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A bi‐directional fractional‐order derivative mask for image processing applications

2020; Institution of Engineering and Technology; Volume: 14; Issue: 11 Linguagem: Inglês

10.1049/iet-ipr.2019.0467

1751-9667

Meriem Hacini, Fella Hachouf, Abdelfatah Charef,

Fractional Differential Equations Solutions

IET Image ProcessingVolume 14, Issue 11 p. 2512-2524 Research ArticleFree Access A bi-directional fractional-order derivative mask for image processing applications Meriem Hacini, Corresponding Author Meriem Hacini meriem.hacini@yahoo.fr Laboratoire d'analyse des signaux et systémes, Département d' Electronique, Université Mohamed Boudiaf, M'sila, Algeria Laboratoire d'Automatique et de Robotique, Département d' Electronique, Université des Fréres Mentouri, Constantine, AlgeriaSearch for more papers by this authorFella Hachouf, Fella Hachouf Laboratoire d'analyse des signaux et systémes, Département d' Electronique, Université Mohamed Boudiaf, M'sila, Algeria Laboratoire d'Automatique et de Robotique, Département d' Electronique, Université des Fréres Mentouri, Constantine, AlgeriaSearch for more papers by this authorAbdelfatah Charef, Abdelfatah Charef Laboratoire de Traitement du Signal, Département d' Électronique, Université des Fréres Mentouri Constantine1, AlgeriaSearch for more papers by this author Meriem Hacini, Corresponding Author Meriem Hacini meriem.hacini@yahoo.fr Laboratoire d'analyse des signaux et systémes, Département d' Electronique, Université Mohamed Boudiaf, M'sila, Algeria Laboratoire d'Automatique et de Robotique, Département d' Electronique, Université des Fréres Mentouri, Constantine, AlgeriaSearch for more papers by this authorFella Hachouf, Fella Hachouf Laboratoire d'analyse des signaux et systémes, Département d' Electronique, Université Mohamed Boudiaf, M'sila, Algeria Laboratoire d'Automatique et de Robotique, Département d' Electronique, Université des Fréres Mentouri, Constantine, AlgeriaSearch for more papers by this authorAbdelfatah Charef, Abdelfatah Charef Laboratoire de Traitement du Signal, Département d' Électronique, Université des Fréres Mentouri Constantine1, AlgeriaSearch for more papers by this author First published: 14 August 2020 https://doi.org/10.1049/iet-ipr.2019.0467Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Fractional computation has been recently designed as a major mathematical tool in image and signal processing fields. This study presents a novel operator established for two-dimensional fractional differentiation. It is developed based on the one-dimensional Charef fractional differentiation extension. A new multi-directional mask is proposed and a new adaptive fractional-order computation is introduced. The proposed method uses the gradient computation properties. It has been applied in edge detection and de-noising problems using real and synthetic images. Obtained results have been compared to those given by integer and fractional useful operators. Results demonstrate that the fractional edge images obtained using the proposed operator has more complete and clear contour information and more abundant texture detail information. The performances have been improved by the proposed method. 1 Introduction Fractional computation is for the mathematical research, the universe that generalises the traditional definition of integer order derivative operators to fractional ones [1–3]. Fractional calculus dates from l'Hopital and Leibniz exchange 300 years ago [4]. Since then, fractional differentiation has been studied by many researchers in different areas. Also, the most extensively adopted definitions have been elaborated by Riemann–Liouville (R–L), Grümwald–Letnikov (G–L), and Caputo [5–7]. Fractional calculus impressed researchers. So, they experimented with it in multidisciplinary domains, such as mechanics, chemistry, economics, and especially control theory and robotics [8–13]. All the developed fractional forms have influenced the fractional calculations domain and scientists studying them. Thereafter, many applications have been developed rapidly. In image processing, researches have noticed that the fractional calculus has an extraordinary capability to strengthen high- and low-frequency parts of an image [14]. Subsequently, this field has been developed rapidly due to the fact that fractional-order derivative operators have a great capacity in treating with fine details such as edges and textures [14–16]. Fractional derivatives have almost been used in all image processing applications. For instance, in image de-noising, a multitude of development have been carried out. In [17], fractional-order has been used in diffusion formulation. The obtained equation has been applied for image de-noising on various digital databases. In [18], the fractional diffusion equation has been used for non-local filtering in speckle interferometry fringers. Besides, the fractional diffusion has been applied for seismic data noise attenuation [19], and so on. Moreover, variational models have been concerned by fractional order. Hacini et al. [20] have elaborated on a new fractional-order variational model for ultrasound image de-noising. In [21], Dazi et al. proposed weighted split Bregman iterations for an adaptive fractional-order total variation model. Also, for variational models, fractional derivatives have recently proven their efficiency in deep learning. In [22], Chen et al. proposed a fractional-order variational residual convolutional neural network for low dose computer tomography (CT) image de-noising. Likewise, in edge detection, a multitude of development have been carried out. For example, in [23], a hybrid method using fractional derivative and genetic algorithms has been proposed for a medical image application. In [24], a fractional edge detection has been applied on field programmable gate array. Moreover, an identification algorithm has been developed using a fractional derivative for a medical image application [25]. Also, in [26], an effective algorithm for benign brain tumours detection has been developed using fractional calculus. Indeed, the fractional-order derivative helps in considering more neighbouring pixels information and extracting more image texture details, which encouraged more applications using this tool as well as in image enhancement [27–30], image segmentation [31, 32], image registration [33, 34], in-painting [35], and neural networks [36, 37]. The most used formulation of the fractional derivative is the G–L definition [6]. Also, fractional differential equations have been the focus of many mathematicians. Consequently, considerable attention has been given to their numerical solutions. However, these methods may not be interesting from an engineering approach, at least, in terms of simulation and implementation of fractional systems. Charef's formulation [38, 39] has a proven record of accomplishment in signal processing and control domains. However, it has never been used in the image processing domain. Consequently, considerable attention has attracted us to their numerical solutions. That led us to adapt Charef's formulation to the image-processing formulation. Obtained results have been convincing to deep our research studies and extend them to more applications. In this work, new numerical calculations and simulations have been elaborated. First, a new fractional-order equation based on Charef's model has been formulated. Next, a multi-directional mask has been developed. Then, an introduction of an adaptive fractional-order computing is proposed. The local variance was used to represent the local image features. These improvements make the proposed operator more effective in extracting edges and low image features than classical methods. The rest of this paper is organised as follows: Section 2 is dedicated to a background theory for the fractional calculus. Section 3 introduces the implementation of one-dimensional fractional Charef derivative (1D-FCD). In Section 4, a generalisation to a 2D-formulation of the 1D-FCD operator is expressed. Then, the proposed operator is used to construct a multi-directional mask. The properties of the proposed operator are analysed in Section 5. It has been integrated into a variational de-noising algorithm and in the well-known Canny operator for edge detection. Section 6 provides numerical results and their analysis. The main concluding remarks are mentioned in Section 7. 2 Background theory for the fractional operators In this section, the basic knowledge of well-known fractional calculus is introduced, including the definitions and several simple properties used in this study. Different from integer calculus, up to now, the fractional derivative does not have a unified temporal definition expression. The commonly used definitions are G–L and R–L derivatives [5, 6]. Fractional derivative G–L definition is as follows: (1)where denotes the fractional differential operator based on G–L definition. denotes a differ-integral function, is the fractional order, is the domain of , is the Gamma function, and [.] is the rounding function. The fractional derivative R–L definition is as follows: (2)where denotes the fractional differential operator based on G–L definition, . Moreover, the G–L fractional derivative can be deduced from the definition of the R–L fractional derivative. Fractional calculus is more difficult to compute than integer calculus. Several mathematical properties used in this study are given here. The fractional differential of a linear combination of differ-integral functions is as follows: (3)where and are differ-integral functions and and are constants. The fractional differential of constant function (C is a constant) is different under different definitions: for G–L definition (4)for R–L definition (5)According to (4) and (5), we can know that the fractional differential of constant function is not equal to 0. 