Image artifacts and noise reduction algorithm for cone-beam computed tomography with low-signal projections

Abstract

This study aims to investigate and test a new image reconstruction algorithm applying to the low-signal projections to generate high quality images by reducing the artifacts and noise in the cone-beam computed tomography (CBCT). For the low-signal and noisy projections, a multiple sampling method is first utilized in projection domain to suppress environmental noise, which guarantees the accuracy of the data for reconstruction, simultaneously. Next, a fuzzy entropy based method with block matching 3D (BM3D) filtering algorithm is employed to improve the image quality to reduce artifacts and noise in image domain. Then, simulation studies on polychromatic spectrum were performed to evaluate the performance of the proposed new algorithm. Study results demonstrated significant improvement in the signal-to-noise ratios (SNRs) and contrast-to-noise ratios (CNRs) of the images reconstructed using the new algorithm. SNRs and CNRs of the new images were averagely 40% and 20% higher than those of the previous images reconstructed using the traditional algorithms, respectively. As a result, since the new image reconstruction algorithm effectively reduced the artifacts and noise, and produced images with better contour and grayscale distribution, it has the potential to improve image quality using the original CBCT data with the low and missing signals.

Keywords

Low-signal projections noise reduction fuzzy entropy block matching 3D filtering

1 Introduction

Cone-beam computed tomography (CBCT) plays a critical role in both medical imaging and industrial non-destructive testing with high efficiency and precision [1 –3], which acquires several projections by scanning an object with X-rays sent from different angles. With the impressive improvements in scanning speed and space resolution, CBCT has been developed extensively and widely.

However, to the turbine blades or other high-density shaped parts in industrial computed tomography (ICT) area, the projection signal received on the detector is relatively low even the maximum scanning voltage and power settings (such as the maximum voltage is 450 kV). The projections data from the industrial shaped parts are severely affected and degraded by kinds of factors, including Poisson noise [4, 5], logarithmic transformation of scaled measurements [6], and the attenuation based on structure and material. And in some cases, X-ray can penetrate the object hardly [7], or the signal is submerged in the noise [8, 9] etc., all the factors attached to the projection will lead to severe artifacts in reconstruction domain, which renders challenging work for noise reduction to maintain a high image quality. So it is necessary to improve the image by maintaining the original information integrity and to remove unnecessary artifacts. Up to now, many noise-reduction strategies have been proposed and utilized to improve the signal-to-noise ratio (SNR) of the CT slices in image domain through various algorithms [10 –12]. However, the noise that corresponding to the calibrated projection data after logarithm transform is signal-dependent [13]. One approach utilizes nonlinear filter to smooth the noise, though the low-pass filter might hold the contour of the image, the details of the image might vanish, and noise by spatially-invariant low-pass filters has shown ineffectiveness [14]. Moreover, the sophisticated noise filtering were proposed to improve the results [15, 16], but they are incapable of eliminating the artifacts perfectly, which usually gains noise reduction at the cost of resolution, such as Median filter, Gaussian filtering, Bilateral filtering, Gabor filter, Wiener filter, No-Local Means filter, etc. Another approach employs a penalty-weighted least squares (PWLS) model [17, 18] to reduce streak artifacts caused by limited CT projections and noise in either sinogram space or image domain [19, 20]. Although the result is improved reasonably, the penalty in space needs to be deliberately designed based on the investigation, and the model parameters are difficult to selected, which leads to more limitations. As the proceedings during a course of CT scanning, the noise-induced artifacts caused by the low-signal projections reconstruction are relative to both sinogram domain and image domain. The denoising algorithms mentioned above are simply operating the relations of different pixels in image domain, or focus on the model, the fidelity of low-signal projection is distorted and the details of image are also smoothed. Thus, there is a need for developing a denoising method which combines with regional features to/and hold image details in the domain.

To reduce the noise for low-signal projections of CBCT image rather than simply smooth the one in image domain, in this work, the noise suppression in both sinogram and image domain is investigated. And three steps are performed between the projection and image domain, including multiple sampling for average, fuzzy entropy and Block-Matching 3D (BM3D) filter [21, 22] algorithm. Based on the randomness of the noise and its irrelevant, the multiple sampling for average process can effectively suppress the noise without destroying the low-signal projection, which assures the data fidelity in projection for reconstruction. Besides, the fuzzy entropy method with Block Matching 3D (BM3D) filtering algorithm is used to suppress artifacts and noise in image domain, which is employed to distinguish the edge and flat area when the denoising is performed, simultaneously. The simulation experimental results have shown the effectiveness and feasibility of our proposed method.

