Prospect of Stein's Unbiased Risk Estimate as Objective Function for Parameter Optimization in Image Denoising Algorithms – A Case Study on Gaussian Smoothing Kernel

Stein's Unbiased Risk Estimate (SURE) is considered as an indirect method for predicting Mean Squared Error (MSE) in the absence of ground-truth, as its computation requires only noisy observation and denoised image. SURE is usually used as an objective function for optimizing the operational parameters of denoising algorithms, adequate for real-time images. Hence, a close analysis of the performance of SURE on standard test images is worthy of investigation. Pearson's Correlation (r) of SURE with Mean Absolute Error (MAE) between denoised images and ground-truth is analyzed in this paper, on Shepp-Logan Phantom and simulated Magnetic Resonance (MR) images, at different noise levels. Denoised images which differ in terms of MAE against ground-truth are produced by varying the standard deviation of a Gaussian smoothing kernel
$(\mathbf{0.01\leq\sigma\leq 0.04})$
of fixed dimension,
$\mathbf{9\times 9}$
. Values of correlation between SURE and MAE on Shepp-Logan and simulated MR images are
$\mathbf{r}=-\mathbf{0.99\pm 0.02}$
and
$\mathbf{r}= \mathbf{0.48\pm 0.36}$
, respectively. Concordance of SURE with MAE is observed to be poor on simulated MR images, especially at higher noise levels. SURE is suitable for optimizing the parameters of denoising kernels only when the underlying function used to compute the kernel is fully differentiable by the noisy observation.