Spatially Adaptive Kernel Regression Using Risk Estimation

An important question in kernel regression is one of estimating the order and bandwidth parameters from available noisy data. We propose to solve the problem within a risk estimation framework. Considering an independent and identically distributed (i.i.d.) Gaussian observations model, we use Stein's unbiased risk estimator (SURE) to estimate a weighted mean-square error (MSE) risk, and optimize it with respect to the order and bandwidth parameters. The two parameters are thus spatially adapted in such a manner that noise smoothing and fine structure preservation are simultaneously achieved. On the application side, we consider the problem of image restoration from uniform/non-uniform data, and show that the SURE approach to spatially adaptive kernel regression results in better quality estimation compared with its spatially non-adaptive counterparts. The denoising results obtained are comparable to those obtained using other state-of-the-art techniques, and in some scenarios, superior.