2 Materials and methods

2.1 Noise model

The mission is to obtain the attenuation coefficients μ(s), where s represents the spatial location. The mono-energetic X-ray CT projection process is modelled by the Beer’s law [23]: $I_{i} = I_{0} e^{- \int_{L_{i}} μ (s) dl}, \begin{matrix} i = 1, \dots, N_{d} \end{matrix}$ (1) where I _i is the mean detected intensity, I ₀ is the initial intensity of X-ray beam, L _i states the path of the ray goes through the object and N _d is the total number of rays.

In CT imaging, Poisson statistics describes the behavior of the majority of the measurement noise. Thus, the complicate noise on projections of CT can be approximated by a Poisson model as: $Y_{i} \sim Poisson {{\hat{y}}_{i}} = Poisson {I_{0} e^{- \int_{L_{i}} μ (s) dl}}$ (2)

After the effective data correction algorithm, including scatter and beam hardening corrections, implemented on CBCT system, the residual errors on the projections are expected to be small as compared to the Poisson noise. In order to make reconstruction unimpeded, it is necessary to discretise a continuous object μ (s) by line integral values for each ray, which is equivalent to the post-log measurement. $y_{i} = log (\frac{I_{0}}{I_{i}}) = \int_{L_{i}} μ (s) dl \approx \sum_{j = 1}^{N_{d}} x_{i} \int_{L_{i}} μ_{j} (s) dl = \sum_{j = 1}^{N_{d}} p_{ij} x_{j}, \begin{matrix} i = 1, \dots, N_{d} \end{matrix}$ (3)

In the study, we investigated a limited number of projections here, and given image x with N × N pixels. Thus, the problem is equivalent to an undetermined linear system, where we need to apply iterative algorithm and find sensing matrix Φ as the partial Radon Transform matrix to recover it. In general, the mean and variance of the noise on projection are great, therefore, the Gaussian distribution is used instead of the Poisson distribution. Assuming that the condition is true, the post-log observation can be modelled by a weighted linear additive Gaussian noise model [24]: $y_{i} = {[Φ x]}_{i} + η_{i} \begin{matrix} , & η_{i} \end{matrix} \sim N (0, σ_{i}^{2})$ (4) where y _i is observation data, Φ means the system matrix of X-ray with the object, η_i states the Gaussian noise distribution, and σ² gives the noise variance.

2.2 Multiple projections in averaging process

Due to the influence of circuit noise, quantization error and other factors, a measurement value actually is the sum of the intensity of the ray itself and the noise signal, where the noise amplitude value is basically unchanged under different circumstances. Sometimes the information can be submerged by noise. It is necessary to take effective ways to make the useful signals appeared. The multi-image averaging process makes it possible to obtain useful information and submerge the noise down to the desired signal [25]. By accumulating the signal several times, the signal emerges.

Since the noise η (x, y) is independent of the image signal itself f (x, y) during the CT scanning, the noisy image g (x, y) uses the additive noise model, which can be expressed as Equation (5), where the coordinate (x, y) represents the pixel position of the image, that is: $g (x, y) = f (x, y) + η (x, y)$ (5)

Denoising is the process of approximating f (x, y) from the known g (x, y). To the numbers of same scene images, f_i (x, y) is the same, but the noise η_i (x, y) is random and unrelated to each other. Assuming that the noise at each coordinate point is irrelevant and its mean is zero, then the same scene of different samplings can be averaged to form the image $\bar{g} (x, y)$ : $\bar{g} (x, y) = \frac{1}{K} \sum_{i = 1}^{K} [f_{i} (x, y) + η_{i} (x, y)] = f (x, y) + \frac{1}{K} \sum_{i = 1}^{K} η_{i} (x, y)$ (6)

Since the noise is random and uncorrelated, the average image can be expected as $E {\bar{g} (x, y)} = f (x, y)$ (7) and the variance of the mean image becomes $σ_{\bar{g} (x, y)}^{2} = \frac{1}{K} σ_{η (x, y)}^{2}$ (8) where $E {\bar{g} (x, y)}$ is the expected value of $\bar{g} (x, y)$ , $σ_{\bar{g} (x, y)}^{2}$ and $σ_{η (x, y)}^{2}$ are the variance of $\bar{g} (x, y)$ and η_i (x, y), respectively. The standard deviation in the mean image at any point is as following: $σ_{\bar{g} (x, y)} = \frac{1}{\sqrt{K}} σ_{η (x, y)}$ (9)

From Equation (7), it is clearly that the mean value of the same scene is noise-free image, but there are some perturbations. The standard deviation of these disturbances in Equation (9) determines the intensity of noise. It is known that the noise can be reduced by increasing the K value (that is, increasing the average number of images). But it can be found that $σ \propto \frac{1}{\sqrt{K}}$ , and $\frac{\partial σ}{\partial K} = - \frac{1}{2 \sqrt{K^{3}}}$ , that is, the standard deviation σ changes smaller and smaller with the K value increased, so it is not enough for the role only by increasing the number of images.

