Research Article  Open Access
Texture Enhancement Based on the SavitzkyGolay Fractional Differential Operator
Abstract
Texture enhancement for digital images is the most important technique in image processing. The purpose of this paper is to design a texture enhancement technique using fractional order SavitzkyGolay differentiator, which leads to generalizing the SavitzkyGolay filter in the sense of the SrivastavaOwa fractional operators. By employing this generalized fractional filter, texture enhancement is introduced. Consequently, it calculates the generalized fractional order derivative of the given image using the sliding weight window over the image. Experimental results show that the operator can extract more subtle information and make the edges more prominent. In general, the capability of the generalized fractional differential will be high because it is sensitive to the subtle fluctuations of values of pixels.
1. Introduction
Texture is an important feature of natural images; hence, a variety of image texture applications has been intensively studied by many researchers [1]. Image texture is defined as a function of the spatial variation in pixel intensities (gray values). Smith and Chang [2] have defined texture as visual patterns, which have properties of homogeneity and not resulting from only a single color or intensity.
In the image, texture features capture information about repeating patterns. Texture analysis can be classified into three models: structural, statistical, and signal theoretic methods [3]. Therefore, the analysis of texture parameters is a useful approach for increasing the information accessible from images. In texture enhancement technique, which is based on mask operation, each pixel is modified according to value of the neighbourhood around the pixel of interest. One important aspect of an image, which enables us to perform this, is the notion of frequencies. Fundamentally, the frequencies of an image are the amount, by which the gray values change with distance. Highfrequency components are characterized by huge changes in gray values over small distances; examples of high frequency components are edges and noise. On the other hand, lowfrequency components are parts of the image, which are characterized by little change in the gray values [4]. Fractional differential mask can further preserve the lowfrequency contour feature in those smooth areas, and nonlinearly keep the highfrequency marginal feature in those areas, where the graylevel changes heavily, and also enhances texture details in those areas, where the graylevel does not change evidently.
Fractional integration and fractional differentiation are generalizations of notions of integerorder integration and differentiation and include th derivatives and nfold integrals as particular cases [5, 6]. Many applications of fractional calculus in physics have replaced the time derivative in an evolution equation with a derivative of fractional order [7–11]. Fractional calculus has been applied to a variety of physical phenomena, including anomalous diffusion, transmission line theory, problems involving oscillations, nanoplasmonics, solid mechanics, astrophysics, and viscoelasticity. Currently, fractional calculus (integral and differential operators) is heavily used in control design [12, 13], Furthermore, in image processing [14–19], all results that are based on the fractional calculus operators (differential and integral) show that this method not only is effective, but also has good immunity.
The digital fractional order differentiator is an important topic in fractional calculus that can estimate the fractional order derivative of any given digital signals, without known function. The SavitzkyGolay filter is a simplified digital differentiator that is implemented by a local polynomial regression technique [20, 21]. Now, SavitzkyGolay digital differentiator has been one of the most popular numerical differentiation methods, due to its high computing speed and strong antinoise ability.
Recently, the interest in using texture enhancement technique based on mask operation has grown in the field of image processing. Pu and Zhou [22] have implemented multiscale texture segmentation by fractional differential. They have proposed two fractional differential masks and presented the structures and parameters of each mask, respectively. Then they have discussed the multiscale texture segmentation based on the fractional mask. Pu [23] has proposed a fractional calculus approach to enhance the texture of digital image. He has found that the textural detail enhancing capability of fractional derivativebased texture operator is much better than integer derivative. Zhang et al. [24] have used the fractional differential masks based on the classical RiemannLiouville definition. They have concluded that the fractional order between 1 and 2 can enhance the texture and edges in multiscale, by controlling the fractional order.
In this paper, we have used a generalized fractional differential based on the generalized SavitzkyGolay filter in sense of SrivastavaOwa fractional operators for image texture enhancement. The SavitzkyGolay filter has become a powerful signal and image processing tool, which has found application in many scientific areas. Moreover, the SavitzkyGolay filter method is considered to be a good approach in image texture enhancement, which is used as an alternative to classical techniques. The rest of the paper is organized as follows: Sections 2 and 3 explain the fractional calculus and the generalized fractional integral operator, respectively, Section 4 describes the construction of fractional differential SavitzkyGolay filter, Section 5 elucidates the experimental results, and Section 6 concludes the paper.
2. Fractional Calculus
The idea of the fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) was found over 300 years ago. Abel in 1823 scrutinized the generalized tautochrone problem and for the first time applied fractional calculus techniques in a physical problem.
2.1. The RiemannLiouville Operators
The RiemannLiouville fractional derivative strongly poses the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. Moreover, this operator possesses advantages of fast convergence, high stability, and higher accuracy to derive different types of numerical algorithms [6].
