Fractional-order TV-L 2 model for image denoising

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1 Cent. Eur. J. Phys. 11(10) DOI: /s Central European Journal of Physics Fractional-order TV-L 2 model for image denoising Research Article Dali Chen 1, Shenshen Sun 1, Congrong Zhang 1, YangQuan Chen 2, Dingyu Xue 1 1 Information Science and Engineering, Northeastern University, Wenhua Road 3-11, Heping Districe Shenyang, Liaoning, CHINA 2 MESA Lab, University of California, Merced, 5200 North Lake Road Merced, CA 95343, USA Received 31 January 2013; accepted 26 April 2013 Abstract: This paper proposes a new fractional order total variation (TV) denoising method, which provides a much more elegant and effective way of treating problems of the algorithm implementation, ill-posed inverse, regularization parameter selection and blocky effect. Two fractional order TV-L 2 models are constructed for image denoising. The majorization-minimization (MM) algorithm is used to decompose these two complex fractional TV optimization problems into a set of linear optimization problems which can be solved by the conjugate gradient algorithm. The final adaptive numerical procedure is given. Finally, we report experimental results which show that the proposed methodology avoids the blocky effect and achieves state-of-the-art performance. In addition, two medical image processing experiments are presented to demonstrate the validity of the proposed methodology. PACS (2008): y, Xx, Tt Keywords: image denoising fractional calculus total variation majorization-minimization algorithm Versita sp. z o.o. 1. Introduction Since the work of Rudin-Osher-Fatemi [1], the TV-L 2 model has been used in many applications such as image restoration [2], image decomposition [3], image decompression [4] and image segmentation [5]. The typical TV-L 2 model, for example ROF model, has been proved to be able to achieve a good trade-off between edge preservation and noise removal [1]. However, it tends to produce the so-called blocky (staircase) effects on the images because it favors a piecewise constant solution in bounded chendali@ise.neu.edu.cn variation (BV) space [6]. In order to deal with blocky effects, the modification of TV-L 2 model, which generalizes the differential order in regularization term, has aroused the more and more attentions of numerous scholars. There are two types of generalization of the differentiation in regularization term. The first one deals with higherorder differentiation. In 2000, You and Kaveh introduced a fourth-order partial differential equation (PDE)-based denoising model in which the regularized solution is obtained by solving the minimization of potential function of second-order derivative of the image [7]. Although this model reduces the blocky effects, it tends to cause the sign of uplifting effect and formation of artifacts around edges. For this problem, some improved fourth-order PDE models were proposed by replacing the Laplacian operator of 1414

2 Dali Chen, Shenshen Sun, Congrong Zhang, YangQuan Chen, Dingyu Xue diffusivity function by the different operators such as the gradient [8]. In this paper, our interest focus on the second generalization which deals with fractional order differentiation. Fractional calculus is a rapidly growing mathematical discipline, which provides an important tool for nonlocal field theories [9]. Recently, fractional calculus has been greatly studied in computer vision [10 14]. The main reason for this development is the expectation that the use of this theory will lead to a much more elegant and effective way of treating problems of blocky effect and detail information protection. Specially, the fractional order total variation (TV) models play an important role for image denoising, inpainting and motion estimation [15 19]. However, the algorithms to compute these TV-based models are developed on the continuous domain. Some of them use the methods based on duality and others solve the associated Euler-Lagrange equation, which is a non-linear partial differential equation (PDE). Note that these methods have to be discretized for the application on digital image, and the discretization method thus needs to be considered. The majorization-minimization (MM) algorithm is an iterative optimization method which is formulated in the discrete domain. The main idea of MM is to solve a sequence of simpler optimization problems instead of a complex one [20]. Recently, the majorization-minimization algorithm has been used to solve the TV deconvolution and denoising optimization problems [21, 22]. In our paper, a new image denoising method is proposed, in which the MM algorithm is used to solve the fractional order TV-L 2 (TV α -L 2 ) denoising model. The proposed denoising method is able to avoid the blocky effect and achieve state-of-the-art performance. This paper is organized as follows. Section 1 introduces prior work and our motivation, and some preliminary knowledges used in this paper are presented in Section 2. In Section 3, we firstly introduce two fractional order TV-L 2 denoising models, which are based on the different TV regularizers. Then, the majorizors of two fractional order TV regularizers are given in one uniform formula. Finally, the numerical procedure is shown for the readers. Experimental evaluation is presented in Section 4 and the paper is concluded in Section Preliminaries 2.1. Majorization-Minimization algorithm The Majorization-Minimization algorithm is an iterative optimization method which decomposes an arbitrarily complex optimization problem into a set of simpler ones. We consider the optimization problem described by arg min E(x). (1) x RN Instead of solving Eq. (1) directly, the Majorization- Minimization algorithm solves a sequence of simpler optimization problems x k+1 = arg min x R N H k(x), (2) where the functions H k (x), k = 0, 1, 2,... are a sequence of majorizors of the cost function E(x), which should satisfy the following two conditions: H k (x) E(x), H k (x k ) = E(x k ). In addition, the Majorization-Minimization algorithm has a useful property [22]. Consider E(x) = E 1 (x) + E 2 (x), and Q k (x) is the majorizor of E 1 (x). Then the majorizor of E(x) can be obtained by the function Q k (x) + E 2 (x). x 2.2. Matrix approximate method An efficient matrix approximate method for fractional order derivative was proposed by [23], which leads to significant simplification of the numerical solution. For easy reading, we reproduce the relevant results. Let us assume that the given signal f is sampled from its continuous version at an uniformly spaced grid size of h. Thereby, the discrete form f i = f(i h), for i = 0, 1,..., m. In image processing applications, the grid size h is one, so we set h = 1 for easy description. From Grünwald-Letnikov fractional derivative definition [24], the discrete formula of the fractional order derivative of the digital signal can be defined by the following formula D α x m k=0 w (α) k f i k, (3) ( ) where w (α) α k = ( 1) k represent the coefficients of the k polynomial (1 z) α. The coefficients can also be obtained recursively from w (α) 0 = 1, w (α) k = (1 α + 1 k ) w (α) k 1, k = 1, 2,.... (4) Using the matrix approximate method, we can rewrite Eq. (4) by the following form D α x f ψ f, (5) 1415

3 Fractional-order TV-L 2 model for image denoising where f = [f 0, f 1,..., f m ] T and ψ is a matrix defined by the following formula w (α) w (α) 1 w (α) ψ = (6).. w m (α) w (α) m 1... w (α) 0 When α = 1, the matrix ψ is a sparse banded matrix which consists of only two diagonals, the main diagonal and one lower diagonal. So, only two points are used to measure the gradient information. When α is not integer, the matrix ψ is a lower triangular matrix and all the points before k are used to calculus the fractional order derivative of the kth point. This long-term memory is an important characteristic of fractional differentiation and also is a prominent difference between fractional differentiation and integer order differentiation. Because of the long-term memory, fractional differentiation becames an important tool for modelling characteristic phenomena in various application. 