2.3 Local fuzzy entropy process

Fuzzy entropy as a physical measure which reflects the fuzzy degree quantitatively is proposed by Shannon [26]. To apply Shannon’s entropy in information theory, the concept of image entropy is proposed, which is used to determine the edge and smooth area of single material object. For the entropy of the image, there are two definitions [27]: (1) histogram entropy (HE), which is defined according to the probability that the gray value appears; (2) gray entropy (GE), which is defined according to the distribution probability of the gray value in the total gray value. If a region of the image is defined, the local entropy is fixed and determined.

Taking a window with size n × n centered at (x, y) for the Universe, then the local coordination of the window can be denoted as: $W_{n} (x, y) = {[\begin{matrix} ⋮ & ⋮ & ⋮ \\ \dots & t (x - 1, y - 1) & t (x, y - 1) & t (x + 1, y - 1) & \dots \\ \dots & t (x - 1, y) & t (x, y) & t (x + 1, y) & \dots \\ \dots & t (x - 1, y + 1) & t (x, y + 1) & t (x + 1, y + 1) & \dots \\ ⋮ & ⋮ & ⋮ \end{matrix}]}_{n \times n}$ (10)

Based on the local information of the image, the GE has been used to deal with the image in the process, which is defined as follows [28]: $H (μ (x, y)) = μ_{M} (t (x, y)) \cdot {log}_{2} (μ_{M} (t (x, y))) + (1 - μ_{M} (t (x, y))) \cdot {log}_{2} (1 - μ_{M} (t (x, y)))$ (11)

For calculating the fuzzy entropy quickly and effectively, a fuzzy set with image local characteristics must be defined, that is, the subordinate degree. The grayscale of the local region is chosen as the feature of the gray distribution, which reflects the integrated characteristics of the image: $μ_{M} (t (x, y)) = \frac{1}{(1 + \frac{| t (x, y) - M |}{Const})}$ (12)

Moreover, to measure the properties of the fuzzy set defined on the image, a fuzzy entropy measure (FEM) is calculated, that is: $E_{M} (μ (x, y)) = \frac{1}{n \times n} \sum_{x = 1}^{n} \sum_{y = 1}^{n} H (μ (x, y))$ (13) according to the theory, the entropy of the image in each neighborhood is calculated as the contribution to the FEM, then the value E_M (μ (x, y)) is assigned to the pixel which is located at (x, y) as the result of image. The process is showed in Fig. 1.

where:

t (x, y) is the grayscale which is located at (x, y); M = max(max(t (x, y))) is the maximum grayscale, n is the window patches in row and column of the local neighborhood

H (•) defines the gray entropy operator based on the subordinate degree;

μ( t (x, y)) denotes the subordinate degree operator, which reflects the degree of neighborhood pixel t (x, y) that belonging to fuzzy set; Const is an empirical constant that is used to adjust the degree of attachment.

Fig.1

Fuzzy entropy image processing.

2.4 Architecture of BM3D algorithm

The input for the BM3D is the observed image f (x, y) with the standard deviation σ_n of the White Gaussian Noise (WGN) η (x, y). BM3D filter divides the image into small blocks and groups the blocks according to its similarity. For blocks in the same group, the filter enhances the similarity between each blocks and calculates the value of pixel by weighted averaging all blocks which contains that pixel [29].

With a fixed block size N_mat , block-matching searches and groups the blocks by whose distance to the reference one is smaller than the fixed threshold, which is calculated as following: $R (P) = {Q : d (P, Q) \leq τ^{hard}}$ (14)

Then 3D transform T_3D is applied on the matched groups G_R and the estimate image blocks X_basic can be obtained as $X_{basic} = T_{3 D}^{- 1} (Θ_{f} (T_{3 D} (G_{R})))$ (15) $Θ (x) = {\begin{matrix} 0 \begin{matrix} if | x | \leq γ^{hard} σ \end{matrix} \\ x \begin{matrix} \begin{matrix} otherwise \end{matrix} \end{matrix} \end{matrix}$ (16)

