Savitzky-Golay Filters and Application to Image and Signal Denoising

Abstract

We explore the applicability of local polynomial approximation of signals for noise suppression. In the context of data regression, Savitzky and Golay showed that least-squares approximation of data with a polynomial of fixed order, together with a constant window length, is identical to convolution with a finite impulse response filter, whose characteristics depend entirely on two parameters, namely, the order and window length. Schafer’s recent article in IEEE Signal Processing Magazine provides a detailed account of one-dimensional Savitzky-Golay (SG) filters. Drawing motivation from this idea, we present an elaborate study of two-dimensional SG filters and employ them for image denoising by optimizing the filter response to minimize the mean-squared error (MSE) between the original image and the filtered output. The key contribution of this thesis is a method for optimal selection of order and window length of SG filters for denoising images. First, we apply the denoising technique for images contaminated by additive Gaussian noise. Owing to the absence of ground truth in practice, direct minimization of the MSE is infeasible. However, the classical work of C. Stein provides a statistical method to overcome the hurdle. Based on Stein’s lemma, an estimate of the MSE, namely Stein’s unbiased risk estimator (SURE), is derived, and the two critical parameters of the filter are optimized to minimize the cost. The performance of the technique improves when a regularization term, which penalizes fast variations in the estimate, is added to the optimization cost. In the next three chapters, we focus on non-Gaussian noise models. In Chapter 3, image degradation in the presence of a compound noise model, where images are corrupted by mixed Poisson-Gaussian noise, is addressed. Inspired by Hudson’s identity, an estimate of MSE, namely Poisson unbiased risk estimator (PURE), which is analogous to SURE, is developed. Combining both lemmas, Poisson-Gaussian unbiased risk estimator (PGURE) minimization is performed to obtain the optimal filter parameters. We also show that SG filtering provides better lowpass approximation for a multiresolution denoising framework. In Chapter 4, we employ SG filters for reducing multiplicative noise in images. The standard SG filter frequency response can be controlled along horizontal or vertical directions. This limits its ability to capture oriented features and texture that lie at other angles. Here, we introduce the idea of steering the SG filter kernel and perform mean-squared error minimization based on the new concept of multiplicative noise unbiased risk estimation (MURE). Finally, we propose a method to robustify SG filters, robustness to deviation from Gaussian noise statistics. SG filters work on the principle of least-squares error minimization, and are hence compatible with maximum-likelihood (ML) estimation in the context of Gaussian statistics. However, for heavily-tailed noise such as the Laplacian, where ML estimation requires mean-absolute error minimization in lieu of MSE minimization, standard SG filter performance deteriorates. `1 minimization is a challenge since there is no closed-form solution. We solve the problem by inducing the `1-norm criterion using the iteratively reweighted least-squares (IRLS) method. At every iteration, we solve an l`2 problem, which is equivalent to optimizing a weighted SG filter, but, as iterations progress, the solution converges to that corresponding to `1 minimization. The results thus obtained are superior to those obtained using the standard SG filter.