Differentiation by integration using orthogonal polynomials, a survey <s> The Savitzky-Golay paper and its follow-up <s> Improved lead-in device passing a conductor through the cover of an electrical precipitation apparatus down to its spray system. The improved device is characterized in that a porous insulator is used as the insulator and a gas-tight insulating cylinder concentrically surrounds the porous insulator. The device is further fitted with a tubular gas inlet opening into the porous insulating cylinder. The upper and lower ends of the insulator and the insulating cylinder, respectively, are connected together by annular metal plates so as to form a space therebetween, and the space is occupied by a sealing gas maintained at a relatively higher pressure than that prevailing inside the electrical precipitation apparatus. <s> BIB001 </s> Differentiation by integration using orthogonal polynomials, a survey <s> The Savitzky-Golay paper and its follow-up <s> The convolution approach to least-squares smoothing and differentiation is extended to remove the data truncation problem of the original Savitzky and Golay algorithm. A formalism based on the recursive properties of Gram polynomials provides a simple and direct means of calculating the convolution weights for all cases and thus enables a short but completely general routine to be written <s> BIB002 </s> Differentiation by integration using orthogonal polynomials, a survey <s> The Savitzky-Golay paper and its follow-up <s> We present here an extension of the convolution technique to include nonuniform data. Although the computation involved is still much greater than that for uniform data, it offers a great reduction on that required by the traditional least-squares method <s> BIB003 </s> Differentiation by integration using orthogonal polynomials, a survey <s> The Savitzky-Golay paper and its follow-up <s> Computation of the derivatives of an image defined on a lattice structure is of paramount importance in computer vision. The solution implies least square fitting of a continuous function to a neighborhood centered on the site where the value of the derivative is sought. We present a systematic approach to the problem involving orthonormal bases spanning the vector space defined over the neighborhood. Derivatives of any order can be obtained by convolving the image with a priori known filters. We show that if orthonormal polynomial bases are employed the filters have closed form solutions. The same filter is obtained when the fitted polynomial functions have one consecutive degree. Moment preserving properties, sparse structure for some of the filters, and relationship to the Marr-Hildreth and Canny edge detectors are also proven. Expressions for the filters corresponding to fitting polynomials up to degree six and differentiation orders up to five, for the cases of unweighted data and data weighted by the discrete approximation of a Gaussian, are given in the appendices. <s> BIB004
The first instance of an approximation of first and higher derivatives by formula (4.1) with possibly n > m was given by Savitzky & Golay BIB001 in 1964. They only dealt with the case of constant weights on {−N, −N + 1, . . . , N }, they used only very special N , n and m, and they did not explicitly mention or use the corresponding orthogonal polynomials. They were motivated by applications in spectroscopy. Their paper had an enormous impact, for instance 5432 citations in Google Scholar in January 2012. Some corrections to BIB001 were given by Steinier et al.  in 1972. Probably, Gorry BIB002 (1990) was the first who gave (4.1) in a more structural form in the case of constant weights on an equidistant set using centered Gram polynomials. Next, in BIB003 (1991) he considered (4.1) on a finite non-equidistant set, still with constant weights. Meer & Weiss BIB004 (1992) gave (4.1) for orthogonal polynomials on a set {−N, −N + 1, . . . , N } with respect to general weights. They made this more explicit in the cases of centered Gram polynomials and of centered Krawtchouk polynomials with symmetric weights. We recommend Luo et al.  as a relatively recent survey of follow-up to the Savitzky-Golay paper.