3 1D digital fractional-order Charef differentiator The transfer function of the fractional-order differentiator has been performed by the following irrational function in the frequency domain as (6)where is the complex frequency and is a positive real number such that . For a practical frequency interest band . The fractional-order operator is formed by a fractional power zero transfer function given as follows: (7)suppose that for , we have . Therefore (8)with and is the dB frequency corner of the fractional power zero, which is obtained from the low-frequency , as , where is the maximum allowed error between the slopes of the fractional-order differentiator of (1) and the fractional power zero of (7) in the given frequency interval So to get the fractional power zero of (7), and therefore the fractional differentiator, using a linear time-invariant system model. An approximation of the irrational transfer function by a rational one is needed. An approximation of the 20 dB/dec slope on the Bode plot of the fractional power pole by a number of alternate slopes of 20 and 0 dB/dec coinciding to alternate zero and poles of the negative real axis of the s-plane such that . Thus, the approximation is as follows: (9) The approximation of the fractional power pole is done based on the method proposed by Charef et al. [38]. Zeros and poles have been found in the form of a geometric progression. This graphical approximation method started with a stated error y in decibels and an approximation frequency band , which can be , are confirmed. The parameters , and N can be computed as follows: (10)The zeros and poles of (4) are derived from (5) as for and for , . Hence, the fractional-order differentiator is approached by a rational function in a given frequency band of practical interest by (11)the frequencies and are such that and . The rational fraction of (11) can be modelled as (12)where the residues are calculated as (13)The transfer function of the digital fractional order differentiator is then obtained using the Tustin generating function (T is the sampling period) in (12) as (14)The above equation is interpreted in a simplified form as (15)where and are given, for , as (16)The impulse response of the digital fractional-order differentiator is obtained using the inverse z transform of (15) as (17)By truncation, the sequence for , we obtain the digital fraction order differentiator impulse response of length L in closed form as (18)Hence, the representation of the digital fractional-order differentiator in a transfer function is as follows: (19) Let , we obtain the following equation: (20)The inverse Laplace transform of (20) is (21) The Charef order derivative of function is defined as (22)where is the step size, T is the sampling step set to 1, , and are approximative parameters defined by (13) and (16). , N is the number of approximation parameters to enhance approximation. It is defined in (10). In practice, the Dirac function is represented by (23)where is the Heaviside function. For an instance t and considering numerical calculation, the discrete approximation at point t of one-dimensional fractional differentiation is given by (24)where T is the sampling step, , for () in the interval , . is the approximation coefficient (25) 4 The 2D-adaptive fractional charef derivative (AFCD): bi-dimensional fractional derivative mask 4.1 Generalisation In this section, the 2D generalisation method of the 1D-Charef fractional differentiator is introduced. Some basic definitions must be known at first to describe the generalisation process clearly. Assume the filtering window's size , m is a positive integer. Using (21), the middle point th-order derivative in the filtering window can be estimated by (26)Suppose the grey value at the (x,y) position in image f is denoted by . Thus, using the th-order derivatives of , based on (24) and (25) of the previous section, the filter of size in the and directions can be computed by (27) (28)From (27) and (28), we can observe that image features extraction is done in horizontal and vertical directions using the same coefficients. Thus, is isotropic, it is used to calculate the fractional-order derivative in one direction. This mask form is not efficient enough to detect a big amount of information on an image. So, an extension of the proposed mask to other directions will perform better. 4.2 Bi-dimensional fractional derivative mask An edge pixel is a characterisation of high-gradient magnitude pixel values. Inspired by Sobel [40], Prewit [41], Xu et al. [42] configurations, a multi-directional mask is presented. The proposed filter has been tested and cross-validated in order to obtain better and more significant results. The mask is proposed as an invariant window. We get the fractional gradients , as follows (Fig. 1): (29) (30)The magnitude and direction of the fractional gradients are given as (31) (32) Fig. 1Open in figure viewerPowerPoint The proposed x- and y-direction masks for fractional-order differential (a) x-direction, (b) y-direction 5 Determination of the fractional-order differential operator The choice of the fractional-order parameter is adaptively done to improve the performance of the proposed operator. This parameter is based on the image aspects such as homogeneous (low frequency) and non-homogeneous (high frequency) regions. By definition, the variance is normally used to find how each pixel varies from the neighbouring pixel and is used to classify images in different regions. It allows identifying sharp details such as edges; that is why local variance is considered as the best tool to express the image region complexity. The local variance is relatively large in high textured images and edge regions and it is small in fine-textured regions. Therefore, the local mean value is calculated. The local variance of each pixel of an image is computed as follows: (33)where represents the local mean value, W is a rectangular window with (x,y) as the centre. is the local mean value (34)Fig. 2 illustrates the relationship between local variance and fractional-order. Results shown in this figure demonstrate that the higher is, the complexity of the local region (edge, textures etc.) the larger is, the local variance, and so on. Thus, the choice of large fractional-order is required for a local region containing edges and small details, while a small order for flat and homogeneous areas is sufficient. Therefore, the choice of fractional-order can affect or improve the performance of the proposed algorithm. In this study, some tests have been carried to choose locally the best possible equation to compute the fractional derivative as (35) Fig. 2Open in figure viewerPowerPoint Illustration of properties of the fractional order computation (a) Adaptive fractional-order on homogeneous and textured region, (b) The result of the proposed new adaptive-order using the Charef derivative mask 6 Experimental results and analysis In this part, obtained experimental results using the proposed method on synthetic and real images are reported. To illustrate the performance of the proposed mask, 2D-AFCD has been tested in different situations. Therefore, two parts are considered in this study; the first one to demonstrate the findings of the proposed 2D-AFCD in terms of edge detection. In this subsection, synthetic, noisy, and medical images have been considered. To support the results obtained in the first part, we use the proposed 2D-AFCD in speckle reduction of ultrasound images. The most well-known derivative operators such as Sobel [40], Laplace [43], R–L [5] and G–L [6] have been used for comparison. All experiments and implementation have been performed using Matlab 2015a on an Intel Core i5 personal computer of 2.2 GHz, 4 GB RAM, using 64 bits Windows10. 6.1 Multi-directional mask advantage In this section, the advantage of using the multi-directional mask is illustrated. Fig. 3a shows the obtained results on the original image (Fig. 3b) by the 2D-FCD operator and its improved version 2D-AFCD in Fig. 3c. We can observe on cameraman and clock images that lines are doubled for the camera tripod. Camera and cameraman's face are badly detected and not well-designed compared to results of Fig. 3c where hand, face details, camera, and its tripod are highly detected. The same remarks can be formulated on the clock image. We observe that numbers are doubled and the edges are disconnected in Fig. 3d. We can confirm that the improvement done to the proposed operator have given more accuracy and precision. Fig. 3Open in figure viewerPowerPoint Results of edge extraction on two different views using the proposed operator (a) Original image, (b) 2D-FCD, (c) Improved 2D-AFCD 6.2 Evaluation of the 2D-AFCD operator on different scenes In this subsection, synthetic and real images have been used to illustrate the performance of the 2D-AFCD. The used images are in grey-scale with different sizes. The proposed masks have been compared to existing methods. To ensure the experiments' quality, optimal parameters such as fractional order have been exactly chosen for the compared methods. An evaluation of the CPU time has been achieved. 