Following the basic estimation obtained above, the denoising can be improved by performing grouping within basic estimate and collaborative empirical Wiener filtering. And the estimate image blocks X_wiener is produced as $X_{wiener} = T_{3 D}^{- 1} (Θ_{wiener} (T_{3 D} (G_{R})))$ (17)

Final estimate X_final is obtained from the block-wise estimates by a weighted average of all local block estimates.

where:

τ ^hard is the distance threshold for d (•) under which two patches are assumed similar;

$d (P, Q) = \frac{{∥ Θ_{r} (T_{2 D} (P)) - Θ_{r} (T_{2 D} (Q)) ∥}_{2}^{2}}{N_{mat}^{2}}$ is the normalized quadratic distance between blocks P and Q which located at x _P and x_Q ;

Θ _r d enotes the thresholding operator with hard threshold $γ_{2 D}^{hard} σ$ ; Θ _f denotes the thresholding operator with hard threshold $γ_{3 D}^{hard} σ$ ; Θ _wiener is the empirical Wiener shrinkage coefficients;

T_2D is the sparse transform operator such as discrete wavelet transform (DWT), discrete cosine transform (DCT) and discrete Fourier transform (DFT); T_3D is the 3D transform which is made up of two transforms: 2D transform denoted by T_2D applied on each block, and 1D transform applied along the third dimension of the 3D group.

σ ² is the variance of the zero-mean Gaussian noise.

Thus, the algorithm is divided in two major steps, which is summarized in Fig. 2.

Fig.2

Scheme of the BM3D algorithm.

Step1: Estimates the denoised image using hard thresholding during the collaborative filtering;

Step2: Based both on the original noisy image and the estimate image X_basic obtained in step 1, the Wiener filtering is used.

To sum up, the whole process for the low-signal projections can be summarized as follows:

Obtaining the projection information, and determining the low-signal projections position by a setting threshold which usually is the three times the standard deviation;

Based on the position knowledge of low-signal projections, resampling the projection K times at the same position, then summing and averaging them;

Reconstructing the projection which has been denoised and obtaining the slice image;

Calculating the local entropy H(x, y) of the slice images (formula (11), and filtering the images with BM3D filter.

For the Fuzzy entropy process, it is used to determine the edge and smooth area of single material object, thus it controls the anisotropic degree of noise in image. And BM3D algorithm takes the combined filtering method to estimate the noise standard deviation and use it as the filter parameter, and it can achieve the purpose of removing noise and preserving the details of the image. The combination of the methods can improve the quality and reduce the noise performance reasonably.

3 Results and discussion

In this section, numerical studies and simulations are carried out on polychromatic spectrum to evaluate the performance. Considering the conditions as in practical circumstance, a polychromatic spectrum is taken into account, and the tube voltage is set to be 140 kV. The projection image is of 256×256 array and the Gaussian noise has been included in the simulation. According to the “3σ principle”, the projection value which is less than 3σ (σ is the standard deviation of background) is considered as an abnormal value (low-signal projections). Therefore, the low-signal position is determined with P < 3σ as the threshold. In the process, the number of the sampling projections at the position where the ray can hardly goes through (low-signals) is 20. And the distance from the detector to the ray source is 600 mm and to the rotation center is 100 mm. Table 1 illustrates the simulation parameters used in CBCT system.

Table 1
Imaging and reconstruction parameters used in CBCT system

Scan Parameters Term Value

Material Iron-blade

X-ray energy 140 kV (polychromatic spectrum)

K(sampling numbers) 20

Distance from source to detector 600 mm

Distance from center to detector 100 mm

Rotation circular, 360 deg.

Number of views 90

Reconstruction voxel size 0.4 mm

Scan Parameters	Term Value
Material	Iron-blade
X-ray energy	140 kV (polychromatic spectrum)
K(sampling numbers)	20
Distance from source to detector	600 mm
Distance from center to detector	100 mm
Rotation	circular, 360 deg.
Number of views	90
Reconstruction voxel size	0.4 mm

Fig. 3 gives the performance of the experiment which is using industrial parts with single-material of iron, and the number of views is 90 which evenly spanned on [0 π). Fig. 3 (a) shows the sampled diagram rwhere the range from the position 1 to the position 2 is part of the low-signal projections sampling angles. The sinogram that low-signal projections below the Gaussian noise in local position 1 to 2 is shown in Fig. 3(b) rand there is a missing data for the continuous sampling position in the domain rwhich reflects that the blade projection information is buried at successive sampling angles.

