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1. Received October 17, 2016, accepted November 10, 2016, date of publication December 1, 2016, date of current version January 4, 2017. Digital Object Identifier 10.1109/ACCESS.2016.2633272 A Novel Fractional-Order Differentiation Model for Low-Dose CT Image Processing YANLING WANG1,2, YANLING SHAO3, ZHIGUO GUI1,2, QUAN ZHANG1,2, LINHONG YAO3, AND YI LIU1,2 1National Key Laboratory for Electronic Measurement Technology, School of Information and Communication Engineering, North University of China, Taiyuan 030051, China 2Key Laboratory of Instrumentation Science and Dynamic Measurement, School of Information and Communication Engineering, North University of China, Taiyuan 030051, China 3School of Science, North University of China, Taiyuan 030051, China Corresponding author: Y. Liu (liuyi1987827@gmail.com) This work was supported in part by the National Nature Science Foundation of China under Grant 61271357, in part by the National Key Scientific Instrument and Equipment Development Project under Grant 2014YQ24044508, in part by the Opening Project of the State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, under Grant KFJJ13-11M, in part by the Natural Science Foundation of Shanxi Province under Grant 2015011046, in part by the Shanxi Province Science Foundation for Youths under Grant 201601D021080, and in part by the Research Project supported by the Shanxi Scholarship Council of China under Grant 2016-085. ABSTRACT Low-dose CT (LDCT) images tend to be degraded by excessive mottle noise and steak artifacts. In this paper, we proposed a novel fractional-order differentiation model that can be applied to LDCT image processing as a post-processing technique. The anisotropic diffusion model (proposed by Perona and Malik, i.e., PM model) has good performance in flat regions, total variation (TV) model works better in edge preservation, and fractional-order differentiation models can mitigate block effect while preserving fine details and more structure. The proposed model is based on the weighted combinations of the fractional- order PM model and the fractional-order TV model, which maintains the advantages of PM model, TV model, and fractional-order differentiation models. Moreover, the local intensity variance was added to both weighted coefficient and diffusion coefficient of the proposed model to properly preserve edges and details. A variety of simulated phantom data, including the Shepp–Logan head phantom, the pelvis phantom, and the actual thoracic phantom, were used for experimental validation. The results of numerical simulation and clinical data experiments demonstrate that the proposed approach has a better performance in both noise suppression and detail preservation, when compared with several other existing methods. INDEX TERMS Low-dose CT, image processing, fractional-order differentiation model, edge and detail preservation. I. INTRODUCTION Although X-ray computed tomography (CT) have gained widely applications in the medical field, the concern on X-ray dose is emerging as high-dose radiation may increase stochastic risks during radiological procedure. The radiation dose delivered to patients during CT examinations therefore needs to be reduced [1]. Among all the methods for reducing radiation dose (such as reducing tube current, tube voltage, and scanning time. etc.), the most simple way is to lower mA (milliampere)/mAs (milliampere second). This way, however, often leads to degraded reconstruction images with increased mottle noise, non-stationary streak artifacts, and decreased CNR (contrast-to-noise ratio) [2]. Streak artifacts occur most frequently in the bony structures at the base of the skull and petrous bone regions, because the very dense structures are only partially included in the slice, resulting in high contrast errors [3]. Many techniques have been proposed to remove noise and artifacts in LDCT. They are generally divided into three categories: projection processing methods, iterative reconstruction algorithms, and post-processing methods. The first category treats the projection data as an image (called sinogram) and reconstructed images can be obtained from the processed projections [4], [5]. Noise reduction in sinogram space before filtered back projection (FBP) is an effective way to obtain high quality reconstructed LDCT images. Nonlinear filtering [6], penalized-likelihood filtering [7], and fuzzy filtering [3] were respectively proposed to VOLUME 4, 2016 2169-3536 2016 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 8487

2. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing suppress excessive quantum noise and keep edges in the sinogram. The second one, i.e., iterative reconstruction algorithm, achieves noise suppression in the procedure of reconstruction. Specifically speaking, it looks for an optimal solution by maximizing or minimizing a prior-regularized cost function that is constructed according to the noise properties of the projections [8]–[18]. In the past decade, studies in this area mainly focus on the design of priors. Many valuable priors have been proposed, such as the anisotropic prior [8], the TV based priors [9]–[11], and the nonlocal priors [12], [13]. Although yielding excellent results through incorporating image prior information into optimization, these algorithms can’t be broadly used due to the requirement of more detailed information for reconstruction, such as scanning geometry, correction physics, and photon statistics [14]. The third one, i.e., post-processing method, by contrast, is more repro- ducible and can be performed on different scanning sys- tems. Since noise and streak artifacts seriously damage the structures, the key issue of these methods is to keep the structures well when reducing noise and artifacts, and in the meantime no new artifacts and blurred details are introduced. A few of outstanding filters considering both noise suppression and edge preservation have been proposed to improve the quality of LDCT images, for instance, nonlocal means filtering [14], [19], dictionary learning based filtering [20], [21], and partial differential equations (PDEs) based filtering [22], [23]. Over the past two decades, extensive research has been conducted on PDEs for image denoising. Two well-known second-order PDEs are the PM model and the TV model. The PM model proposed by Perona and Malik in 1990 [24] is based on the following partial differential equation ∂u ∂t = div[c(|∇u|) · ∇u], (1) where div is the divergence operator, |∇u| is the absolute value of the gradient of image u, and the diffusion coefficient c(·) is a monotone decreasing smooth function of the magnitude of local image gradient ∇u. A possible diffusion coefficient function is given by c(|∇u|) = 1/[1 + |∇u|2 /k2 ], (2) where k is the gradient threshold. The TV model proposed by Rudin, Osher, and Fatemi in 1992 [25] has the following energy function E(u) = Z (|∇u| + λ 2 |u − u0|2 )dxdy, (3) where the first term is a regularization term that denotes the total variation of denoised image u, the second term is a fidelity term, u0 is the noisy image, and λ is the regularization parameter. By using the gradient descent method, the TV denoising model was obtained as follows: ∂u ∂t = ∇ · ( ∇u |∇u| ) − λ(u − u0). (4) The PM model has good performance in flat regions with uniform intensity distribution, and the TV model works better in preserving edges. Zhang et al. [26] proposed a novel model (i.e., PMTV model) by weighted combinations of PM model and TV model. Yahya et al. [27] proposed a new denoising technique by blending isotropic diffusion, PM model, and TV model. Although the above second-order PDEs can reduce noise level while preserving the image features, they tend to make the processed image look ‘‘blocky’’, because the images used by second-order PDEs to approximate an observed image are often piecewise constant. In order to reduce blocky effect, a class of four-order PDEs were introduced by You and Kaveh in 2000 [28], but these methods often lead to speckle effect. To overcome those aforementioned limitations, fractional- order PDEs have recently been researched and applied to the field of image processing and computer vision. For example, Bai and Feng [29] proposed a class of FPM models for image denoising, in which the energy function is defined as E(u) = Z f (|Dα u|)d, (5) where is the image support region, Dαu = (Dα x u, Dα y u), |Dαu| = q (Dα x u)2 + (Dα y u)2, and f (|Dαu|) ≥ 0 is an increasing function associated with the diffusion coefficient shown as c(t) = f 0( √ t) √ t . (6) When α = 1, equation (5) is precisely PM model in [24]; when α = 2, equation (5) is precisely the fourth-order PDE in [28]; when 1 ≤ α ≤ 2, the FPM model (5) can be considered as an interpolation between the second- order and the fourth-order anisotropic diffusion equations. Zhang and Wei [30] developed a FTV model as follows: E(u) = Z (|Dα u| + λ 2 |u − u0|2 )dxdy. (7) The FTV model can be seen as the generalization of TV model, and the variation order is changed from integer (one) in TV model to fraction (α) in FTV model. Zhang et al. [31] first applied fractional calculus to medical image processing. They proposed two fractional-order equations for CT metal artifacts reduction [31], [32], and gave two novel fractional- order models for CT image reconstruction [33], [34]. Hu [35] proposed a fractional-order diffusion scheme for sinogram restoration of LDCT. Although the above fractional-order differentiation methods can, to a certain extent, reach a good trade-off between noise removal and edge preservation, some noise and steak artifacts still exist. According to what is mentioned above, PM model has good performance in flat regions, TV model works better in edge preservation, and fractional-order differentiation models can mitigate block effect while preserving fine details and more structure. In order to keep the advantages of these models, we integrated FPM model and 8488 VOLUME 4, 2016

3. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing FTV model to obtain our new model (namely, FPMTV model). Additional, the local intensity variance usually can be used to distinguish desirable image features from artifacts. In our study, the local intensity variance was added to weighted coefficient of the proposed model to properly adjust the weights of FPM model and FTV model. On the other hand, both intensity variance and gradient were treated as two local pixel characteristics in the diffusion coefficient of the proposed model to further preserve edges and details. The proposed model is appropriate for applications in LDCT image processing because it can effectively preserve edges and fine details while removing mottle noise and steak artifacts. The remainder of this paper is organized as follows. Section 2 first briefly overviews the definitions of fractional-order calculus. The proposed fractional-order differentiation model and its numerical computation are then discussed. Section 3 presents experimental results from multiple samples including the Shepp-Logan head phantom, the pelvis phantom, and the actual thoracic phantom. In the end, Section 4 gives brief conclusion for this study. II. MATERIALS AND METHODS A. FRACTIONAL-ORDER CACULUS The fractional-order calculus has been researched for cen- turies. Though defined in a number of ways, most of the definitions are classified into two categories, the time domain ones and the frequency ones. A discrete Fourier transform was used to calculate the fractional differential values in [29], but the frequency definition often causes high computational cost. The Grümwald-Letniklv (G-L) and the Riemann- Liou- ville (R-L) definitions are the most famous and universal time domain definitions. The G-L definition expresses a function by weighted sum around the function, which makes it suitable for applications in image processing. According to [36], α-order differential of signal f (x) was defined by the G-L as: aG Dα t f (x) , lim h→0 1 hα t−a h X m=0 (−1)m α m f (x − mh), (8) where α ∈ R, the duration of f (x) is [a, t] (a t, a ∈ R, t ∈ R), G denotes G-L definition, G a Dα t denotes G-L-based fractional-order differential operator, h = (t − a) /n is the step size, and the formula α m is the binomial coefficient defined as α m = 0(α + 1) 0(m + 1)0(α − m + 1) , (9) where 0(m) = (m−1)! is the gamma function of independent variable m. B. THE PROPOSED FRACTIONAL-ORDER DIFFERENTIATION MODEL In this section, we present a modified fractional-order differentiation model based on the weighted combinations of FPM model and FTV model. Moreover, the local intensity variance in [37] is introduced in both weighted coefficient and diffusion coefficient of the proposed model to properly preserve edges and details. The local intensity variance can distinguish desirable image features from artifacts because desirable image features in the neighborhood of the image usually have larger intensity variance than the artifacts. The energy function of the proposed model is given by (10): E(u) = Z [ϕ|Dα u| + γ f (|Dα u|) + λ 2 |u − u0|2 ]dxdy (10) where ϕ = η(2 − η), γ = (η − 1)2, (11) and η = e −1 σ2 t,N /L , (12) here L is a positive constant. σt,N keeps changing as the increase of iteration times, and σt,N in a fine detail area or edge is usually larger than that in the noisy background or flat region. η can be used to determine whether the region is the flat area or the edge because the larger σt,N , the larger η. The local intensity variance σt,N in each region of the image is computed by using (13) and (14) as a measure of the number of image details. σ2 t (x, y) = 1 9 1 X i=−1 1 X j=−1 [ut(x + i, y + j) − ūt(x, y)]2 , (13) σ2 t,N (x, y) = 1 + σ2 t (x, y) − Minσ2 t Maxσ2 t − Minσ2 t · 254, (14) where Maxσ2 t and Minσ2 t denote the maximal and minimal intensity variances of the image at iteration t. ūt(x, y) is the mean of gray levels in a 3 × 3 neighborhood window. The local intensity variance was added to weighted coefficient of the proposed model to properly adjust the weights of FPM model and FTV model. In the flat region where σt,N is small, η is also small according to (12), and when η is close to zero, the proposed model will highlight the importance of FPM model. In the fine detail area or edge where σt,N is large, η is also large according to (12), and when η is close to one, the proposed model will emphasize the role of FTV model. It is very difficult to calculate fractional-order differential equations directly by the G-L definition. In this paper, we establish a fractional order differential model by combin- ing the G-L definition with convolution integral. According to [38] and (8), we can obtain the discrete fractional-order differential equation Dα f = X k≥0 (−1)k α m f (x − k). (15) Let convolution kernel function vα (z) =    (−1)z α z , z ≥ 0, 0, z 0. (16) VOLUME 4, 2016 8489

4. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing We have vα ∗ f = Z +∞ −∞ vα (z)f (x − z)dz. (17) Equation (15) can be viewed as an approximate discretization of convolution integral (17), namely, equation (17) may be treated as an approximate representation of the G-L definition (8). Equation (17) is easier to calculate and analyze than (8). Replacing the fractional gradient Dαu of (10) with convolution integral vα ∗ u, we obtain the fractional-order differential model based on convolution integral as follows E(u) = Z [ϕ|vα ∗ u| + γ f (|vα ∗ u|) + λ 2 |u − u0|2 ]dxdy, (18) where

6. vα ∗ u

8. = q (vα ∗ u)2 x + (vα ∗ u)2 y, (vα ∗ u)x = Z +∞ −∞ vα (z)u(x − z, y)dz, (vα ∗ u)y = Z +∞ −∞ vα (z)u(x, y − z)dz. (19) To solve the problem (18), we take any test function φ ∈ C∞() and define g (ε) := E(u + εφ) = Z [ϕ

10. vα ∗ (u + εφ)

