Low-dosed x-ray computed tomography imaging by regularized fully spatial fractional-order Perona-Malik diffusion.
Perona-Malik diffusion (PMD) proposed in 1990 is a popular technique in image denoising and it is defined as 
[[partial derivative]u(x, y, t)/[partial derivative]t] = div[c([partial derivative][nabla]u(x, y, t)[parallel])[nabla]u(x, y, t)], (1)
where u(x, y, 0) is the initial gray scale image, u(x, y, t) is the smoothed gray scale image at time t, [nabla] denotes the gradient, div(x) is the divergence operator, and c(x) is the diffusion coefficient.
In 1992, Catte et al. indicated that PMD is ill-posed and they propose a new well-posed method named regularized Perona-Malik diffusion (RPMD), by replacing the gradient [nabla]u in diffusion coefficients by the smoothed version [G.sub.[sigma]] x [nabla]u . Thus, the RPMD can be represented as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
which is a Gaussian function and C is a constant.
In order to eliminate undesired "staircase" of PMD and RPMD, high-order PDEs (typically fourth-order PDEs) for image restoration have been introduced in [3, 4]. Though these methods can eliminate the staircase effect efficiently, they often leave the image with isolated black and white speckles (so-called speckle effect) .
Recently, fractional-order PMD (FOPMD) has been studied in image denoising [5-14], whose fractional order is [alpha], 0 [less than or equal to] [alpha] [less than or equal to] 2, which is a "natural interpolation" between PMD and fourth-order PDEs. Therefore, it has the benefits of both of PMD and high order PDEs.
Bai and Feng proposed a FSFOD method for image denoising with Euler-Lagrange equations of a cost functional and using Fourier-domain to compute the fractional derivative . Jun and Zhihui develop a class of fractional-order multiscale variational model using G-L definition of fractional-order derivative and propose an efficient condition of the convergence of the model .
We also reported an experimental study that used FOPMD for sinogram restoration of low-dosed computed tomography (LDCT) based on fully spatial FOPMD using GL definition . In , we observe that FOPMD experiments with different fractional orders show different diffusion behaviors, which leads us to study FOPMD further.
In a most recent study, Hu proposed a new FOPMD by diffusing only on external gradient vector and reported good performance in LDCT imaging .
Although above FOPMD have reported on good performance of preserving edges, suppressing staircase and speckle effects, the resulting images of these FOPMD methods still have some artifacts and speckle effect which hamper them to be used in LDCT imaging.
Minimizing the radiation exposure to patients has been one of the major efforts in modern clinical X-ray CT radiology [15-17]. However, the presentation of strong noise degrades the quality of LDCT images dramatically and decreases the accuracy of diagnosis.
Filtering noise from clinical scans is a challenging task, since these scans contain many structures with different shapes, sizes, and contrasts, which should be preserved for making correct diagnosis. In addition, LDCT imaging also requires no artifacts and speckle effect to emerge while denoising because artifacts also increase chances of misdiagnosis. Many strategies have been proposed to reduce noises, but few of them discuss how to avoid artifacts and speckle effect [8, 18-28].
The artifacts of existing FOPMD methods used for LDCT imaging are generated by strong noises of LDCT sinogram, which leads to falsely located positions of edges. Thus, the resulting images produce some undesired weak edges, which form the artifacts.
Therefore, one valid method to suppress artifacts and speckle effect of existing FOPMD methods is by smoothing the fractional-order gradient (FOG) in the diffusion coefficients to avoid error-detected edges of the noisy LDCT.
Following the above discussion, we propose a regularized FOPMD, named regularized fully spatial FOPMD (RFS-FOPMD), by replacing FOG of diffusion coefficients with its smoothing counterparts while keeping "external" FOD unchanged. Here "fully spatial" represents all the derivatives of right-hand side of PMD equation and "external" indicates the spatial derivatives except for the derivatives used in diffusion coefficients. Since locations of edges can be detected correctly by our new scheme, FOPMD can preserve edges well and avoid artifacts.
The arrangement of this paper is as follows: in Section 2, the EFOGV-PMD is introduced, and then the the numerical scheme is given in Section 3, the experiment results are shown and discussed in Section 4, the final part is the conclusions and acknowledgments.
2. Regularized Fully Spatial Fractional-Order Perona-Malik Diffusion
In this paper, we use G-L definition defined as follows [29, 30]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where g(x) is a real function, [alpha] > 0 is a real number, [C.sup.[alpha].sub.k] = [GAMMA]([alpha] + 1)/[[GAMMA](k + 1)[GAMMA]([alpha] - k + 1)] is the generalized binomial coefficient, and [GAMMA](x) denotes the gamma function.