The fractional (arbitrary) order integral of the function of order is defined by When , we write , where denoted the convolution product, , and , , and as , where is the delta function.
The fractional (arbitrary) order derivative of the function of order is defined by When , we have The Caputo fractional derivative of order is defined, for a smooth function , by where (the notation stands for the largest integer not greater than ). Note that there is a relationship between the RiemannLiouville differential operator and the Caputo operator: and they are equivalent in a physical problem (i.e., a problem which specifies the initial conditions).
2.2. The SrivastavaOwa Operators
In [25], Srivastava and Owa defined and studied fractional operators (derivative and integral) in the complex plane for analytic functions.
The fractional derivative of order is defined, for a function by where the function is analytic in simplyconnected region of the complex plane containing the origin and the multiplicity of is removed by requiring to be real when . Furthermore, for , the fractional differential operator is defined as
The fractional integral of order is defined, for a function , by where the function is analytic in simply connected region of the complex plane () containing the origin and the multiplicity of is removed by requiring to be real when : Note that the real case of the SrivastavaOwa operators is equivalence to the RiemannLiouville operators.
3. Generalized Fractional Integral Operator
This section briefly describes the mathematical background for the fractional integral operator that has been used by the proposed algorithm. The usual way of representing the fractional derivatives is by the RiemannLiouville formula . Another way to represent the fractional derivatives is by the GrünwaldLetnikov formula [23]. The discrete approximations derived from the GrünwaldLetnikov fractional derivatives present some limitations, such as the following[26]:(i)they frequently originate unstable numerical methods;(ii)the order of accuracy of such approaches is never higher than one.
To implement the generalized fractional integral method, Ibrahim in [27] has imposed a formula for the generalized fractional integral. Consider, for natural and real , the fold integral of the form Applying the Cauchy formula for iterated integrals implies Repeating the above step times we obtain which yields the fractional operator type where and are real numbers and the function is analytic in the simply connected region of the complex plane containing the origin, and the multiplicity of is removed by requiring to be real when . When , we arrive at the standard SrivastavaOwa fractional integral, which is used to define the SrivastavaOwa fractional derivatives.
Corresponding to the generalized fractional integrals (13), we define the generalized differential operator of order by where the function is analytic in the simply connected region of the complex plane containing the origin and the multiplicity of is removed by requiring to be real when .
Proposition 1 (see [27]). The generalized derivative of the function , is given by the following: which is later used to compute the coefficient matrix .
4. Construction of the Fractional Differential SavitzkyGolay Filter
The SavitzkyGolay filter has been introduced for computing the numerical derivatives and is also called a digital smoothing polynomial filter. The SavitzkyGolay method is often used to preserve higher moments in the data, thus reducing the distortion of essential features of the data.
In this section, we willgeneralize this filter for calculating the fraction derivatives which will be utilized by the proposed algorithm.
Assume a uniformly sampled signal, our aim is to estimate its th order derivative using point filtering window and an degree polynomial [21]: which is used to fit the given signal . In matrix notation, (16) is reduced to the system where is the estimate error, is the coefficient matrix and is the Vandermonde matrix defined by The coefficients of the bestfit polynomial can be obtained by minimizing the sum of the squared errors between the actual data and fitting points. Thus, implies where denotes the moving window’s coefficients matrix. Consequently, the th order derivative can be estimate by
Now, in view of Proposition 1, we have where
Note that when , we have the RiemannLiouville differential operator. Moreover, when , the Vandermonde matrix is a square matrix. The purpose of the SavitzkyGolay filter is to estimate , which can be used to calculate the th order derivative of any given signal [21]. The coefficient matrix can be computed by where where is the sampling interval.
The matrix can be assumed as the formula of computation time; therefore, the generalized SavitzkyGolay filter can be viewed as the generalization of the differentiator. However, digital image is a function of two variables, so, we can generalize these definitions to include both the and values.
The mask is designed into size matrix which has layers ( is odd). The window’s size can be an arbitrary odd number, and a larger window can improve the accuracy of fractional differential, but increases the computational time. Therefore, we proceed to use moving window with size as shown in Table 1.

The fractional differential operator can enhance edges and contours as well as reserve the texture details. The nine values output of each fractional differential window is performed by sliding the mask window over the image . Generally one can start at the top left corner of the image block through all the pixels, where the fractional differential mask fits entirely within the boundaries of the image. The output of each image block is nine values, which represent the texture information in each image block, that takes the following formula: where is the value of an image pixel and is the value of filter mask.
5. Experimental Results and Discussion
The reason of this experiment is to validate the correctness of the proposed algorithm.
Performance tests for the algorithm proposed by this paper were implemented using Matlab 2010a on Intel (R) Core i7 at 2.2 GHz, 4 GB DDR3 Memory, and system type 64bit, Window 7. The computation time per image differs for each image and depends mainly on the window’s size as well as image size.