Fig.3

CT scan simulation. (a) is the CT simulated geometry rand the position 1 to 2 shows the multiple sampling ranges; (b) is the sinogram and the low-signal projections in the region.

Based on the presented studies rthe results are obtained from the described algorithm. Fig. 4 shows the results of the 60th projection with multiple sampling for average. The white region inner the projection in Fig. 4 (a) is the part that the ray goes through the piece hardly rthat is rthe projection has been buried and the noise is higher than the projection value. According to the algorithm mentioned in section 2.2 rthe low-signal projections are multiple sampled and averaged. Fig. 4 (b) ∼ (d) show the images before and after the processing. Fig. 4 (b) is the noiseless projection in area-Ω rand Fig. 4 (c) is the noisy image rFig. 4 (d) gives the results based on the method. It is clear that the projection information is preserved and the projection noise is suppressed. And the standard deviation of the projection image down to $1 / \sqrt{20}$ times (K = 20) rwhen the projection data is preserved rcompletely.

Fig.4

Projection from multiple sampling for average. (a) shows the low-signal projections distribution in one sampling; (b) is the no noise projection in domain Ω; (c) is the low-signal projections with noise in the domain; (d) shows the multiple average projection with noise in the domain.

To further improve the image quality rthe image is reconstructed and is filtered in reconstruction domain. Images in different positions are selected to show the results. Fig. 5 shows the slice with CS-based algorithm. The denoising algorithm BM3D and the Entropy have been used in the process rwhich produces the state-of-art recovery for images. To illustrate the effectiveness of the algorithm rtwo slices are selected randomly and arbitrarily rwhich reflects the thickness and structure at different positions.

Fig.5

CBCT image reconstruction. Display window: –0.02, 0.8]. (a),(e) are the reconstructed image with low-signal mixed noise in some ranges; (b),(f) are the reconstructed image with multiple projections process; (c),(g) show the image with fuzzy entropy method; (d),(h) show the image in (c)and(g) with the BM3D filter. (a) ∼ (d) show the 128th slice image and (e) ∼ (h) show the 240th slice image.

In Fig. 5, (a) ∼ (d) state the 128th slice images, and Fig. 5 (a) gives the result from the projection with adding Gaussian noise, it is found that there are serious artifacts in the low-signal projections sampling position, and the contour edge is poor in some region, which conforms to the result of the low-signal projections. Fig. 5 (b) is the result from multiple projections process, and it is easy to see that the image quality has been improved. Fig. 5 (c) is the image using fuzzy entropy (FE) to Fig. 5 (b) according to the algorithm in section 2.3, and the noise of the filtered image is well suppressed. The contours are maintained, but the reconstructed artifacts are not removed enough. Thus, the image after fuzzy entropy processing is subjected to BM3D filtering again, and the artifacts of the contours are improved obviously. Fig. 5 (d) shows the final result. The second row in Fig. 5 (e) ∼ (h) shows the 240th slice images with the same pattern.

The zoomed details are illustrated in Figs. 6 and 7. It can be seen that both FE and BM3D-FE yield a better result of the structure for the image than the one that uses the low-signal projections to reconstruct directly. Fig. 6 (a) ∼ (d) are the zoomed map of region A, and Fig. 7 (a) ∼ (d) are the zoomed map of region B in Fig. 5 (a). Line AB and CD are the profile location, the values have been shown in (e) both Figs. 6 and 7. It is obvious that the quality of the processed image is better than that of the previous one; meanwhile, the artifacts have been suppressed and the contour clarity has also been improved.

Fig.6

Image detail in domain A. (a) ∼ (d) are correspond to (a) ∼ (d) in Fig. 5.

Fig.7

Image detail in domain B. (a) ∼ (d) are correspond to (a) ∼ (d) in Fig. 5.

In order to assess the quality of denoising, four typical filtering methods, including Gaussian filtering, Bilateral filtering, NLM filtering and BM3D filtering, are compared to process the reconstructed image. Figs. 8 and 9 show comparisons of the 128th and 240th slice image, respectively.

Fig.8

Comparison of the 128th slice images with different filter. (a)∼(c) are the slice images with projection denoising by multiple sampling for average, and the images filtered by Gaussian filtering, bilateral filtering and NLM filtering, respectively. (d) is the image of proposed method with BM3D filtering. (e) is the profile comparison along the line EF. Display window: [–0.02, 0.7].

Fig.9

Comparison of the 240th slice images with different filter. (a)∼(c) are the slice images with projection denoising by multiple sampling for average, and the images filtered by Gaussian filtering, bilateral filtering and NLM filtering, respectively. (d) is the image of proposed method with BM3D filtering. (e) is the profile comparison along the line MN. Display window: [–0.02, 0.7].