6.2.1 Comparative study with integer differential operators Most typical first-order differential operators such as Sobel [40], Prewit [41], Roberts [44], and the second-order differential operators as Laplace [43], are in fact integer differential operators. They are constructed to perform in well in high frequency regions. Fig. 4 exhibits a comparison of results obtained by our approach and classical integer-differential operators. Figs. 4b and c show, respectively, the results of Laplace and Sobel operators. It is observed that low-frequency characteristics are poorly detected whereas an excessive recognition of the high-frequency characteristics is noticed by white edges. 2D-AFCD takes into consideration low- and high-frequency components compared to other integer differential operators. This avoids the overly emphasis on high frequencies. Therefore, results of the proposed 2D-AFCD operator surpasses integral differential operators ones (see Fig. 5). Fig. 4Open in figure viewerPowerPoint Integral differential operators and proposed operator comparison (a) Original, (b) Laplace, (c) Sobel, (d) Proposed -AFCD result Fig. 5Open in figure viewerPowerPoint Another comparison of integral differential operators and the proposed operator (a) Original, (b) Laplace, (c) Sobel, (d) Proposed 2D-AFCD result 6.2.2 Comparative study with fractional differential operators Fractional differential operators preserve the complex texture details of an image. However, traditional methods ignore them in most cases. The proposed 2D-AFCD is compared to G–L [6] and R–L [5] operators (Fig. 6). Fig. 6Open in figure viewerPowerPoint (First column) original, (second column) G-L, (third column) R-L, (fourth column) proposed 2D-AFCD result (a) Fractional differential operator results on natural images, (b) Fractional differential operator results on medical images Fractional operators' results on natural images: As it can be observed in Fig. 6a the G–L operator results shown in the second column of this figure preserve well low-frequency features in different areas. Nevertheless, the loss of details is observed on the overall image as parrot's pupil, the little girl's facial features, her mouth and lips, and her eyes contours with details. The third column of this figure shows the R–L differential operator gradient results. We can also note that the complexity of local texture patterns was not taken into account effectively. Thus, leads to only exterior edges. Neither the parrot's beak nor its feathers and pupil were detected. The same remarks can be formulated for the girl's figure where the detection was badly done. In the second and third columns, the loss of information is obvious. Finally, the fourth column presents the proposed 2D-AFCD results. It has been observed that the detection is very rich. All the important details have been detected. Even more, the paint on the little girl's face, the texture of her ribbon, her nostrils, lips, eyelashes, and eyebrows were detected. Fractional operator results on medical images: To confirm our results, we tested our operator on medical images. As we know, medical images contain a lot of structures and information characterised by low contrast and resolutions. Fig. 6b presents the performance of the proposed 2D-AFCD operator and two other methods. In this part, magnetic resonance imaging (MRI) and CT-scan have been selected from other tested modalities supporting the previous tests. we observed that the 2D-AFCD outperforms the R–L and G–L operators on all sides. Eyes, brain, tumours, bones, and hand articulations have been well detected in contrast to the R–L and G–L operators, where a lot of information was lost. CPU process time comparison: In this part, the CPU processing time of the compared differentiators is analysed. It is summarised in Table 1. It can be seen that for an image of size , the proposed differentiator takes at least half the time (4.565 s) than the other two methods. At the same time, it is noted that by increasing image size the obtained results by the proposed method are more satisfactory and lower, compared to the results obtained from the other operators. This demonstrates and confirms the effectiveness of the proposed 2D-AFCD compared to the other fractional differentiator. Table 1. The CPU process time of the compared fractional differential operators Image test Image size, pixels G–L operator, s T–L operator, s 2D-AFCD, s Parrot 8.824 8.065 4.565 Head MRI 64.544 52.050 11.026 Hand Scan 98.366 85.036 14.044 Girl 129.071 121.386 15.066 6.3 Sensitivity and robustness against noise The performance of G–L, R–L, and the proposed 2D-AFCD operators against noise are presented in Fig. 7. The noise has been added increasingly with variances 0.1–0.3 with a 0.1 step (Fig. 7a). In the first column on Fig. 7, the noise was added with a variance equal to 0.1. We can observe G–L's results in the second row, R–L's in the third row, and the proposed 2D-AFCD operator in the fourth