Isotropic diffusion will damage the image features such as edges, lines, and textures. To avoid the damage, the smoothing has to be adaptively controlled by the amount of smoothing or the direction of smoothing. A classic example of adaptive smoothing is the anisotropic diffusion scheme proposed by Perona and Malik , in which the smoothing process is formulated by a partial differential equation (PDE). PMD is formulated in (1).
However, PMD methods suffer from their "staircase" effects. Therefore, FOPMD is proposed to suppress the staircase of PMD.
The fractional-order gradient vector with [alpha] order is defined as
[[nabla].sup.[alpha]]u(x, y, t) = [[nabla].sup.[alpha].sub.x]u(x, y, t), [[nabla].sup.[alpha].sub.y]u(x, y, t)], (5)
where a is a positive real, [[nabla].sup.[alpha].sub.x]u (x, y, t) represents the partial fractional-order derivative of u(x, y, t) with respect to the variable x whose order is [alpha], and [[nabla].sup.[alpha].sub.y]u(x, y, t) represents the partial fractional-order derivative of u(x, y, t) with respect to the variable y whose order is [alpha].
According to , FOPMD is defined as
[partial derivative]u(x, y, t)/[partial derivative]t = [div.sup.[alpha]][c([parallel][[nabla].sup.[alpha]]u(x, y, t)[parallel])[[nabla].sup.[alpha]]u(x, y, t)], (6)
where [div.sup.[alpha]] denotes the [alpha]-order divergence. For the vector [[nabla].sup.[beta]]v = [[v.sup.[beta].sub.x], [v.sup.[beta].sub.y]] where [v.sup.[beta].sub.x], [v.sup.[beta].sub.y] represent the partial fractional-order derivative of u(x, y, t) with whose order is [beta] respect to the variable and respectively, its [alpha]-order divergence is defined as:
[div.sup.[alpha]][[nabla].sup.[beta]]v = [v.sup.[beta]+[alpha].sub.x] + [v.sup.[beta]+[alpha].sub.y]. (7)
However, FOPMD defined by (3) will produce some artifacts for sinogram restoration of LDCT, which increases the probability of error diagnosis. In order to avoid artifacts produced in sinogram restoration of LDCT using FOPMD, we propose a new diffusion model, named regularized fully spatial fractional-order PMD (RFS-FOPMD), where "fully spatial" indicates all derivatives of the right-hand side of (6). That is, the FOD in diffusion coefficient is replaced by its smoothed version.
Therefore, the RFS-FOPMD is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
with the observed image as the initial condition and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined in (3).
When [alpha] = 1, (6) is precisely the PMD and (8) is precisely the RPMD; when [alpha] = 2, (6) is precisely the fourth-order anisotropic diffusion equation. In this paper, we are interested in 0.5 [less than or equal to] [alpha] [less than or equal to] 1.5 since Bai and Feng in  suggest that [alpha] = 1.5 in their model has the best performance.
3. The Numerical Scheme
An image U will be a 2-dimensional matrix of size N x N. In order to get the aim of anisotropic diffusion along different directions and because the discrete -order gradient [[nabla].sup.[alpha]]u is an 8-dimensional vector
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where T represents the transpose of the vector and [[nabla].sup.[alpha]][u.sub.k](i, j), k = 0, ..., 7 are defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where represents the transpose of the vector. From (4), we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
g = ([g.sub.0], [g.sub.1], [g.sub.2], [g.sub.3], [[gamma].sub.4], [[gamma].sub.5], 06' 07)T' (13)
where T represents the transpose of the vector and [g.sub.k], k = 0, ..., 7 are defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [[nabla].sup.[alpha].sub.k](i, j), k = 0, ... 7 defined in (9)are the components of vector [[nabla].sup.[alpha]]u(i, j) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the normalized constant, is the decreasing function of absolute value of [[nabla].sup.[alpha].sub.k]u(i, j), k = 0, ... 7, and g([parallel][[nabla].sup.[alpha].sub.k]u(x, y, t)[parallel]) is defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
g([parallel][[nabla].sup.[alpha].sub.k]u(x, y, t)[partial derivative]) = 1/1 + [([parallel][[nabla].sup.[alpha].sub.k]u(x, y, t)[partial derivative]/[sigma]).sup.2], k = 0, ..., 7, (16)
where [parallel]x[parallel] is the module of the fractional-order vector and the constant [sigma] controls the sensitivity to edges.
The new FOPMD based on G-L fractional-order derivative is defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
where [g.sub.k], k = 0, ..., 7 defined in (14) are the components of g in (13).
The above equation can be represented as
[partial derivative]u(i, j, t)/[partial derivative]t = [7.summation over (k=0)][g.sub.k][[nabla].sup.2[alpha].sub.k]u(i, j, t), (18)
where [[summation].sup.7.sub.k=0][g.sub.k] = 1 and [[nabla].sup.2[alpha].sub.k]u(i, j, t) can be computed according to (8).