Figs. 8 and 9 illustrate that the slice images in (c) and (d) have a higher edge contrast than that of in (a) and (b). For the denoising ability, all filtering methods can reduce the noise level to the images, but BM3D filtering has better visual results, and the artifacts decrease obviously, compared with (a), (b) and (c) in Figs. 8 and 9. Moreover, it can be seen from the curve that the grayscale obtained by new algorithm is even and smooth, in contrast, other algorithms have a large grayscale fluctuations. Combined with the property of single material blade to be measured, it is well known that the more uniform the grayscale, the high quality the image. To evaluate the reconstructed image quality and illustrate the effectiveness and practicability of this algorithm, a further more quantitative description and comparison are given. In this paper, Contrast to Noise Ratio (CNR) [30] and Average Gradient (AG) [31] in region C are computed, and Signal Noise Rate (SNR) in region D is performed, which are used the similar definition as in the literature, respectively. To evaluate the performance, the local area for SNR is selected inside of the object, and the whole image quality is based on the local one. SNR reflects the noise level of the reconstructed image. The larger the SNR, the better the image quality. AG is used to characterize the contour clarity of the part, the larger the AG is, the clearer the detail of the image.

Table 2 shows the numerical results that both SNRs and CNRs increase obviously, and CNRs are improved sharply. Because the AG cannot distinguish the noise and details clearly, AGs have shown different value in different method. What is more, SNR and CNR of BM3D filtering are both higher than that of the others, which not only decreases the noise, but also holds the detail by block matching. Therefore, BM3D filtering has better performance in denoising process. While, the reconstruction results are slightly different for slices at different position. The factors are complicated, including scattering, beam hardening and the scanning settings (current, voltage and exposure), which are closely related to the material and structure of the object. But those questions are not studied in this paper.

Table 2

Numerical comparison of image with different methods

		Evaluation
Slice	Filtering Method	SNR		CNR		AG
		Value	Improved/%	Value	Improved/%	Value	Improved/%
128th	Previous	8.7280	/	4.3227	/	0.0547	/
	Gaussian	9.9511	14.01	4.2214	–2.34	0.0421	–23.03
	Bilateral	11.2671	29.09	4.4507	2.96	0.0453	–17.18
	NLM	12.4022	42.10	5.2781	22.10	0.0497	–9.14
	BM3D	13.6464	56.35	5.5783	29.05	0.0489	–10.01
240th	Previous	11.0527	/	4.7341	/	0.0514	/
	Gaussian	11.7718	6.50	4.7823	1.02	0.0352	31.51
	Bilateral	12.8791	16.52	4.8746	2.97	0.0401	21.98
	NLM	13.9473	26.19	5.3157	12.29	0.0474	7.78
	BM3D	15.2245	37.74	5.5925	18.13	0.0429	16.53

What needs to be reiterated here is that multiple sampling process is an effective means of reducing noise to guarantee the original projection information. Theoretically, the more samplings, the more noise reduction, but it does not means that there needs a large number of projections. It can be found that the standard deviation $σ \propto \frac{1}{\sqrt{K}}$ , that is, the σ changes smaller and smaller with the K value increasing. Moreover, the stepper rotary table in arc trajectory can be commanded by the code, and based on the detector integration mode, the integration timing can be set in hundreds of milliseconds which are almost unaffected. Therefore, the time spent by its multiple sampling is small, and it does not increase the extra time due to the scan sample. The feasibility of the method might be developed in practical use.

As we know, the projection sampling by CBCT system is associated with many factors. From projection domain to reconstruction domain, various factor have affected the quality of the image. It may be different but important, such as X-ray dose, structure, materials, as well as sampling strategy, etc., all of which can be an area of research. Here we only study an effective method to improve the noise projections and artifacts image for high quality image in certain conditions. And, the image has been significantly improved with our proposed method on CBCT, which shows the proposed method has good robustness and practicality.

4 Conclusion

In this article, we proposed a high quality image acquisition algorithm for CBCT using low-signal projections which has been masked by noise. The improvements mainly include three aspects. First, the multiple sampling for average process has been utilized for low-signal projections to suppress the noises in projection domain, which is to guarantee the accuracy of the data source. Second, the fuzzy entropy is used as a function for keeping the detail of reconstructed image. The greater the local entropy, the more likely it will be in the smooth region. The smaller the local entropy is, the more likely the edge is. Third, BM3D filter denoising step for slice images is performed, which improves denoising performance when the image detail is preserved. The experimental results show that the proposed method can achieve a good quality of low-signal projections, which is based on grayscale, SNRs and CNRs.