The proposed texture features enhancement algorithm includes the following steps:(i)read the original grayscale image;(ii)set the value of and (≥1);(iii)set the value of the image sampling interval ;(iv)set the values of the fractional power parameters ();(v)compute SavitzkyGolay moving window as in (24);(vi)compute the Vandermonde matrix as in (25);(vii)apply the SavitzkyGolay fractional differential mask with the corresponding image pixels by sliding the window over the image.
By varying both the fractional powers and , keeping , and fixed (, ), the elements of the SavitzkyGolay moving window have been computed as shown in Tables 2 and 3.


Tables 2 and 3. show the coefficients of the fractional differential moving window for different values of . All coefficient values are not equal to zero, which implies that the magnitude response of SavitzkyGolay filter is not also zero in the image region. This will likelyimprove the texture detail. However, the qualities of texture is defined by the spatial distribution of gray values for this reason, we have used grayscale images for testing, which are shown in Figures 1(a) and 1(b).
(a)
(b)
In order to illustrate the efficiency of the proposed algorithm in Figure 2, we have presented an illustration of the obtained results for the texture enhancement of the original images of Figure 1.
(a)
(b)
The proposed enhancement algorithm shows good enhancement performance for both, testing images by different degrees of fractional power values and which are experimentally fixed at , 0.5 and , and the value of the image sampling interval . It is seen that, the proposed enhancement algorithm using fractional differential masks, can extract more texture information and sharpen edges more efficiently. The eye’s qualitative analysis of the proposed algorithm acts as one of the important parameters to judge its performance.
Other metrics used to judge the algorithm performance are the statistical measures. In this paper, among the statistical features, the following secondorder statistics are used as texture features in representing images. The graylevel cooccurrence matrix (GLCM) is a statistical method used to describe textures in an image, by modeling texture as a twodimensional gray level variation [28]. Four statistical measures are extracted to evaluate the images texture enhancement; these are entropy, homogeneity, contrast, and energy.(1) Entropy measures the amount of information, and the larger value of entropy is the greater amount of information carried by image, but inversely correlated to energy. Entropy feature of grayscale cooccurrence matrix is one of the features having the best discriminatory power, which is given in following equation: where is the probability for grayscale and and occurs at two pixels.(2) Homogeneity measures the closeness of the distribution of elements in the GLCM to the GLCM diagonal: (3) Contrast measures the intensity contrast between a pixel and its neighbour over the whole image: (4) Energy measures the sum of squared elements in the graylevel cooccurrence matrix (GLCM):
Figure 2 shows the results of the proposed enhancement algorithm for with (a) , and (b) . The variation of the image texture is observed when is increased from 0.4 to 0.5. So, the selection of differential order is important.
Tables 4 and 5 and Figures 3 and 4 show the performance evaluation of the proposed algorithm for image (a) and image (b) according to those four statistical measures of graylevel cooccurrence matrix (GLCM). It can be clearly seen that there has been a large increase in the value of entropy, which means the greater amount of information is carried by image due to texture enhancement. The entropy values for image (a) climbed to approximately 0.224 for and to 0.26 for for all texture enhancement cases. Moreover, for image (a), it inversely correlated to energy, which decreases to approximately 0.881 for and to 0.860 for and from its value of the original testing image. The homogeneity steadiness is reduced with the increase of texture enhancement, which means more divergence of the distribution of elements of information carried by image due to texture enhancement process. While the contrast showed diverse tendencies for all texture enhancement cases, it is conclude that the intensity contrast between a pixel and its neighbour over the whole image are changed too. This variation in the statistical measures makes the proposed algorithm capable to control the degree of texture enhancement of the image by controlling the fractional order parameters and .


6. Conclusion
In this paper, a texture enhancement technique using fractional order SavitzkyGolay differentiator, which leads to generalize SavitzkyGolay filter in sense of SrivastavaOwa fractional operators, have been introduced. The new algorithm presented in this paper can control the degree of texture enhancement of the image with the fractional power values. The new approach can control the degree of texture enhancement of the image with fractional order of the parameters , and . However, the technique is by no means limited only to images, instead, it can be applied in the setting of different image applications, taking into consideration the limitations of each imaging method. Furthermore, our goal is to keep away from the effect of the noise that caused in the texture enhancement of the image and to control the degree of texture enhancement of the image with the filter mask parameters. The experiment results had demonstrated the efficacy of this algorithm according to the metrics used to judge the algorithm performance.
Acknowledgments
The authors would like to thank the reviewers for their comments on earlier versions of this paper. This research has been funded by university of Malaya, under Grant no. UMRG 10412ICT.
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Copyright
Copyright © 2013 Hamid A. Jalab and Rabha W. Ibrahim. 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.