Sinogram Restoration for Low-Dosed X-Ray Computed Tomography Using Fractional-Order Perona-Malik Diffusion

Existing integer-order Nonlinear Anisotropic Diffusion NAD used in noise suppressing will produce undesirable staircase effect or speckle effect. In this paper, we propose a new scheme, named Fractal-order Perona-Malik Diffusion FPMD , which replaces the integer-order derivative of the Perona-Malik PM Diffusion with the fractional-order derivative using G-L fractional derivative. FPMD, which is a interpolation between integer-order Nonlinear Anisotropic Diffusion NAD and fourth-order partial differential equations, provides a more flexible way to balance the noise reducing and anatomical details preserving. Smoothing results for phantoms and real sinograms show that FPMDwith suitable parameters can suppress the staircase effects and speckle effects efficiently. In addition, FPMD also has a good performance in visual quality and root mean square errors RMSE .


Introduction
Radiation exposure and associated risk of cancer for patients receiving CT examination have been an increasing concern in recent years.Thus minimizing the radiation exposure to patients has been one of the major efforts in modern clinical X-ray CT radiology 1-8 .
A simple and cost-effective means to achieve low-dose CT applications is to lower Xray tube current mA as low as achievable 6, 7 .However, the presentation of strong noise degrades the quality of low-dose CT images dramatically and decreases the accuracy of the diagnosis dose.
Filtering noise from clinical scans is a challenging task, since these scans contain artifacts and consist of many structures with different shape, size, and contrast, which should be with an analytical formula between the sample mean and sample variance, that is, the noise is a signal-dependent Gaussian distribution 20 .In this section, we will introduce signal-independent Gaussian noise SIGN , Poisson noise, and signal-dependent Gaussian noise.

Signal-Independent Gaussian Noise (SIGN)
SIGN is a common noise for imaging system.Let the original projection data be {x i }, i 1, . . ., m, where i is the index of the ith bin.The signal has been corrupted by additive noise {n i }, i 1, . . ., m and one noisy observation where y i , x i , n i are observations for the random variables Y i , X i , and N i where the uppercase letters denote the random variables and the lower-case letters denote the observations for respective variables.

Poisson Model and Signal-Dependent Gaussian Model
The photon noise is due to the limited number of photons collected by the detector 36 .For a given attenuating path in the imaged subject, N 0 i, α and N i, α denote the incident and the penetrated photon numbers, respectively.Here, i denote the index of detector channel or bin, and α is the index of projection angle.In the presence of noises, the sinogram should be considered as a random process and the attenuating path is given by where N 0 i, α is a constant and N i, α is Poisson distribution with mean N. Thus we have Both its mean value and variance are N.
Gaussian distributions of ploy-energetic systems were assumed based on limited theorem for high-flux levels and followed many repeated experiments in 20 .We have where μ i is the mean and σ 2 i is the variance of the projection data at detector channel or bin i, γ is a scaling parameter, and f i is a parameter adaptive to different detector bins.
The most common conclusion for the relation between Poisson distribution and Gaussian distribution is that the photon count will obey Gaussian distribution for the case Mathematical Problems in Engineering with large incident intensity and Poisson distribution with feeble intensity 20 .In addition, in 36 , the authors deduce the equivalency between Poisson model and Gaussian model.Therefore, both theories indicate that these two noises have similar statistical properties and can be unified into a whole framework.

Perona-Malik Diffusion
In image smoothing, Nonlinear Anisotropic Diffusion NAD , also called Perona-Malik diffusion PMD , is a technique aiming at reducing image details without removing significant parts of the image contents, typically edges, lines, or textures, which are important for the image 50 .
With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the heat equation, which is equivalent to Gaussian blurring.This is ideal for smoothing details but also blurs edges.When the diffusion coefficient is chosen as an edge seeking function, the resulting equations encourage diffusion hence smoothing within regions and stop it near strong edges.Hence the edges can be preserved while smoothing from the image 50 .
Formally, NAD is defined as ∂u x, y, t ∂t div g x, y, t ∇u x, y, t , where u x, y, 0 is the initial gray scale image, u x, y, t is the smooth gray scale image at time t, ∇ denotes the gradient, div • is the divergence operator, and g x, y, t is the diffusion coefficient.g x, y, t controls the rate of diffusion and is usually chosen as a monotonically decreasing function of the module of the image gradient.Two functions proposed in 50 are g ∇u x, y, t e − ∇u x,y,t /σ 2 , 3.2 where • is the module of the vector and the constant σ controls the sensitivity to edges.Perona and Malik propose a simple method to approach the modules of gradients, which is called PM diffusion 50 .Its discretization for Laplacian operator is where According to 3.2 -3.3 , the diffusion coefficient is defined as a function of module of the gradient.However, computing a gradient accurately in discrete data is very complex and the module of the gradient is simplified as the absolute values of four directions and diffusion coefficients are where | • | is the absolute value of the number and g • is defined in 3.2 or 3.3 .
The main default for PM diffusion is that it will lead to staircase effect or sometimes details oversmoothing.In order to eliminate the staircase effects and preserve anatomical details, we propose to replace the first-order and the second-order derivative of the PM Diffusion with the fractional-order derivative using G-L fractional derivative.The new diffusion model will be introduced in the next section.