Footnotes

Appendix: The evaluation to the reconstructed image

The signal to noise ratio (SNR), contrast to noise ratio (CNR), and the average gradient (AG), are measured as the evaluation of the reconstructed image in selected regions, respectively.

Signal to noise ratio (SNR):

SNR is commonly used in image quality, which reflects the ratio of signal intensity to noise intensity in image. It is mainly used to evaluate the degree of image noise. The higher the value, the better the image quality. The mathematical definition is as follows: (A1) $SNR = 10 \times {log}_{10} (\frac{μ_{ROI}}{S_{ROI}})$

The μ_ROI and S_ROI are calculated as (A2) ${\begin{matrix} μ_{ROI} = \frac{1}{N} \sum_{i = 1}^{N} A (r_{i}) \\ S_{ROI} = \frac{1}{N} \sum_{i = 1}^{N} [A (r_{i}) - μ_{ROI}]^{2} \end{matrix}$ where μ_ROI and S_ROI represent the Mean and Standard Deviation of the region of interest, respectively. A (r_i) is the gray value of the region of interest, and N is the number of pixels in the region.

Contrast-to-Noise Ratio (CNR):

CNR is the ratio of the difference of tissue to the noise standard deviation in background, which reflects the contrast of image in ROI and background. The higher the value, the clearer the image, which means the larger the difference between the background area and the image in ROI. The mathematical definition is as follows: (A3) $CNR = \frac{| μ_{ROI} - μ_{background} |}{\sqrt{\frac{S_{ROI}^{2} + S_{BG}^{2}}{2}}}$ where μ_ROI and μ_background represent the Mean of ROI and background region, respectively. S_ROI and S_BG represent the Standard Deviation of ROI and background region, respectively.

Average Gradient (AG):

AG is used to characterize the contour clarity of the part. The larger the AG value is, the more detailed the information is. For the region of N×M array, AG is defined as (A4) $AG = \frac{\sum_{y = 1}^{M} \sum_{x = 1}^{N} \sqrt{\frac{Δ I_{x} (x_{i}, y_{j})^{2} + Δ I_{y} (x_{i}, y_{j})^{2}}{2}}}{M \times N}$ where ΔI_x (x_i, y_j) and ΔI_y (x_i, y_j) represent the first derivative of the pixel (x_i, y_j) along the x- and y direction, which is computed as (A5) $\begin{matrix} Δ I_{x} (x_{i}, y_{j}) = I (x_{i}, y_{j}) - I (x_{i}, y_{j}) \\ Δ I_{y} (x_{i}, y_{j}) = I (x_{i}, y_{j + 1}) - I (x_{i}, y_{j}) \end{matrix}$

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 51675437, 51605389), the Natural Science Foundation research project of Shaanxi Province (Grant No. 2016JM5003), and the Graduate Starting Seed Fund of Northwestern Polytechnical University (Grant No. Z2017022).

References

Huang

K.D.

, Zhang

, Shi

Y.K.

, Zhang

and Xu

, Scatter correction method for cone-beam CT based on interlacing-slit scan, Chin Phys B 23(9) (2014), 098106–098107.

Liu

, Huang

and Li

, Cone-beam CT reconstruction along any orientation of interest, Journal of X-ray Science and Technology 23(6) (2015), 773–782.

Alsbou

, Ahmad

and Ali

, A motion algorithm to extract physical and motion parameters of mobile targets from cone-beam computed tomography images, Journal of X-ray Science and Technology 24(4) (2016), 599–613.

Zhao

M.L.

and Li

H.W.

, Image denoising based on Poisson-like noise model, Chinese Journal of Stereology & Image Analysis (2012).

Kais

, Noureddine

and Chedly

F.M.

, Comparative analysis between a variational method and wavelet method PURE-LET to remove poisson noise corrupting CT images, International Multi-Conference on Systems, Signals & Devices. IEEE (2016), 287–294.

H.B.

, Li

, Xiao

Y.C.

and Liang

Z.R.

, Analytical noise treatment for low dose CT projection data by weighted least square smoothing in the K-L domain, Chinese Journal of Medical Physics 19(4) (2002), 201–208.

Yang

F.Q.

, Zhang

D.H.

, Huang

K.D.

, Wang

and Xu

, Review of reconstruction algorithms with incomplete projection data of computed tomography, Acta Phys Sin 63(5) (2014), 058701.