Thus, the explicit form for solving (18) is
u (i, j, t + 1) = u (i, j, t) + [lambda][7.summation over (k=0)][g.sub.k][[nabla].sup.2[alpha].sub.k]u(i, j, t), (19)
where u(i, j, t + 1) is the gray level of (i, j) at time t + 1 and [lambda] is the integration constant (0 [less than or equal to] [lambda] [less than or equal to] 1/7).
To summarize, our sinogram restoration approach is done in the following steps:
(1) let the input sinogram be U and set t = 1, [U.sub.t] = U, input iterative numbers n, Gaussian deviations [[sigma].sub.1] of regularized Gaussian kernel in (3), fractional order [alpha], integration constant [lambda] in (19), gradient modulus threshold [sigma] that controls the conduction used in (15) or (16), and choose (15) or (16) as diffusion coefficients;
(2) compute a-order gradient vector [[nabla].sup.[alpha]]u using (9);
(3) compute diffusion coefficients vector g using (13)-(16);
(4) compute 2[alpha]-order gradient vector [[nabla].sup.2[alpha]]u using (11);
(5) compute [U.sub.t+1] using (19), and set t = t + 1, if t = n, output sino gram [U.sub.t]; else goto step 2;
(6) return back-project sinogram [U.sub.t] into the image I.
4. Experiments and Discussion
The main objective for LDCT imaging is to delete the noise and avoid artifacts while preserving anatomy details for the back-projection images.
Two abdominal CT images of a 58-year-old man and two abdominal CT images of a 62-year-old woman with different doses were scanned from a 16-multi detector-row CT unit (Somatom Sensation 16; Siemens Medical Solutions) using 120 kVp and 5 mm slice thickness. Other remaining scanning parameters are gantry rotation time, 0.5 second; detector configuration (number of detector rows section thickness), 16 x 1.5 mm; table feed per gantry rotation, 24 mm; pitch, 1: 1, and reconstruction method, back projection (FBP) algorithm with the soft-tissue convolution kernel "B30f." Different CT doses were controlled by using two different fixed tube currents 60 mAs and 150 mAs (60 mA or 150 mAs) for LDCT and standard-dose CT (SDCT) protocols, resp.). The CT dose index volume (CTDIvol) for LDCT images and SDCT images is in positive linear correlation to the tube current and is calculated to be approximately ranging between 15.32 mGy and 3.16 mGy  (see Figures 1(a)-1(d)
In order to compare our method with classical PM and other state-of-art FOPMD methods, three compared methods: PMD , regularized PMD (RPM) , and FOPMD are proposed in . According to the numerical scheme of PMD and RPMD, they used half-point central difference discretization scheme, while FOPMD in  and RFS-FOPMD use integer-point unilateral difference discretization scheme.
In order to ensure that the comparison is put on a fair level, the common used parameters are set to the same value. The common used parameters for four methods include gradient modulus threshold [sigma] that controls the conduction, integration constant [lambda], and iteration number t. Due to numerical stability, [lambda] is set to its maximum value 1/100 and [sigma] is set to 30 to reduce iteration number.
The iteration number t is very important in all comparison methods. That is, big t will make smooth image while small will still leave a lot of noise. In order to study the performance of four compared methods with different iteration numbers t and other fixed parameters, t is set to 20, 50, and 100, respectively.
The standard deviation of smoothed Gaussian kernel for the image [[sigma].sub.1] used for RPMD is set to 1 since, in , the authors suggest that I should be a small number.
On sinogram space, FOPMD with [alpha] = 0.5, [alpha] = 0.8 and [alpha] = 1.2 is carried on two image collections.
Since bigger iteration number leads to smoother denoised results sometimes, it also leads to dark processed images by posing too big integration constant [lambda]. In order to observe the behaviors of big iteration number clearly, Figure 1 sets iteration number = 100. Comparing all the original SDCT images in Figures 1(a) and 1(c), LDCT images Figures 1 (b) and 1(d) were severely degraded by nonstationary noise. All denoised images in Figure 1 can suppress most of noises. Particular, FOPMD and RFS-FOPMD can provide very satisfied images with little noise and preserving all useful anatomy structures. However, denoised images of PMD and RPMD are oversmooth, which lost a lot of details.
In order to test the consistency of the definitions of different integer order or fractional order, we set fractional-order [alpha] = 1, in which a two fractional-order PMD should have same forms and they also correspond to the order of PMD and RPMD (see Figures 1(e)-1(1)). Observing Figures 1(h) and 1(k) and we can find that the denoised images are identical, which demonstrate that the fractional-order definitions between  and RFS-FOPMD are identical when [alpha] = 1.
