Digital Fractional Order Savitzky-golay Differentiator and Its Application

Firstly the one-dimension digital fractional order SavitzkyGolay differentiator (1-D DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order, is proposed to estimate the fractional order derivative of the noisy signal. The polynomial least square fitting technology and the Riemann-Liouville fractional order derivative definition are used to ensure robust and accuracy. Experiments demonstrate that 1-D DFOSGD can estimate the fractional order derivatives of both ideal signal and noisy signal accurately. Secondly, the two-dimension DFOSGD is obtained from 1-D DFOSGD by defining a group of direction operators, and a new image enhancing method based on 2-D DFOSGD is presented. Experiments demonstrate that 2-D DFOSGD has very good performance on image enhancement. INTRODUCTION In 1695, Leibniz generalized the differentiation operation to non-integer order and proposed a new branch of calculus, Fractional Calculus [1]. Because of some of its characteristics superior to the integral order, Fractional Calculus has been widely used in many engineering fields. Many definitions of fractional differentiation and integration have been proposed and the most popular definitions among them involve: Riemann-Liouville def∗Address all correspondence to this author. inition [2], Grünwald-Letnikow definition [3] and Caputo definition [4]. In this paper, we aim our interests at the digital implement of fractional order derivative. The digital fractional order differentiator (DFOD) is a very useful tool to estimate the fractional derivatives of a given signal. In 2001, an excellent survey, which summarized and compared some fractional order operators for implementing fractional order controllers, was presented in [5]. In 2002, some DFODs were discussed for the continuous models in [6]. These methods included Carlson’s method [7], Roy’s method [8], Chareff’s method [9], Matsuda’s method [10] and Oustaloup’s method [11]. For discrete time case, some FIR filters were presented in [12, 13]. However, the FIR approximation may lead to very high order of FIR filters. Hence, some infinite impulse response(IIR) filters were proposed [14–19]. These methods included fractional differencing formula, Tustin method, Taylor series expansion method, continued fraction expansion, least-squares method and generalized mean method. More recently, many DFODs have been widely applied in many different fields, such as fractional order control [20], image signal processing [21], electrocardiogram signal detecting [22], chaotic oscillations suppression [23], biological signal processing [24] and etc. However, the existing digital fractional order differentiators are sensitive to the noisy signal [25]. When the given signal is contaminated by the noise, the effect of the DFOD will become unsatisfying. In this paper, we proposed a new digital fractional order differentiator based on 1 Copyright c © 2011 by ASME Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA