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

Authors: Shaoxiang Hu; Zhiwu Liao; Wufan Chen
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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 FPMD with 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).


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## Body

## 1. 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 X-ray 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 preserved for making correct diagnosis. Many strategies have been proposed to reduce the noise, for example, by nonlinear noise filters [8–20] and statistics-based iterative image reconstructions (SIIRs) [21–29].The SIIRs utilize the statistical information of the measured data to obtain good denoising results but are limited for their excessive computational demands for the large CT image size. Although the nonlinear filters show effectiveness in reducing noise both in sinogram space and image space, they cannot handle the noise-induced streak artifacts. Some nonlinear filters, such as Nonlinear Anisotropic Diffusion (NAD), even produce new artifacts in L-CT denoising [30–37].To eliminate the undesirable staircase effect, high-order PDEs (typically fourth-order PDEs) for image restoration have been introduced in [38–43]. Though these methods can eliminate the staircase effect efficiently, they often lead to a speckle effect [44].Recently, fractional-order PDEs have been studied in many fields [38–49]. The fractional derivative can be seen as the generalization of the integer-order derivative. It has been studied by many mathematicians (e.g., Euler, Hardy, Littlewood, and Liouville) [47]. Not until Mandelbrot found fractals and applied the G-L fractional derivative to the Brownian motion did the fractional derivative cause great attention. There are many methods that can define the fractional derivative. The usual definitions among them involve G-L fractional derivative, Cauchy-integral fractional derivative, frequency-domain (Fourier-domain) fractional derivative.Li and Zhao investigate relation between the data of cyber-physical networking systems and power laws and then suggest that power-law-type data may be governed by stochastically differential equations of fractional order [45]. They also propose that one-dimensional random functions with long-range dependence (LRD) based on a specific class of processes called the Cauchy-class (CC) process maybe a possible model of sea level data [46].You and Kaveh 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 for the model [38]. The experiments show that the model can improve the peak signal-to-noise ratio, preserve texture, and eliminate the stair effect efficiently.Bai and Feng proposed a class of fractional-order anisotropic diffusion equations based on PM equation for image denoising using Fourier-domain fractional derivative in [49]. The numerical results showed that both of the staircase effect and the speckle effect can be eliminated effectively by using the fractional-order derivative.Inspired from previous works and 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. It should be indicated that the method proposed in this paper, which is carried on the sinogram space directly, is different to the method proposed in [49], which is carried on the Fourier space.The arrangement of this paper is as follows. In Section2, the noise model of Low-dosed CT (L-CT) is introduced; and then the PM diffusion is given in Section 3, new fractional-order PM method is developed using G-L fractional definition in Section 4; the experiment results are shown and discussed in Section 5; the final part is the conclusions and acknowledgement.


## 2. Noise Models

Based on repeated phantom experiments, low-mA (or low-dose) CT-calibrated projection data after logarithm transform were found to follow approximately a Gaussian distribution 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.


### 2.1. Signal-Independent Gaussian Noise (SIGN)

SIGN is a common noise for imaging system. Let the original projection data be{xi},i=1,…,m, where i is the index of the ith bin. The signal has been corrupted by additive noise {ni},i=1,…,m and one noisy observation(2.1)yi=xi+ni,
where yi, xi, ni are observations for the random variables Yi, Xi, and Ni where the uppercase letters denote the random variables and the lower-case letters denote the observations for respective variables. Xi is normal N(0,σX2); Ni is normal N(0,σN2) and independent to the Gaussian random variable Xi. Thus Yi is normal N(0,σX2+σN2).


### 2.2. 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, N0(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(2.2)ri=-ln⁡[N(i,α)N0(i,α)],
where N0(i,α) is a constant and N(i,α) is Poisson distribution with mean N.Thus we have(2.3)N(i,α)=N0(i,α)exp⁡(-ri).Both its mean value and variance areN.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(2.4)σi2(μi)=fiexp⁡(μiγ),
where μi is the mean and σi2 is the variance of the projection data at detector channel or bin i, γ is a scaling parameter, and fi 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 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.


## 2.1. Signal-Independent Gaussian Noise (SIGN)

SIGN is a common noise for imaging system. Let the original projection data be{xi},i=1,…,m, where i is the index of the ith bin. The signal has been corrupted by additive noise {ni},i=1,…,m and one noisy observation(2.1)yi=xi+ni,
where yi, xi, ni are observations for the random variables Yi, Xi, and Ni where the uppercase letters denote the random variables and the lower-case letters denote the observations for respective variables. Xi is normal N(0,σX2); Ni is normal N(0,σN2) and independent to the Gaussian random variable Xi. Thus Yi is normal N(0,σX2+σN2).


## 2.2. 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, N0(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(2.2)ri=-ln⁡[N(i,α)N0(i,α)],
where N0(i,α) is a constant and N(i,α) is Poisson distribution with mean N.Thus we have(2.3)N(i,α)=N0(i,α)exp⁡(-ri).Both its mean value and variance areN.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(2.4)σi2(μi)=fiexp⁡(μiγ),
where μi is the mean and σi2 is the variance of the projection data at detector channel or bin i, γ is a scaling parameter, and fi 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 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.


## 3. 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(3.1)∂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(3.2)g(‖∇u(x,y,t)‖)=e-(‖∇u(x,y,t)‖/σ)2,(3.3)g(‖∇u(x,y,t)‖)=11+(‖∇u(x,y,t)‖/σ)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
(3.4)u(i,j,t+1)=u(i,j,t)+14[cN⋅∇N2u(i,j,t)+cS⋅∇S2u(i,j,t)+cE⋅∇E2u(i,j,t)+cW⋅∇W2u(i,j,t)],
where(3.5)∇N2u(i,j,t)=u(i-1,j,t)-u(i,j,t),∇S2u(i,j,t)=u(i+1,j,t)-u(i,j,t),∇E2u(i,j,t)=u(i,j+1,t)-u(i,j,t),∇W2u(i,j,t)=u(i,j-1,t)-u(i,j,t).
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(3.6)cN(i,j,t)=g(|∇N2u(i,j,t)|),cS(i,j,t)=g(|∇S2u(i,j,t)|),cE(i,j,t)=g(|∇E2u(i,j,t)|),cW(i,j,t)=g(|∇W2u(i,j,t)|),
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.


## 4. The Fractional-Order PM Diffusion (FPMD)

The FPMD is developed using G-L fractional-order derivative, which is defined as [38](4.1)Dαg(x)=lim⁡h→0+⁡∑k≥0(-1)kCkαg(x-kh)hα,α>0,
where g(x) is a real function, α>0 is a real number, Ckα=Γ(α+1)/[Γ(k+1)Γ(α-k+1)] is the generalized binomial coefficient and Γ(·) denotes the Gamma function. If h=1, the finite fractional difference is(4.2)∆αg(x)=∑k=0K-1(-1)kCkαg(x-k).An imageU will be a 2-dimensional matrix of size N×N and its discrete fractional-order gradient ∇αu is an 8-dimensional vector:(4.3)∇αu(i,j)=(∇0αu(i,j),∇1αu(i,j),∇2αu(i,j),∇3αu(i,j),∇4αu(i,j),∇5αu(i,j),∇6αu(i,j),∇7αu(i,j))T,
where T represents the transpose of the vector and ∇αuk(i,j),k=0,…,7 are defined as(4.4)∇0αu(i,j)=∑k=0K-1(-1)kCkαu(i,j+k),∇1αu(i,j)=∑k=0K-1(-1)kCkαu(i-k,j+k),∇2αu(i,j)=∑k=0K-1(-1)kCkαu(i-k,j),∇3αu(i,j)=∑k=0K-1(-1)kCkαu(i-k,j-k),∇4αu(i,j)=∑k=0K-1(-1)kCkαu(i,j-k),∇5αu(i,j)=∑k=0K-1(-1)kCkαu(i+k,j-k),∇6αu(i,j)=∑k=0K-1(-1)kCkαu(i+k,j),∇7αu(i,j)=∑k=0K-1(-1)kCkαu(i+k,j+k).Thus(4.5)∇2αu(i,j)=(∇02αu(i,j),∇12αu(i,j),∇22αu(i,j),∇32αu(i,j),∇42αu(i,j),∇52αu(i,j),∇62αu(i,j),∇72αu(i,j))T,
where T represents the transpose of the vector. From (4.3), we have(4.6)∇02αu(i,j)=∑k=0K-1(-1)kCkα∇0αu(i,j+k),∇12αu(i,j)=∑k=0K-1(-1)kCkα∇1αu(i-k,j+k),∇22αu(i,j)=∑k=0K-1(-1)kCkα∇2αu(i-k,j),∇32αu(i,j)=∑k=0K-1(-1)kCkα∇3αu(i-k,j-k),∇42αu(i,j)=∑k=0K-1(-1)kCkα∇4αu(i,j-k),∇52αu(i,j)=∑k=0K-1(-1)kCkα∇5αu(i+k,j-k),∇62αu(i,j)=∑k=0K-1(-1)kCkα∇6αu(i+k,j),∇72αu(i,j)=∑k=0K-1(-1)kCkα∇7αu(i+k,j+k).Let(4.7)g=(g0,g1,g2,g3,g4,g5,g6,g7)T,
where T represents the transpose of the vector and gk, k=0,…,7 is defined as(4.8)gk=g(|∇kαu(i,j)|)∑n=07g(|∇nαu(i,j)|),k=0,1,…,7,
where ∇kαu(i,j),k=0,…,7, defined in (4.3) are the components of vector ∇αu(i,j) and ∑n=07g(|∇nαu(i,j)|) is the normalized constant, g is the decreasing function of absolute value of ∇kαu(i,j),k=0,…,7. Following (2.2) and (2.3), g(|∇ukα(x,y,t)|) can be defined as(4.9)g(|∇ukα(x,y,t)|)=e-(|∇ukα(x,y,t)|/σ)2,k=0,…,7
or(4.10)g(|∇ukα(x,y,t)|)=11+(|∇ukα(x,y,t)|/σ)2,k=0,…,7,
where |·| is the absolute value of the number and the constant σ controls the sensitivity to edges.The new FPMD based on G-L fractional-order derivative is defined as(4.11)∂u(i,j,t)∂t=div⁡⁡(g0∇0αu(i,j,t)g1∇1αu(i,j,t)g2∇2αu(i,j,t)g3∇3αu(i,j,t)g4∇4αu(i,j,t)g5∇5αu(i,j,t)g6∇6αu(i,j,t)g7∇7αu(i,j,t)),
where the ∇kαu(i,j,t), k=0,…,7, are the components of vector ∇αu(i,j,t) in (4.3) and gk, k=0,…,7, defined in (4.8) are the components of g in (4.7).The above equation can be represented as(4.12)∂u(i,j,t)∂t=∑k=07gk∇k2αu(i,j,t),
where ∑k=07gk=1 and ∇k2αu(i,j,t) can be computed according to (4.5).Thus the explicit form for solving (4.12) is(4.13)u(i,j,t+1)=u(i,j,t)+∑k=07gk∇k2αu(i,j,t),
where u(i,j,t+1) is the gray level of (i,j) at time t+1 and gk, ∇k2αu(i,j,t) are the same as in (4.12).


## 5. 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-dimensional256×256 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 fi=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(5.1)1256×256∑i=1256∑j=1256(frecon(i,j)-fPh(i,j))2,
where frecon(i,j) denotes the reconstructed value on position (i,j) while fPh(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),16×1.5mm; 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 withα=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.Table1 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.Table 1
RMSE of different smoothing methods.
NoisyMedianWlenerGaussianPMDFPMDFPMDFPMDimageFilterFilterFilterα=0.2α=0.5α=1.5RMSE0.09620.08040.06340.09630.07740.06030.07350.0752Comparing all the original SDCT images and L-CT images in Figures1 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.Shepp-Logan phantoms. (a) Original ideal reconstructed phantom. (b) Simulated LDCT reconstructed phantom. (c) LDCT reconstructed phantom processed by5×5 median filter. (d) LDCT reconstructed phantom processed by 5×5 wiener filter. (e) LDCT reconstructed phantom processed by Gaussian smoothing with [σ=1.8,μ=0]. (f) LDCT reconstructed phantom processed by PMD with [σ=2]. (g) LDCT reconstructed phantom processed by FPMD with [σ=2,α=0.2]. (h) LDCT reconstructed phantom processed by FPMD with [σ=2,α=0.5]. (i) LDCT reconstructed phantom processed by FPMD with [σ=2,α=1.5].
(a)(b)(c)(d)(e)(f)(g)(h)(i)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 by5×5 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].
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 6.  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).



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