A New Definition of Fractional Derivatives Based on Truncated Left-Handed Grünwald-Letnikov Formula with0<α<1and Median Correction

Abstract

We propose a new definition of fractional derivatives based on truncated left-handed Grünwald-Letnikov formula with0&lt;α&lt;1and median correction. Analyzing the difficulties to choose the fractional orders and unsatisfied processing results in signal processing using fractional-order partial differential equations and related methods; we think that the nonzero values of the truncated fractional order derivatives in the smooth regions are major causes for these situations. In order to resolve the problem, the absolute values of truncated parts of the G-L formula are estimated by the median of signal values of the remainder parts, and then the truncated G-L formula is modified by replacing each of the original signal value to the differences of the signal value and the median. Since the sum of the coefficients of the G-L formula is zero, the median correction can reduce the truncated errors greatly to proximate G-L formula better. We also present some simulation results and experiments to support our theory analysis.

A Comparative Analysis of Laguerre-Based Approximators to the Grünwald-Letnikov Fractional-Order Difference

“…The FFD has been analyzed in some papers under the heading of a practical implementation of FD [26][27][28], or a truncated/finite difference [21,23,29], or a short-memory difference [30].…”

“…An excellent approximator to FD, which is a combination of the classical finite fractional difference (FFD) [8] and finite Laguerre-based difference (FLD) [16], is found superior to the celebrated Al-Alaouibased approximator. We advocate the contribution of the FFD on the one hand since in the high frequency range it is 2 Mathematical Problems in Engineering identical to the original FD [8,21,22]. On the other hand, we advantage approximating the medium/low-frequency "tale" of the FD by means of the Laguerre filter FLD [16,18].…”

Optimized Parallelization for Nonlocal Means Based Low Dose CT Image Processing

“…This optimization strategy can be easily used to accelerate other reconstruction or restoration tasks using the patch similarity type metrics [ 24 – 26 ]. We can also consider improving the performance of the NLM filtering by incorporating the fractional metric into the calculation of patch similarity [ 27 ]. Evaluation of the potential accuracy enhancement in segmentation/registration (related with CT images) that can be brought by the proposed processing also needs to be performed [ 28 – 30 ].…”

A Fractional Derivative-Based Lateral Preview Driver Model for Autonomous Automobile Path Tracking

“…where x ∈ [a, b], is the order of the fractional derivative, h represents the space step, and ( ) = (−1) ( ) denotes the normalized weights [24], which can be directly obtained by the following recursive formula:…”