3. Fractional-order TV-L 2 model 3.1. Model description Let f(x, y) = u(x, y) + v(x, y) denote the observed noisy image, where (x, y) T denotes the location with a rectangular image domain Ω R 2 and v is white Gaussian noise. TV-L 2 denoising model estimates the desired clean image u(x, y) by solving the following finite-dimensional optimization problem û = arg min u { E(u) = f u λtv(u)}, (7) where f u 2 2 is the data fidelity term, TV(u) is the regularization term, u ν is ν-norm of u, and λ is regularization parameter which controls the degree of smoothing. In this paper, we consider a fractional order TV p -L 2 (TV α p- L 2 )model, defined as û = arg min u { E(u) = f u λtvα p(u) }, (8) where p = 1 or 2. For better description, the discrete definition of the regularization term TV α p(u) is used, which is obtained by the following formula TV α p(u) = n Ω D α u(n) p, (9) where D α is the fractional-order derivative operator. When p = 1, D α u(n) 1 = D α hu(n) + D α v u(n) ; when p = 2, D α u(n) 2 = (D α hu(n)) 2 + (D α v u(n)) 2. D α h and D α v are linear operators corresponding to horizontal and vertical fractional order derivative Majorizor of TV α p(u) The key to apply Majorization-Minimization algorithm to the TV α p-l 2 denoising model is to find the majorizor of TV α p(u). Using the fact that t t2 2 t k + t k 2, t, t k R, (10) it follows that the function Q (k,1) (u) defined by Q (k,1) (u) = 1 2 TVα 1 (u k ) + 1 ( (Dα h u(n))2 2 D α n Ω h u k(n) + (Dα v u(n)) 2 Dv α u k (n) ) (11) satisfies Q (k,1) (u) TV α 1 (u) and Q (k,1) (u k ) = TV α 1 (u k ). Therefore, Q (k,1) (u) is the majorizor of TV α 1 (u). For easy description, we let D α h and D α v denote matrices such that D α hu and D α v u respectively yield the fractional order horizontal and vertical derivative. Based on this, we can rewrite Eq. (11) by the following compact formula Q (k,1) (u) = 1 2 TVα 1 (u k ) ut (D α ) T V 1 (k,1)d α u = C α 1 (u k ) ut (D α ) T V 1 (k,1)d α u, (12) where D α = [(D α h) T, (D α v ) T ] T, V (k,1) = diag(λ h k, Λv k ), Λh k = diag( D α hu k ) and Λ v k = diag( Dα v u k ). In the same way, we define the function ( Q (k,2) (u) =TV α 2 (u k ) + 1 (D α hu(n)) 2 (D α hu k (n)) 2 2 n Ω (D α hu k (n)) 2 + (D α v u k (n)) 2 ) + (Dα v u(n)) 2 (D α v u k (n)) 2. (13) (D α hu k (n)) 2 + (D α v u k (n)) 2 Since the inequality t t k + (t t k ) 2 t k for any t 0, the function Q (k,2) (u) satisfies Q (k,2) (u) TV α 2 (u) and Q (k,2) (u k ) = TV α 2 (u k ). Therefore, Q (k,2) (u) is the majorizor of TV α 2 (u) and has a compact form described by Q (k,2) (u) = TV α 2 (u k ) ut (D α ) T V 1 (k,2)d α u = C α 2 (u k ) ut (D α ) T V 1 (k,2)d α u, (14) 1416

4 Dali Chen, Shenshen Sun, Congrong Zhang, YangQuan Chen, Dingyu Xue where V (k,2) = diag(v k, v k ) and v k = diag( (D α hu k ) 2 + (D α v u k ) 2 ). Integrating Eq. (12) and Eq. (14) into one uniform formula, we can obtain the majorizor of TV α p(u) by the following function Q (k,p) (u) = C α p(u k ) ut (D α ) T V 1 (k,p)d α u. (15) 3.3. Numerical scheme This section gives details on the employed numerical procedure for the proposed TV α p(u)-l 2 denoising approach. Based on the property of Majorization-Minimization algorithm, the majorizor for the cost function E(u) is constructed by adding the data term f u 2 2 to Eq. (15), which is given by the function H (k,p) (u) = f u ut (D α ) T V 1 (k,p)d α u+c α p(u k ). (16) Since C α p(u k ) is a constant independent of u, it is irrelevant for the optimization problem. We thus estimate u by solving a sequence of optimization problems u k+1 = arg min u { f u λ 2 ut (D α ) T V 1 (k,p)d α u }. (17) Since Eq. (16) is a quadratic function, an explicit solution to Eq. (17) leads to a linear system ( I + λ(d α ) T V 1 (k,p)d α) u k+1 = f, (18) which can be solved by the conjugate gradient algorithm. However, some values of D α u p will go to zero as the iterations progress, and therefore the matrix V(k,p) 1 will go to infinity. This makes the solution of Eq. (18) more difficult. We tackle this difficulty by