row. It can be observed that the proposed operator in the last row has well-done. All objects were detected in contrast to the R–L and G–L operators (second and third rows) where some edges are lost. In the second column (Fig. 7), where noise of variance equal to 0.2 was added to the figure, the proposed operator can detect all structures on the figure in contrast to the two other operators. The same observations can be done for the last test. We can conclude that the increase of noise intensity does not affect the 2D-AFCD's performance in terms of gradient detection and, therefore, in edge detection unlike traditional methods, it is insensitive to noise whatever its intensity. Fig. 7Open in figure viewerPowerPoint Performance of the proposed method against noise sensitivity (a) Gaussian noise with 0 mean (first column) variance 0.1, (second column) variance 0.2, (third column) variance 0.3. Edge detection results using, (b) G–L, (c) R–L, (d) 2D-AFCD method 6.4 Using the 2D-AFCD operator in speckle reduction Speckle noise has been defined as a pattern of a multiplicative form. It depends upon the tissue structures of the image and different imaging parameters. Speckle has a negative effect on medical ultrasound images. It has a tendency to reduce the contrast hence making image details obscure and blur, affecting in this way the human ability in identifying normal and abnormal tissues [45]. In this section, the proposed 2D-AFCD has been applied to both natural and real ultrasound images. Tested operators have been introduced into a de-noising algorithm. A comparative study has been carried. The considered classical significant operators are G–L [6] and Sobel [40]. 6.4.1 De-noising algorithm [45] In this section, the first-order gradient is replaced by the fractional-order gradient in the regularisation equation of the model in [41]. Thus, the fractional-order variational de-noising model is described as follows: details of the used algorithm are in [45]. Noise model of speckled images cannot be easily described. It is assumed as (36)where is the original image, is the showed image, and is a zero-mean Gaussian noise. This choice is done for its flexibility, it is restrictive than the usual radio frequency model. Let be the image domain in . In this work, the minimisation of a fractional-order weighted total variation function is proposed as (37)where is the area of the domain . The weighted function is defined, respectively, as follows: (38)The factor in (38) is introduced for restoring differentiability to the TV-factor. In particular, is chosen as follows: (39)where represents an optimal solution to our problem. Considering the noise model in (36), the fractional-order total variation function is minimised taking into account a multiplicative regularisation function in the following way: (40)with (41)Assuming that d represents the observed data and f an optimal solution to our problem, the cost functional to be minimised is (42) 6.4.2 Evaluation of the proposed method The performance of the proposed approach has been studied using different quality parameters and time computing. For a given X, representing the observed image, and its reconstructed image, experimentations, for edge preservation, speckle reduction and structural similarity (SSIM) and image quality parameters are defined as follows. Edge preservation experimentation: Pratt's figure of merit (FOM) is used to compare the edge preservation performance of the different filtering approaches. The FOM is given by [46] (43)where and are the number of detected and ideal edge pixels, is the Euclidean distance between the detected edge pixel and the nearest ideal edge pixel and is a constant typically set to 1/9. FOM ranges between 0 and 1, with unity for the ideal edge detection. Speckle reduction experimentation: The peak signal-to-noise ratio (PSNR) is used for each filtering operation. This parameter is able to measure the ability of filtering algorithm to reduce the speckle noise. It is defined by the following form: (44)where n is the number of bits used in representing an image pixel. For greyscale image, n is 8. The mean-squared error (MSE) of the reconstructed image is defined as (45)where is the original image, is the reconstructed image, and is the image size. SSIM experimentation: The mean SSIM index is a metric based on a similarity measure between two images. It is proposed to evaluate the de-noising filters [47]. The mean structural similarity (MSSIM) index is described as a function of three factors: the luminance, contrast, and SSIM. It is defined as (46)where X and are, respectively, the original and reconstructed images, and are the image contents at the local window and M is the number of windows. The MSSIM has values in , with unity representing structurally identical images. The S