Sure-optimal two-dimensional Savitzky-Golay filters for image denoising

Savitzky-Golay (SG) filters are linear, shift-invariant lowpass filters employed for data smoothing. In their pathbreaking paper published in Analytical Chemistry, Savitzky and Golay mathematically established that polynomial regression of data over local intervals and evaluation of their values at the center of the approximation window is equivalent to convolution with a finite impulse response filter. In this paper, we expound SURE (Stein´s unbiased risk estimate) based adaptive SG filters for image denoising. Our goal is to optimally choose SG filter parameters, namely, order and window length, the optimality defined in terms of the mean squared error (MSE). In practical scenarios, only a single realization of the noisy image is available and the ground truth is inaccessible. Hence, we propose SURE, which is an unbiased estimate of MSE, to solve the parameter selection problem. It is observed that bandwidth of the minimum MSE (MMSE)-optimum SG filter is small at relatively slowly varying portions of the underlying image, and vice versa at abrupt transitions, thereby enabling us to trade off bias and variance to obtain near-optimal performance. The denoising results obtained exhibit considerable peak signal-to-noise-ratio (PSNR) improvement. At low SNRs, the filter performance is further enhanced by using a regularized cost function.