adopting the spatial smoothing strategy which has been largely used to overcome the ill-posed problem. It is common to smooth the image prior to differentiation, e.g. by convolving the updated image with the Gaussian kernel K σ (x, y) of standard deviation σ, u (x, y) = (K σ u)(x, y). The regularization parameter λ in Eq. (18) is related to the noise statistics and controls the degree of filtering of the obtained solution. When λ is too large, details and texture can be over smoothed; when λ is too small, it yields overly oscillatory estimates owing to either noise or discontinuities. Therefore, the selection of the regularization parameter is a critical issue to which much attention has been devoted. Many popular approaches have been proposed in the regularization and Bayesian frameworks. In this paper, the method proposed in [22] is used to adjust λ, λ p k = MNσ 2 /2TV α p(u k ). Here M and N are respectively the row and column of given image, and σ is the standard deviation of noise. To summarize, our entire noise removal algorithm in a form of a pseudo-code is done in following 1. Let the input image be f and set k = 0 and u k = f. Initialize K σ, σ, M, N and iteration number K iter. 2. Set u k = K σ u k and λ p k = MNσ 2 /2TV α p(u k ). 3. Compute A k = ( I + λ p k (Dα ) T V 1 (k,p) Dα) according to Eq. (12) to (15). 4. Compute u k+1 by using the conjugate gradient algorithm to solve the linear system A k u k+1 = f. 5. If k = K iter, stop; else, set k = k +1 and go to step Experiments and analysis 4.1. Restraint of block effect In this section, some experiments are given to assess the capability of reducing block effect of our proposed fractional order TV-L 2 model. Consider a one-dimension signal, y(t) = 4 sin(2πt) + 8 sin(3πt), which is contaminated by the additive white Gaussian noise (AWGN) with standard deviation (SD) of 0.5. The corrupted signal is shown in Fig. 1(a). We use the traditional TV-L 2 model, in which the differential order of regular term is one, to process the contaminated signal and the result at the 200th iteration is shown in Fig. 1(b). For comparison, the signal processed by the proposed fractional order TV-L 2 model with α = 2.8 is shown in Fig. 1(c). Note that the TV α 1 and TVα 2 have the same form when the given signal is one dimension signal. In order to ensure the fair comparison, the same iteration number K iter = 100 and regular parameter λ = 10 are used in this experiment. The difference between the traditional TV-L 2 model and proposed fractional order TV-L 2 model is obvious: while the traditional TV-L 2 model approximates the observed signal with a step signal, the proposed fractional order model with a piecewise planar signal which looks more natural and does not produce false edges. In conclusion, the proposed fractional order TV-L 2 model can reduce blocky effect effectively comparing with the traditional TV-L 2 model Analysis of denoising performance The aim of this section is to analyze the denoising performance of the proposed fractional order TV p -L 2 model. For 1417

5 Fractional-order TV-L 2 model for image denoising Figure 1. Noisy signal (a) with 0.5 SD Gaussian noise denoised with the traditional TV-L 2 model (b) and fractional order TV-L 2 model (c). Table 1. PSNR quantitative comparison among five denoising models. determine the quality of a processed image. calculated by the following formula It can be Image SD IP-M F-O-PDE IF-O-PDE ROF TV α 2 -L Barbara Lena Peppers this purpose, three famous test images, named Barbara, Lena and Peppers, are shown in Fig. 2, which have been corrupted by the additive white Gaussian noise (AWGN) with standard deviation (SD) of 10, 20 and 30 respectively. Five diffusion models are used to process the given noisy images, which are the improved Perona and Malik (IP-M) model [25], fourth order (F-O) PDE model [7], improved fourth order (IF-O) PDE model[8], ROF model [1] and proposed fractional order TV 2 -L 2 (TV α 2 - L 2 ) model. In this experiment, we set σ =5and t =0.25 for IP-M, F-O-PDE and IF-O-PDE models, and we set λ =0.07, τ =0.1 and K = 20 for ROF model. These parameters are able to ensure the best denoising performance of the corresponding denoising model. For TV α 2 -L2 model, we set α =1.8. The denoising results are shown in Table 1. In the table, the first column shows three given images and the second column shows their noisy images corrupted by the additive white Gaussian noise (AWGN) with standard deviation (SD) of 10, 20 and 30 respectively. In order to quantify the denoised image, we consider the peak-signal-to-noise ratio (PSNR), which has been largely used in literature and commonly applied to VL 2 PSNR = 10 log MN 10 N M, (19) i=1 j=1 (u(i, j) û(i, j))2 where u is the original image, û is the denoised image and V L is the maximal gray level of the image. The PSNRs of the denoised images processed by the different denoising models are listed under the corresponding denoising model. The bigger PSNR is, the better denoising performance is. For easy observation, we use bold font for the biggest PSNR values in the table. From the table, it is obvious that the PSNR values of our proposed model are bigger than that of the other four models, so we can conclude that our TV α 2 -L2 model outperforms the other models. In order to further verify the denoising performance of our proposed model, Fig. 3 shows the denoised images processed by four denoising models. Here, we use the different iteration number K iter in the different denoising model. For TV α 2 -L2 model, we set K iter = 10 because the method converges at 10 iterations. Similarly, we set K iter = 100 for the ROF model to ensure the converged result. In order to obtain the best result, we set K iter = 13 for the IP-M model and K iter = 10 for the IF-O PDE model. The first figure is the famous image "Goldhill" corrupted by the additive white Gaussian noise with standard deviation (SD) of 10. In order to show the clear comparison, the partial enlarged view of the noisy image is shown in the second figure. The denoised result of our proposed model is shown in the right-top figure. The second row shows the denoised images processed by IP-M model, IF-O PDE model and ROF model. In the left-bottom figure, although the detail information is preserved, there are a lot of noises unremoved. From the middle-bottom figure, it is obvious that the denoised image produces many false edges, and the result in the right-bottom figure looks blocky. Only the result of our model shown in the right-top figure looks 1418

6 Dali Chen, Shenshen Sun, Congrong Zhang, YangQuan Chen, Dingyu Xue Figure 2. Barbara (a), Lena (b) and Peppers (c). Figure 3. Goldhill with 10 SD Gaussian noise (a) and its partial view (b) processed by TVα2 -L2 model (c), IP-M model (d), IF-O PDE model (e) and ROF model (f). natural and does not produce false edges. So we conclude that our proposed fractional order model is able to achieve a good trade-off between edge preservation and noise removal. Table 2 shows the PSNR values of denoised images processed by the TVα1 -L2 and TVα2 -L2 model. The first row is the given image and the second row is the standard deviation (SD) of additive white Gaussian noise of noisy image. The PSNR values obtained by the TVα2 -L2 model are listed in the third row and the values obtained by the TVα1 -L2 model are listed in the last row. Here we set α = 1.6. It is obvious that the PSNR values of TVα2 -L2 model are bigger than that of TVα1 -L2 model. So we conclude that the denoising performance of TVα2 -L2 model is better. From the other α values, we can get the same conclusion. 1419