Liu

, Gui

and Zhang

, Noise reduction for low-dose X-ray CT based on fuzzy logical in stationary wavelet domain, Optik - International Journal for Light and Electron Optics 124(18) (2013), 3348–3352.

Zhang

, Liu

, Han

, Wang

, Ma

J.H.

, Li

L.H.

and Liang

Z.R.

, A comparison study of sinogram- and image-domain penalized re-weighted least-squares approaches to noise reduction for low-dose cone-beam CT, Prof of SPIE Medical Imaging (2013), 86683E.

10.

Mahmood

, Shahid

, Vandergheynst

and Ulf

, Graph Based Sinogram Denoising for Tomographic Reconstructions; 2016.

11.

Karimi

and Ward

, A denoising algorithm for projection measurements in cone-beam computed tomography, Computers in Biology & Medicine 69 (2016), 71–82.

12.

Tang

, Yang

and Tang

, Characterization of imaging performance in differential phase contrast CT compared with the conventional CT–noise power spectrum NPS(k), Medical Physics 38(7) (2011), 4386.

13.

Gui

Z.G.

and Liu

, Noise reduction for low-dose X-ray computed tomography with fuzzy filter, International Journal for Light and Electron Optics 123(13) (2012), 1207–1211.

14.

, Li

, Wang

, Wen

, Lu

, Hsieh

and Liang

, Nonlinear sinogram smoothing for low-dose X-ray CT, IEEE Trans Nucl Sci 51(5) (2004), 2505–2513.

15.

Baek

and Pelc

N.J.

, Effect of detector lag on CT noise power spectra, Medical Physics 38(6) (2011), 2995.

16.

Wang

, Lu

, Li

and Liang

, Sinogram noise reduction for low-dose CT by statistics-based nonlinear filters, Proc SPIE Med Imag 5747 (2005), 2058–2066.

17.

, Tong

Y.L.

and Xiao

X.P.

, Adaptive fuzzy enhancement algorithm of surface image based on local discrimination via grey entropy, Procedia Engineering 15 (2011), 1590–1594.

18.

Gui

Z.G.

and Liu

, Noise reduction for low-dose X-ray computed tomography with fuzzy filter, International Journal for Light and Electron Optics 123(13) (2012), 1207–1211.

19.

Bian

, Ma

, Tian

and Huang

, Penalized weighted alpha-divergence approach to sinogram restoration for low-dose X-ray computed tomography, Nuclear Science Symposium and Medical Imaging Conference. IEEE (2012), 3675–3678.

20.

Wang

, Li

, Lu

and Liang

, Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose X-ray computed tomography, IEEE Transactions on Medical Imaging 25(10) (2006), 1272–1283.

21.

Niu

Y.M.

, Cui

, Yang

and Wu

, RL algorithm for passive millimeter wave imaging based on BM3D, Applied Mechanics & Materials 414 (2013), 1155–1158.

22.

, Chen

and Xing

, Multi-segment limited-angle CT reconstruction via a BM3D filter, (2012), 2390–2394.

23.

Alessandro

and Mike

, Compressive computed tomography image reconstruction with denoising message passing algorithms, European Signal Processing Conference, (2015).

24.

Gonzalez

R.C.

, Woods

Richard E.

, Digital image processing. Beijing: Electronic Industry Press; 2011, 88–91.

25.

Wang

, Li

, Lu

and Liang

, Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose x-ray computed tomography, IEEE Trans on Medical Imaging 25(10) (2006), 1272–1283.

26.

Singh

P.K.

, Cherukuri

A.K.

and Li

, Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy, International Journal of Machine Learning & Cybernetics 6(1) (2015), 1–11.

27.

Chou

, Lookabaugh

and Gray

R.M.

, Entropy-constrained vector quantization, IEEE Transactions on Acoustics, Speech, and Signal Processing 37(1) (2003), 31–42.

28.

Tao

, Jin

and Liu

, Object segmentation using ant colony optimization algorithm and fuzzy entropy, Pattern Recognition Letters 28(7) (2007), 788–796.

29.

Kostadin

, Alessandro

, Vladimir

and Karen

, Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Transactions on Image Processing 16(8) (2007), 2080–2095.

30.

Zhang

, Huang

K.D.

, Shi

Y.K.

and Li

M.J.

, Ring artifact correction method based on air scan for cone-beam CT, Computerized Tomography Theory & Applications (2012).

31.

Huang

K.D.

, Zhang

D.H.

, Li

M.J.

and Zhang

, Image lag modeling and correction method for flat panel detector in cone-beam CT, Acta Physica Sinica 62(21) (2013), 210702–118.