However, the resulting images of PMD and RPMD are quite different to the images denoised by FOPMD and RFS-FOPMD. That is, the images processed by PMD and RPMD are smoother than the images processed by FOPMD and RFS-FOPMD. Just as introduced in the previous paragraph, different discretization schemes lead to this interesting result.
Since two FOMD schemes provide more satisfied results, we only compare two FOPMD methods with different fractional orders (see Figure 2) and different iteration numbers (see Figure 3).
In order to compare denoised results of two FOPMD schemes with different fractional orders, two original LDCT images in Figures 1(b) and 1(d) are used with iteration number t = 100 and fractional-order [alpha] = 0.5, [alpha] = 0.8, and [alpha] = 1.2. From the second and the fourth rows of Figure 2, we can conclude that the resulting images of RFS-FOPMD are very satisfied and they become smoother when a becomes bigger, which is coherent with our intuition, for example, bigger fractional-order smoother resulting images.
However, denoised images in Figures 2(a) and 2(g) with [alpha] = 0.5 using FOPMD in  have many artifacts, which are small black circles in two images. Although big fractional-order FOPMD proposed in  will decrease the artifacts, its denoised images in Figures 2(c) and 2(i) are very dark comparing with the original LDCT images in Figures 1(b) and 1(d). Images in Figures 2(c) and 2(i) also have some isolated artificial white points, which are called speckle effect. It is obvious that resulting images in Figures 2(b) and 2(h) with [alpha] = 0.8 processed by FOPMD in  have the best performance in three image series with different [alpha].
Generally, artifacts in denoised images are oscillations near edges, caused by that the low-passed filtering is not processed correctly near the real edges. That is, some smooth regions near edges are regarded falsely as edges, which makes these error edges preserved. Therefore, improving accuracy of edge detection is a good choice for improving the performance of FOPMD in . In this paper, we use regularization for FOPMD in  to locate edges correctly.
Intuitively, processed images with bigger iteration number correspond to smoother images. In order to check the influence of iteration numbers for FOPMD in  and RFS-FOPMD, processed image series of two original LDCT images, Figures 1(b) and 1(d) with different iteration numbers 20, 50, and 100 are shown in Figure 3.
The resulting images in the first and the third rows of Figure 3 are the processed images using FOPMD in .
Comparing with original LDCT images in Figures 1(b) and 1(d), all resulting images with different iteration numbers have less noise. In addition, the smoother images can be obtained as the iteration number becomes bigger. However, the most undesired default for FOPMD in  is that resulting images become dark as the iteration number becomes big. Moreover, except for Figure 3(g), the resulting images in the third row have some isolated white points, which are the speckle effect.
The resulting images in the second and the fourth rows of Figure 3 show that RFS-FOPMD with different iteration numbers is very satisfied and it becomes smoother when the iteration number becomes bigger, which is coherent with our intuition. Another attractive nature for RFS-FOPMD about iteration is that the smoothing shown in these images is very slow. That is, the resulting images Figures 3(d) and 3(j) with iteration number 20 are slightly different to the images Figures 3(f) and 3(l) with iteration number 100. This nature shows that RFS-FOPMD has good stability. Therefore, it is not sensitive to iteration number.
All existing FOPMD methods at least suffer from speckle effect from the resulting images of these images. Fortunately, RFS-FOPMD can avoid artifacts, dark images, and speckle effect partly, which ensure its applications in sinogram restoration. More important for the new scheme is its stability, which makes it not sensitive to the iteration number.
In this paper, we propose a new FOPMD, RFS-FOPMD, for LDCT sinogram imaging based on G-L fractional-order derivative definition. RFS-FOPMD not only has good ability in preserving edges while denoising, but it also can avoid artifacts, dark images, and speckle effects of FOPMD in  and other existing FOPMD schemes partly by improving the performance of edges locating by regularization, which ensures that RFS-FOPMD can be used for sinogram restoration of LDCT. Of more importance, RFS-FOPMD has good stability for iteration numbers, which makes it not sensitive to the iteration number choice.
This paper is supported by the National Natural Science Foundation of China (no. 60873102), Major State Basic Research Development Program (no. 2010CB732501), and Open Foundation of Visual Computing and Virtual Reality Key Laboratory of Sichuan Province (no. J2010N03). This work was supported by a Grant from the National High Technology Research and Development Program of China (no. 2009AA12Z140).
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Received 19 September 2013; Accepted 2 November 2013
School of Computer Science, Sichuan Normal University, Chengdu, Sichuan 610101, China
Correspondence should be addressed to Zhiwu Liao; email@example.com
Academic Editor: Ming Li
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