7 Fractional-order TV-L 2 model for image denoising Table 2. PSNR quantitative comparison between TV α 1 -L2 model and TV α 2 -L2 model Barbara Lena Peppers Goldhill Model TV L TV L Figure 4. Relation between the iteration number and PSNR value. In this experiment, the proposed TV α 2 -L2 model is used to process the image "Lena" corrupted by the additive white Gaussian noise with standard deviation (SD) of 10, and the PSNR value is recorded at each iteration. Fig. 4 shows the relation between the iteration number and PSNR value. The horizontal axis of this figure is iteration number and the vertical axis is PSNR value. The different colorful curve denotes the result obtained from the different α, for example, the red curve is the result obtained by the TV α 2 -L2 model with α =1.4. From the figure, we can see that the PSNR values are stable after 15 iterations, which indicates our proposed TV α 2 -L2 denoising algorithm is stable. In addition, it can be seen that the steady-state PSNR value of our proposed fractional order TV-L 2 model is bigger than that of the traditional TV-L 2 model, which shows that the fractional order TV-L 2 model has better denoising performance. In addition, in order to decide the value of the fractional order α, we study the relation between the PSNR and α. The Fig. 5 shows the relation between the PSNR and α on Lena and Peppers image corrupted by the additive white Gaussian noise with standard deviation (SD) of 10. The left image is the result of Lena image and the right one is the result of Peppers image. From the figure, we can obtain the following conclusions. Firstly, the PSNR reaches a maximum between α =1and α =2. Secondly, the PSNR at α =1is lower than PSNR at α>1, which is owing to the blocky effect. Finally, the PSNR decreases rapidly as α tends to zero and when α =0, Figure 5. Relation between the fractional order α and PSNR on Lena (left) and Peppers (right) image corrupted by the additive white Gaussian noise with standard deviation (SD) of 10. the PSNR reaches a minimum. According to these facts, we can select the fractional order α between α =1.4 and α = Lung nodule segmentation experiment Image denoising technology not only can improve the observability of the image, but also can increase the accuracy of some postprocessing technologies. In this experiment, the lung nodule CT image is used as the test image, which is downloaded from the LIDC medical imaging database [26]. The given CT image is obtained from Siemens low-dose spiral CT, whose peak voltage ranges from 120 kvp to 140 kvp and tube current ranges from 1420

8 Dali Chen, Shenshen Sun, Congrong Zhang, YangQuan Chen, Dingyu Xue Figure 6. Lung nodule CT image with Gaussian noise and the partial view of the segmented results of the undenoised image and the images denoised by the TV 1 2 -L2 and TV L2 denoising methods. Figure 7. PET image with Gaussian noise and the partial view of the segmented results of the undenoised image and the images denoised by the TV 1 2 -L2 and TV L2 denoising methods. 40 mas to 388 mas. In order to assess our denoising method, the lung nodule CT image is contaminated by the additive white Gaussian noise and the typical regiongrowing segmentation method [27] is used to segment the lung nodule from the noisy image and denoised image. The results are shown in Fig. 6. The first figure shows the given noisy lung nodule CT image. For easy observation, the partial enlarged view of the segmented result of the noisy image is shown in the second figure, from which we can see that the lung nodule cannot be segmented. The third and forth figures show the segmented results of the images denoised by the TV 1 2 -L2 and TV L2 denoising methods, respectively. Although these two segmented results both show the right location of lung nodule, the detected edge of TV L2 method is more accurate than that of TV 1 2 -L2 method. This demonstrates that the proposed fractional order TV-L 2 denoising method is helpful for improving the accuracy of the postprocessing technologies in the lung nodule CT image segmentation Cardiac muscular segmentation experiment In the medical image processing, the segment of the cardiac muscular tissue is an important step. The purpose of this experiment is to assess the effect of our method on the the cardiac muscular segmentation. In this experiment, the test image is the cardiac muscular PET image downloaded from the DICOM7 medical image database ( The additive white Gaussian noise is added in the test image. We use the typical iteration threshold segmentation algorithm [27] to segment the cardiac