12. + γ f (|vα ∗ (u + εφ) |) + λ 2 |u + εφ − u0|2 ]dxdy. (20) When g0 ε(0) = 0, we have ϕ Z (vα ∗ u)x(vα ∗ φ)x + (vα ∗ u)y(vα ∗ φ)y |vα ∗ u| dxdy + γ Z f 0 (|vα ∗ u|) · [(vα ∗ u)x(vα ∗ φ)x + (vα ∗ u)y(vα ∗ φ)y] |vα ∗ u| dxdy + λ Z (u − u0) · φdxdy = 0. (21) According to (6), we get ϕ Z (vα ∗ u)x(vα ∗ φ)x + (vα ∗ u)y(vα ∗ φ)y |vα ∗ u| dxdy + γ Z c(|vα ∗ u|2 ) · [(vα ∗ u)x(vα ∗ φ)x + (vα ∗ u)y(vα ∗ φ)y]dxdy + λ Z (u − u0) · φdxdy = 0. (22) We can enlarge the support region of the image u0(x, y) from to R2 as follows: u0(x, y) = u0(x, y), (x, y) ∈ , 0, (x, y) / ∈ . (23) Using the Parseval equation Z R2 f · gdxdy = Z R2 ˆ f · ¯ ĝdω1dω2, we can obtain ϕ Z R2 ( (vα ∗ u)x |vα ∗ u| · (vα ∗ φ)x + (vα ∗ u)y |vα ∗ u| · (vα ∗ φ)y)dω1dω2 + γ Z R2 c(|vα ∗ u|2) · [ (vα ∗ u)x · (vα ∗ φ)x + (vα ∗ u)y · (vα ∗ φ)y]dω1dω2 + λ Z R2 (¯ û − ¯ û0) · φ̂dω1dω2 = 0. (24) In the frequency domain, the convolution integral (17) is satisfied with vα ∗ u0 = b vα · b u0, where b u0 and b vα denote the Fourier transform of u0 and vα respectively. By using the frequency domain properties of convolution integral, we get ϕ Z R2 [ (vα ∗ u)x |vα ∗ u| · b vα(ω1) · φ̂ + (vα ∗ u)y |vα ∗ u| · b vα(ω2) · φ̂]dω1dω2 + γ Z R2 c(|vα ∗ u|2)·[ (vα ∗ u)x · b vα(ω1) · φ̂ + (vα ∗ u)y · b vα(ω2) · φ̂]dω1dω2 + λ Z R2 (¯ û − ¯ û0) · φ̂dω1dω2 = 0. (25) Taking the conjugate of both sides of (25), we can obtain the Euler-Lagrange equation of (10) as follows: ϕ[ (vα ∗ u)x |vα ∗ u| · b vα(ω1) + (vα ∗ u)y |vα ∗ u| · b vα(ω2)] + γ [ c(|vα ∗ u|2)·[ (vα ∗ u)x · b vα(ω1) + (vα ∗ u)y · b vα(ω2)] + λ(û − b u0) = 0. (26) Since b vα(ω) = b vα(−ω), we have ϕ[ (vα ∗ u)x |vα ∗ u| · b vα(−ω1) + (vα ∗ u)y |vα ∗ u| · b vα(−ω2)] + γ [ c(|vα ∗ u|2)·[ (vα ∗ u)x · b vα(−ω1) + (vα ∗ u)y · b vα(−ω2)] + λ(û − b u0) = 0. (27) Taking the inverse Fourier transform of (27) by using the frequency domain properties of convolution integral, we get ϕ[(vα (−z) ∗ (vα ∗ u)x |vα ∗ u| )x + (vα (−z) ∗ (vα ∗ u)y |vα ∗ u| )y] + γ · c(|vα ∗ u|2 ) · [(vα (−z) ∗ (vα ∗ u)x)x + (vα (−z) ∗ (vα ∗ u)y)y] + λ(u − u0) = 0. (28) 8490 VOLUME 4, 2016

13. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing According to (17), we obtain ϕ · [ Z +∞ −∞ vα (z) · (vα ∗ u)x |vα ∗ u| (x + z, y)dz + Z +∞ −∞ vα (z) · (vα ∗ u)y |vα ∗ u| (x, y + z)dz] +γ · c(|vα ∗ u|2 ) · [ Z +∞ −∞ vα (z) · (vα ∗ u)x(x + z, y)dz + Z +∞ −∞ vα (z) · (vα ∗ u)y(x, y + z)dz] + λ(u − u0) = 0. (29) Since vα(z) = 0 (z 0), we have ϕ · [ Z +∞ 0 vα (z) · (vα ∗ u)x |vα ∗ u| (x + z, y)dz + Z +∞ 0 vα (z) · (vα ∗ u)y |vα ∗ u| (x, y + z)dz] + γ c(|vα ∗ u|2 ) · [ Z +∞ 0 vα (z) · (vα ∗ u)x(x + z, y)dz + Z +∞ 0 vα (z) · (vα ∗ u)y(x, y + z)dz] + λ(u − u0) = 0, (30) where c(|vα ∗ u|2 ) = 1/[1 + |vα ∗ u|2 κ2 ], (31) here κ is a constant and acts as an edge strength threshold in the diffusion coefficient function. In order to further preserve edges and details, both intensity variance and gradient are treated as two local pixel characteristics in the diffusion coefficient c(·) of FPMTV model. According to [37], the diffusion coefficient c(·) is revised as c(|vα ∗ u|2 , σ2 t,N ) = 1/[1 + |vα ∗ u|2 · σ4 t,N k2 1 ], (32) where k1 = k0e−1t(n−1). Here k0 is a positive constant, 1t is the step size, n is the number of iterations. And k1/σ2 t,N can be seen as an adaptive version of κ in (31). Let Uα (x, y) := ϕ · [ Z +∞ 0 vα (z) · (vα ∗ u)x |vα ∗ u| (x + z, y)dz + Z +∞ 0 vα (z) · (vα ∗ u)y |vα ∗ u| (x, y + z)dz] + γ c(|vα ∗ u|2 , σ2 t,N ) · [ Z +∞ 0 vα (z) · (vα ∗ u)x(x + z, y)dz + Z +∞ 0 vα (z) · (vα ∗ u)y(x, y + z)dz]. (33) The Euler-Lagrange equation (27) is written as Uα (x, y) + λ(u − u0) = 0. (34) The Euler-Lagrange equation (34) can be solved through the following gradient descent procedure ∂u ∂t = −Uα (x, y) − λ(u − u0). (35) C. NUMERICAL COMPUTATION To proceed with the numerical computation in solving (35), we assume that both the discrete noisy image u0(i, j) and denoised image u(i, j) are M × N pixels, where i = 0, 1, . . . , M − 1, j = 0, 1, . . . , N − 1. Let N1 = min{M, N}, the numerical computation of the proposed FPMTV algorithm is implemented in (36)-(39). According to (15), we can discretize (vα ∗u)x and (vα ∗u)y as follows:          (vα ∗ u)x(i, j) = N1−1 P k=0 (−1)k α k u(i − k, j), (vα ∗ u)y(i, j) = N1−1 P k=0 (−1)k α k u(i, j − k), (36) where u(i, j) = 0 for iM − 1, jN − 1 or i, j 0. Let        TVDx(i, j) = (vα ∗ u)x(i, j) √ (i, j) + ε , TVDy(i, j) = (vα ∗ u)y(i, j) √ (i, j) + ε , (37)                    PMDx(i, j) = c(|vα ∗ u|2, σ2 t,N ) · (vα ∗ u)x(i, j) = k2 1 · (vα ∗ u)x(i, j) k2 1 + (i, j) · σ4 t,N + ε , PMDy(i, j) = c(|vα ∗ u|2, σ2 t,N ) · (vα ∗ u)y(i, j) = k2 1 · (vα ∗ u)y(i, j) k2 1 + (i, j) · σ4 t,N + ε , (38) where (i, j) = [(vα ∗ u)x(i, j)]2 + [(vα ∗ u)y(i, j)]2 , and ε is a very small positive number. Therefore, the discretization for Uα(x, y) is Uα i,j(u) = N1−1 X k=0 (−1)k α k · {ϕ · [TVDx(i + k, j) + TVDy(i, j + k)] + γ · [PMDx(i + k, j) + PMDy(i, j + k)]}. (39) To summarize, the FPMTV algorithm consists of the following steps. (1) Initialization: u(0) = u0, determine the values of parameter α, λ, L, k0 and the iteration step length 1t. (2) Iteration: For n=1, 2, 3..., compute u(n+1) according to the following steps: Step1: Compute Uα n = ((Uα i,j(u(n)))M−1 i=0 )N−1 j=0 by using (36)-(39); Step2: Compute u(n+1) = u(n) − 1t · [Uα n + λ(u(n) − u0)]. VOLUME 4, 2016 8491

14. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing FIGURE 1. Visualization of the phantoms and images used in our study. (a) original Shepp-Logan head phantom; (b) original pelvis phantom; (c) processed HDCT image by the AS-LNLM method; (d) and (e) are the reconstructed LDCT images from simulated noisy sonograms by FBP (Hanning filter with cutoff at 80% Nyquist frequency); and (f) original LDCT image (30 mAs). If u(n+1) satisfies a given condition, we terminate the iteration and output u(n+1), otherwise, let n := n + 1 and return to Step1. III. EXPERIMENTS AND ANALYSIS In this section, experiments based on both digital phantom simulations and clinical datasets were preformed to validate the proposed FPMTV method. Fig. 1(a) shows the Shepp- Logan phantom that is composed of 256 pixels×256 pixels. Fig. 1(b) shows the pelvis phantom with 256 pixels×252 pixels. Fig. 1(d) and Fig. 1(e) show the LDCT images that are reconstructed from simulated noisy sonograms by using the FBP with Hanning filter (cutoff frequency equal to 80% Nyquist frequency). An anatomical model of a human chest torso was used in our experiments as the thoracic phantom, and CT images were obtained from a multi-detector row Siemens Somatom Sensation 16 CT scanner with a tube voltage of 120 kVp. The original high-dose CT (HDCT) image was collected with a higher tube current of 240 mAs. Fig. 1(c) shows the processed HDCT image by the artifact suppressed large-scale nonlocal means (AS-LNLM) method. The processed HDCT image has a better performance in noise and artifacts suppression than the original one, therefore, the processed HDCT image can be taken as the reference image. Fig. 1(f) illustrates the LDCT image which was obtained with a reduced tube current of 30 mAs. Both Fig. 1(c) and (f) are composed of 512 pixels×512 pixels. All experiments were implemented in MATLAB 2012b on a PC with Intel(R) Pentium(R) CPU 2.60 GHz and 4Gb RAM. A. RELATED PARAMETERS AND ASSESSMENT CRITERIA For quantitative analyses, the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) [39], which have been typically used in CT reconstructed image quality evaluation, were utilized in this paper. The SSIM index was used to measure the structure similarity between two images and can be calculated by SSIM = 2ūoriginalū(σuoriginalu + c2) (ū2 original + ū2 + c1)(σ2 + σ2 original + c2) , (40) where σuoriginalu = Cov uoriginal, u = 1 N − 1 (un − ū) uoriginaln − ūoriginal , ū = 1 N N X n=1 un, σ2 = 1 N − 1 N X n=1 (un − ū)2 , ūoriginal = 1 N N X n=1 uoriginaln , σ2 original = 1 N N X n=1 uoriginaln − ūoriginal 2 , here c1 and c2 are constants set according to [39]. un and uoriginal are the pixel values of the denoised image and the 8492 VOLUME 4, 2016

15. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing FIGURE 2. The SSIM curves of the FPMTV iterative algorithm with different fractional orders α. original image respectively, and N is the total number of pixels in the reconstructed image. The PSNR can be calculated via the mean squared error (MSE) which is computed according to the following formula MSE = 1 N N X n=1 un − uoriginaln 2 . (41) Then PSNR = 10log10 max(un, uoriginaln )2 MSE . (42) The selection of fractional order α is very important for high quality denoised image. In this paper, we deter- mined the value of parameter α by both SSIM curves and the visual effect of denoised image. Take the Shepp-Logan head phantom for example, Fig. 2 shows SSIM curves of the FPMTV iterative algorithm with different fractional orders α that changes from 0.85 to 1.85 with an interval of 0.2. As can be seen from Fig. 2, with α increasing from 1.05 to 1.85, the SSIM value becomes smaller and smaller, and the SSIM value at α =0.85 is smaller than that at α = 1.05. Obviously, the optimal SSIM value is obtained at α = 1.05. On the other hand, the LDCT image (Fig. 1(d)) was processed by FPMTV algorithm with different fractional orders α, and the processed images are shown in Fig. 3. We can see that Fig. 3(a) (α = 0.85) suffers from very obvious blocky effect. From Fig. 3(b) (α = 1.05), we can observe that block effect and streak artifacts are eliminated, and that edges and details are preserved. Fig. 3(c) (α = 1.25) appears only a few streak artifacts. However, we can easily see from Fig. 3(d)-(f) (α = 1.45, α = 1.65, α = 1.85) that streak artifacts become more and more obvious with the increase of α. Judging from both SSIM improvement and the visual effect of denoised image, α = 1.05 is found the best choice in this experiment. The iteration stop criterion was chosen by the maximum SSIM value in our experiments, because the SSIM metric has been widely proven to have much better consistency with the qualitative visual performance. A larger SSIM value indicates better structure similarity between denoised image and original ground truth image. The SSIM curve of the iterative algorithm can be used to select the optimal iteration point to stop the whole iteration process. For instance, we can conclude from the SSIM curve with α = 1.05 in Fig. 2 that the iteration should be stopped at iter=80 in the Shepp- Logan head phantom study. At the same time, Fig. 2 shows FIGURE 3. Comparison of processed images by using the proposed FPMTV algorithm with different fractional orders α on the LDCT image (shown in Fig. 1(d)). (a) α = 0.85, (b) α = 1.05, (c) α = 1.25, (d) α = 1.45, (e) α = 1.65, and (f) α = 1.85. VOLUME 4, 2016 8493

16. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing FIGURE 4. The comparative experiments on a pelvis phantom. (a) original phantom, (b) LDCT image, (c) processed image by TV method (240 steps), (d) processed image by PMTV method (160 steps), (e) processed image by FTV method (250 steps), and (f) processed image by FPMTV method (50 steps). From left to right, the images in the second, third, and fourth columns show the zoomed ROIs specified in (a), and all of the zoomed images are from the corresponding images of the first column. that the best numbers of iterations with different α are almost the same, so we can use the same method to select the iteration stop time in other phantom studies. The other parameters of FPMTV model were set manually by comprehensive analysis of the SSIM index, the PSNR index and the visual effect of processed images. We set 8494 VOLUME 4, 2016

17. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing TABLE 1. PSNR, SSIM values of three ROIs (marked by red squares in the following pelvis phantom) of processed images by TV, PMTV, FTV, and FPMTV. 1t = 0.05, λ = 0.01, L = 0.1, and k0 = 100 for FPMTV model in all phantom studies. B. EXPERIMENTAL RESULTS 1) THE PELVIS PHANTOM STUDY To assess the performance of the proposed FPMTV algorithm, the following methods including TV model [25], PMTV model [26], and FTV model [30] were chosen for the comparative experiments on LDCT images. The parameters in TV, PMTV, and FTV were set according to the suggestions in [25], [26], [30]. The processing results of the LDCT image (Fig. 1(e)) are shown in Fig. 4, in which (c), (d), (e), and (f) are the processed images by TV, PMTV, FTV, FPMTV methods respectively. To further compare the performance of multiple denoising algorithms, three regions of interest (ROIs) and their corresponding zoomed images are shown in Fig. 4. The three ROIs identified by red squares are shown in the original pelvis phantom (Fig. 4 (a)), and their zoomed images are shown in Fig. 4 (a1-a3). Fig. 4 (b-b3) show the LDCT image and its zoomed ROIs. By compare the images in Fig. 4 (a-a3) with (b-b3), we can see that mottle noise and streak artifacts severely degrade the reconstructed images. Fig. 4 (c-c3) processed by TV suffer from obvious blocky effect (pointed by blue arrows). Mottle noise and steak artifacts (pointed by blue arrows) can be clearly seen in Fig. 4 (d-d3) processed by PMTV. Obviously, the processed images by FTV (Fig. 4 (e-e3)) perform better than that by TV and PMTV, but the performance in preserving edges and details is really bad (pointed by blue arrows). In addition, it is easily observed from the processed images by FPMTV (Fig. 4 (f-f3)) that the FPMTV algorithm can robustly reduce mottle noise, streak artifacts and blocky effect while preserving edges and details. With the original phantom as reference, we can see that the processed images by FPMTV are the most close to the reference images. Fig. 4 demonstrates that the proposed FPMTV model perform better than other algorithms. For further quantitative analysis, Table 1 shows the PSNR and SSIM values of three ROIs (marked by red squares in the pelvis phantom in Table 1) of processed images by TV, PMTV, FTV, and FPMTV. In order to intuitively illustrate FIGURE 5. Histogram of PSNR, SSIM values in Table 1. The corresponding algorithms are shown in figure legend. the PSNR and SSIM values of different denoising techniques, Fig. 5 plots the histogram of the PSNR and SSIM values in Table 1. As we can see from Table 1 and Fig. 5, FPMTV has the highest PSNR/SSIM for all of the ROIs. It means that the image processed by FPMTV is the most close to the original phantom. Based on both visual effect and quantitative analysis, the experimental results demonstrate that the FPMTV algorithm can effectively smooth noisy background while preserving edges and details in the LDCT images. VOLUME 4, 2016 8495

18. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing FIGURE 6. The comparative experiments on an actual thoracic phantom. (a) processed HDCT image by the AS-LNLM method, (b) LDCT image, (c) processed image by TV (250 steps), (d) processed image by PMTV (200 steps), (e) processed image by FTV (260 steps), and (f) processed image by FPMTV (60 steps). From left to right, the images in the second, third, fourth, and fifth columns show the zoomed ROIs specified in (a), and all of the zoomed images are from the corresponding images of the first column. 2) THE ACTUAL THORACIC PHANTOM STUDY In this study, we took an anatomical model of a human chest torso as the thoracic phantom, and the parameter settings were the same as that in the pelvis phantom study. Fig. 6 shows processing results of the LDCT image (Fig. 1(f)) by the four discussed algorithms (TV, PMTV, FTV, and FPMTV). The processed HDCT image by AS-LNLM method is shown in Fig. 6(a) as the reference image, in which four ROIs are marked by red squares. Fig. 6 (a1-a4) show the zoomed images of the four ROIs. The original LDCT image (30 mAs) and its zoomed ROIs are shown in Fig. 6 (b-b4), in which there are obvious mottle noise and steak artifacts. It means that LDCT scanning can severely degrade reconstructed images. The images processed by TV (Fig. 6(c-c4)) suffer 8496 VOLUME 4, 2016

19. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing TABLE 2. PSNR, SSIM values of five ROIs (marked by red squares in the following actual thoracic phantom) of processed images by TV, PMTV, FTV, and FPMTV. FIGURE 7. Histogram of PSNR, SSIM values in Table 2. The corresponding algorithms are shown in figure legend. from obvious blocky effect. Fig. 6(d-d4) and Fig. 6(e-e4) are the processed images by PMTV and FTV respectively. By comparison, PMTV and FTV perform better than TV in blocky effect suppression, however, PMTV introduces new artifacts to processed images, whereas FTV blurs edges and details. On the base of the analysis above, TV, PMTV, and FTV are all inappropriate for LDCT images processing because the inpainting ignores detailed features and may lead to misdiagnosis. The image processed by the proposed FPMTV model is illustrated in Fig. 6(f), and its corresponding zoomed images of ROIs are shown in Fig. 6(f1-f4). We can see that mottle noise and steak artifacts are suppressed effectively, and that edges and fine details are preserved commendably. This observation verifies the effectiveness of the FPMTV method in LDCT image processing. Moreover, Fig. 6 confirms the same conclusion as Fig. 4 that the FPMTV method has superior performance over other denoising methods (see the areas pointed by blue arrows in the third column in Fig. 6). For further comparison, Table 2 displays the PSNR and SSIM values of five ROIs (marked by red squares in the actual thoracic phantom in Table 2) of processed images by TV, PMTV, FTV, and FPMTV respectively. Fig. 7 shows the PSNR and SSIM values in Table 2 in a different and more intuitive way. It is worth mentioning that the quantitative results in Table 2 and Fig. 7 have a similar trend as in the pelvis phantom study. The proposed algorithm has the highest PSNR/SSIM for all of the ROIs. IV. CONCLUSION PDEs are generally treated as fine candidates for noise removal in LDCT image processing, however, the tradi- tional integral-order differentiation methods may often cause blocky effect and speckle effect, which inevitably blur edges and fine details of an image. At the same time, these details blurred by integral-order algorithms in medical images may have important clinical application value. To overcome the disadvantages, fractional-order PDEs have been recently researched and applied to medical image processing. Although several fractional-order differentiation methods such as FPM and FTV can, to a certain extent, reach a good trade-off between noise removal and edges preservation, some mottle noise and steak artifacts still exist. To fix this problem, we integrated FPM model and FTV model to obtain FPMTV model. Additionally, the local intensity variance was added to weighted coefficient and diffusion coefficient of the FPMTV model to properly preserve edges and details. Utilizing the weighted coefficient, we can adaptively control the FPMTV model, which alternates between FPM model and FTV model in accordance with the image feature. In the VOLUME 4, 2016 8497

20. Y. Wang et al.: Novel Fractional-Order Differentiation Model for LDCT Image Processing flat region, the FPMTV model highlighted the importance of FPM model. In the fine detail area or edge, the FPMTV model emphasized the role of FTV model. The experimental results show that compared with other denoising algorithms (TV, PMTV, FTV), the proposed FPMTV method achieves superior performance in terms of both noise suppression and edges preservation in LDCT images. The computational cost should be mentioned here. Fractional-order differentiation methods suffer from heavy computational burden because many more pixels are needed by the computation of fractional-order PDEs than that of integral-order PDEs. The FPMTV has the same computational complexity as FTV. However, the computational cost of FPMTV was greatly reduced by cutting down on the number of iterations in our study. From Fig. 4, we can see that 50 steps and 250 steps are required separately by FPMTV and FTV in the pelvis phantom study. Moreover, Fig. 6 shows that 60 steps and 260 steps separately are needed by FPMTV and FTV in the actual thoracic phantom study. In the further study, more ways should be found to reduce the computational cost. For example, the new algorithm may be developed based on the graphics processing unit (GPU) to raise the calculation speed. Many research results have shown the possibility of GPU to enormously accelerate the iterative procedure [40], [41]. With the powerful tool, fractional-order differentiation methods will be more suitable for clinical applications. It is worth mentioning that selection of fractional order α is very important for a high quality processed image. The order in our study was set by both SSIM improvement and the visual effect of denoised image. Further study is needed for the adaptive selection of fractional order. 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