The Fractional-Order PM Diffusion (FPMD)
The FPMD is developed using G-L fractional-order derivative, which is defined as 38 An image U will be a 2-dimensional matrix of size N × N and its discrete fractionalorder gradient ∇ α u is an 8-dimensional vector: where T represents the transpose of the vector and ∇ α u k i, j , k 0, . . ., 7 are defined as Thus where T represents the transpose of the vector.From 4.3 , we have

4.6
Let g g 0 , g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 T , 4.7 where T represents the transpose of the vector and g k , k 0, . . ., 7 is defined as The new FPMD based on G-L fractional-order derivative is defined as where the ∇ α k u i, j, t , k 0, . . ., 7, are the components of vector ∇ α u i, j, t in 4.3 and g k , k 0, . . ., 7, defined in 4.8 are the components of g in 4.7 .
The above equation can be represented as ∂u i, j, t ∂t where 7 k 0 g k 1 and ∇ 2α k u i, j, t can be computed according to 4.5 .Thus the explicit form for solving 4.12 is where u i, j, t 1 is the gray level of i, j at time t 1 and g k , ∇ 2α k u i, j, t are the same as in 4.12 .

Experiments and Discussion
The main objective for smoothing L-CT images is to delete the noise while to preserve anatomy details for the images.In order to show the performance of FPMD, a 2-dimensional 256 × 256 Shepp-Logan head phantom developed in MatLab.The number of bins per view is 888 with 984 views evenly spanned on a circular orbit of 360 • .The detector arrays are on an arc concentric to the X-ray source with a distance of 949.075 mm.The distance from the rotation center to the X-ray source is 541 mm.The detector cell spacing is 1.0239 mm.The L-CT projection data sinogram is simulated by adding Gaussian-dependent noise GDN whose analytic form between its mean and variance has been shown in 2.4 .In this paper, set f i 4.0 and T 2e 4. The projection data is reconstructed by standard Filtered Back Projection FBP .Since both the original projection data and sinogram have been provided, the evaluation based on root-mean-square error RMSE between the ideal reconstructed image is and reconstructed images defined as where f recon i, j denotes the reconstructed value on position i, j while f Ph i, j denotes the ideal reconstructed value on position i, j .Two abdominal CT images of a 62-year-old woman with different doses were scanned from a 16 multidetector 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 × 1.5 mm; table feed per gantry rotation, 24 mm; pitch, 1 : 1 and reconstruction method, Filtered Back Projection FBP algorithm with the soft-tissue convolution kernel "B30f".Different CT doses were controlled by using two different fixed tube current 30 mAs and 150 mAs 60 mA or 300 mAs for L-CT and standard-dose CT SDCT protocols, resp. .The CT dose index volume CTDIvol for LDCT images and SDCT images are in positive linear correlation to the tube current and are calculated to be approximately ranged between 15.32 mGy to 3.16 mGy 51 see Figures 2 a and 2 b .
On sinogram space, FPMD with α 0.2, α 0.5, and α 1.5 is carried on two image collections.Other compared methods include median filter with 5 × 5 window; wiener filter with 5 × 5 window; Gaussian filter whose mean is 0 and its standard deviation is 1.8.The diffusion coefficient for PMD and FPMDs is selected as a Gaussian function whose standard deviation is 2. All smoothed projection data will be reconstructed by standard FBP.
Table 1 summarized RMSE between the ideal reconstructed image and filtered reconstructed image.The FPMD with α 0.2 has the best performance in RMSE, while other FPMD with α 0.5 and α 1.5 also has better performance than almost other comparing methods except for 5 × 5 wiener.In summary, the FPMD has a very good performance in RMSE.Since FPMD provides a more flexible way for diffusion than PMD, FPMD has much good performance in denoising while preserving structures.Comparing all the original SDCT images and L-CT images in Figures 1 and 2, we found that the L-CT images were severely degraded by nonstationary noise and streak artifacts.In Figures 2 g -2 i , for the proposed FPMD approach, experiments with fractional-order α gradually increased will obtain more smooth images.Both in Figure 1 and 2, we can observe better noise/artifacts suppression and edge preservation when α 0.2.Especially, compared to their corresponding original SDCT images, the fine features representing the intrahepatic bile duct dilatation and the hepatic cyst were well restored by using the proposed FPMD.We can observe that, the noise grains and artifacts were significantly reduced for the FPMD processed L-CT images with suitable α both in Figures 1 and 2. The fine anatomical/pathological features can be well preserved compared to the original SDCT images Figures 1 a and 2 a under standard dose conditions.

Conclusions
In this paper, we propose a new fractional-order PMD FPMD for L-CT sinogram imaging based on G-L fractional-order derivative definition.Since FPMD is a interpolation between integer-order Nonlinear Anisotropic Diffusion NAD and fourth-order partial differential equations, it provides a more flexible way to balance the noise reducing and anatomical details preserving.Smoothing results for phantoms and real sinograms show that FPMD with suitable parameters can suppress the staircase effects and speckle effects efficiently.In addition, FPMD also has good performance in visual quality and root mean square errors RMSE .

Figure 2 :
Figure 2: Abdominal CT images of a 62-year-old woman.a Original SDCT image with tube current time product 150 mAs.b Original LDCT image with tube current time product 60 mAs.c LDCT image processed by 5 × 5 median filter.d LDCT image processed by 5 × 5 wiener filter.e LDCT image processed by Gaussian smoothing with σ 1.8, μ 0 .f LDCT image processed by PMD with σ 2 .g LDCT image processed by FPMD with σ 2, α 0.2 .h LDCT image processed by FPMD with σ 2, α 0.5 .i LDCT image processed by FPMD with σ 2, α 1.5 .

Table 1 :
RMSE of different smoothing methods.