far more sophisticated methods reported in the literature. The proposed solution exhibits considerable parallelism, which we exploit in a Graphics Processing Unit (GPU) implementation.