Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 325785, 26 pages
Research Article

Fractional Directional Differentiation and Its Application for Multiscale Texture Enhancement

1College of Information Science and Technology, Chengdu University, Chengdu 610106, China
2School of Computer Science, Sichuan University, Chengdu 610064, China

Received 11 June 2012; Revised 24 July 2012; Accepted 26 July 2012

Academic Editor: Kwok-Wo Wong

Copyright © 2012 Chaobang Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper derives the directional derivative expression of Taylor formula for two-variable function from Taylor formula of one-variable function. Further, it proposes a new concept, fractional directional differentiation (FDD), and corresponding theories. To achieve the numerical calculation, the paper deduces power series expression of FDD. Moreover, the paper discusses the construction of FDD mask in the four quadrants, respectively, for digital image. The differential coefficients of every direction are not the same along the eight directions in the four quadrants, which is the biggest difference by contrast to general fractional differentiation and can reflect different fractional change rates along different directions, and this benefits to enlarge the differences among the image textures. Experiments show that, for texture-rich digital images, the capability of nonlinearly enhancing comprehensive texture details by FDD is better than those by the general fractional differentiation and Butterworth filter. By quantity analysis, it shows that state-of-the-art effect of texture enhancement is obtained by FDD.