muscular tissue from the noisy image and the denoised images respectively. For easy observation, the pseudo-color technology is used on the segmented images, and the results are shown in Fig. 7. The first figure shows the noisy image and the red part of which is the cardiac muscular tissue. The partial enlarged view of the segmented result of the noisy image is shown in the second figure, in which some noises remain. The third and forth figures show the segmented results of the images denoised by the TV 1 2 -L2 and TV L2 denoising methods, respectively. Although the segmented results both are satisfactory, the result of TV L2 denoising method looks much more smooth than that of TV 1 2 -L2 method. Therefore, we can conclude that the proposed fractional order TV-L 2 denoising method is helpful for improving the accuracy of the postprocessing technologies in the cardiac muscular PET image segmentation. 1421

9 Fractional-order TV-L 2 model for image denoising 5. Conclusion A new adaptive fractional order TV-based denoising method was proposed in this paper. The main contributions are as following: (1) two fractional order TV-L 2 (TV α p-l 2 ) models are constructed for image denoising; (2) the majorization-minimization algorithm was used to solve the TV α p-l 2 model, which provides an effective tool to solve fractional TV optimization problem; (3) the majorizors of two fractional order TV regularizers are obtained in one uniform formula; (4) the experiments demonstrate that the proposed methodology is able to avoid the blocky effect and achieve state-of-the-art performance. Future works involve extending the proposed methodology to the other total variational models such as optical flow model. Acknowledgments This work was supported by National Natural Science Foundation of China (No ), Scientific Research Fund of Liaoning Provincial Education Department (L ) and Fundamental Research Funds for the Central Universities (N ). The authors would like to thank Mário A. T. Figueiredo, Ivan Selesnick and Igor Podlubny for their outstanding prior work. References [1] L. Ruding, S. Osher, E. Fatemi, Physica D 60, 259 (1992) [2] J. F. Aujol, J. Math. Imaging Vis. 34, 307 (2009) [3] C. Vogel, M. Oman, IEEE T. Image Process. 7, 813 (1998) [4] F. Alter, S. Durand, J. Froment, J. Math. Imaging Vis. 23, 199 (2005) [5] F. Li, C. Shen, C. Li, J. Math. Imaging Vis. 37, 98 (2010) [6] J. Zhang, Z. Wei, L. Xiao, J. Math. Imaging Vis. 43, 39 (2012) [7] Y. L. You, M. Kaveh, IEEE T. Image Process. 9, 1723 (2000) [8] M. Hajiaboli, IPSJ Transactions on Computer Vision and Application 2, 94 (2010) [9] R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, New Jersey, 2011) [10] S. C. Liu, S. Chang, IEEE T. Image Process. 6, 1176 (1997) [11] S. Didas, B. Burgeth, A. Imiya, J. Weickert, Scale Space and PDE Methods in Computer Vision 3459, 13 (2005) [12] B. Ninness, IEEE T. Inf. Theory 44, 32 (1998) [13] I. Petras, D. Sierociuk, I. Podlubny, IEEE T. Signal Proces. 60, 5561 (2012) [14] Y. F. Pu, J. L. Zhou, X. Yuan, IEEE T. Image Process. 19, 491 (2010) [15] B. Jian, X. C. Feng, IEEE T. Image Process. 16, 2492 (2007) [16] P. Guidotti, J. V. Lambers, J. Math. Imaging Vis. 33, 25 (2009) [17] E. Cuesta, M. Kirane, S. A. Malik, Signal Process. 92, 553 (2012) [18] M. Janev, S. Pilipović, T. Atanacković, R. Obradović, N. Ralević, Math. Comput. Model. 54, 729 (2011) [19] D. Chen, H. Sheng, Y. Q. Chen, D. Y. Xue, Phil. Trans. R. Soc. A., DOI: /rsta [20] D. Hunter, K. Lange, The American Statistician 58, 30 (2004) [21] M. A. T. Figueiredo, J. M. Bioucas Dias, R. D. Nowak, IEEE T. Image Process. 16, 2980 (2007) [22] J. P. Oliveira, J. M. Bioucas Dias, M. A. T. Figueiredo, Signal Process. 89, 1683 (2009) [23] I. Podlubny, Fractional Calculus and Applied Analysis 3, 359 (2000) [24] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999) [25] P. Perona, J. Malik, IEEE T. Pattern Anal. 12, 629 (1990) [26] S. G. Armato, et al., Med. Phys. 38, 915 (2011) [27] R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd edition (Addison-Wesley, Massachusetts, 1992) 1422