The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract

Texture enhancement is an important component of image processing that finds extensive application in science and engineering. The quality of medical images, quantified using the imaging texture, plays a significant role in the routine diagnosis performed by medical practitioners. Most image texture enhancement is performed using classical integral order differential mask operators. Recently, first order fractional differential operators were used to enhance images. Experimentation with these methods led to the conclusion that fractional differential operators not only maintain the low frequency contour Contents C591features in the smooth areas of the image, but they also nonlinearly enhance edges and textures corresponding to high frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we apply the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other first order fractional differential operators, we find that our new algorithms provide higher signal to noise values and superior image quality. Subject class: 26A33, 92C55

A Caputo fractional derivative of a function with respect to another function

“…For more than two centuries, this subject was relevant only in pure mathematics, and Euler, Fourier, Abel, Liouville, Riemann, Hadamard, among others, have studied these new fractional operators, by presenting new definitions and studying their most important properties. However, in the past decades, this subject has proven its applicability in many and different natural situations, such as viscoelasticity [11,26], anomalous diffusion [14,19], stochastic processes [9,29], signal and image processing [31], fractional models and control [24,32], etc. This is a very rich field, and for it we find several definitions for fractional integrals and for fractional derivatives [16,25].…”

An image-enhancement method based on variable-order fractional differential operators

“…We first add white noise to the Lena image with standard variance V = 25, 35, 45, 55. A non-fractional differential based image enhancement method (Tang,et al [14]) and another two fractional based methods (Tian's method [13], FCD-1 method [9]) are used to perform comparisons with our proposed method. The PSNR (peak signal-to-noise ratio) measure is adopted to evaluate the quality of the enhanced image, which is defined as follows: It can be clearly seen from Table 1 that the fractional operator methods have a better performance than the non-fractional differential based method, because they can enhance the edge information and preserve the texture details within the smooth areas nonlinearly in accord with the original nonlinear characteristic of image signal.…”

“…On the base of two commonly used definitions for fractional differential (Grünwald-Letnikov and Riemann-Liouville), Pu, et al [8] developed the so-called YIFEIPU-1 algorithm that adopted the coefficients of factional differential derivative to generate five different kinds of masks for image filtering, but their approach suffers from distortion while dealing with color images in the RGB space. Yu, et al [9] improved the original YIFEIPU-1 method on texture enhancement by using a second-order Riesz fractional differential operator, with obvious improvements both for grayscale image enhancement as well as color image enhancement. However, how to adapt the fractional order according to the complexity of features in different portions of the image is a big challenge at present.…”

A Galerkin FEM for Riesz space-fractional CNLS

“…An explosive interest has been gained to the fractional calculus and its applications during recent years. It has been known as an extension of classical calculus to non-integer orders, but compares favorably with classical calculus in modeling various applications, including image processing [1], coupled pendulums [2], capacitor microphone [3], optimal control [4], and anomalous diffusion [5,6].…”