Fractional Directional Differentiation and Its Application for Multiscale Texture Enhancement

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 p...

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Bibliographic Details
Published in:Mathematical Problems in Engineering 2012-01, Vol.2012 (2012), p.1067-1092-162
Main Authors: Gao, Chaobang, Zhang, Weihua, Zhou, Jiliu
Format: Article
Language:English
Subjects:
Butterworth filters
Control theory
Derivatives
Differentiation
Digital imaging
Euclidean space
Fractals
Image contrast
Image enhancement
Image filters
Masks
Neighborhoods
Power series
Quadrants
Remote sensing
Robotics
Science
Studies
Texture
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Summary: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.
ISSN:1024-123X
1563-5147
DOI:10.1155/2012/325785