Sparse angle CT reconstruction based on group sparse representation

Abstract

OBJECTIVE:

In order to solve the problem of image quality degradation of CT reconstruction under sparse angle projection, we propose to develop and test a new sparse angle CT reconstruction method based on group sparse.

METHODS:

In this method, the group-based sparse representation is introduced into the statistical iterative reconstruction framework as a regularization term to construct the objective function. The group-based sparse representation no longer takes a single patch as the minimum unit of sparse representation, while it uses Euclidean distance as a similarity measure, thus it divides similar patch into groups as basic units for sparse representation. This method fully considers the local sparsity and non-local self-similarity of image. The proposed method is compared with several commonly used CT image reconstruction methods including FBP, SART, SART-TV and GSR-SART with experiments carried out on Sheep_Logan phantom and abdominal and pelvic images.

RESULTS:

In three experiments, the visual effect of the proposed method is the best. Under 64 projection angles, the lowest RMSE is 0.004776 and the highest VIF is 0.948724. FSIM and SSIM are all higher than 0.98. Under 50 projection angles, the index of the proposed method remains achieving the best image quality.

CONCLUSION:

Qualitative and quantitative results of this study demonstrate that this new proposed method can not only remove strip artifacts, but also effectively protect image details.

Keywords

Computed tomography imaging sparse angle group sparse representation dictionary learning

1 Introduction

Computed tomography (CT) reconstruction technology has been widely used in clinical diagnosis. However, the human body is exposed to a large dose of radiation during CT scanning, and excessive radiation dose can negatively affect the health of patients, which may lead to hereditary diseases or cancer [1]. Therefore, people pay more and more attention to the problem of X-ray radiation. Significantly reducing the dose of radiation while ensuring image quality is the focus of current low-dose CT research.

Reducing the projection number of each rotation of the human body [2] is a feasible strategy to reduce the radiation dose of CT, however this will lead to degraded reconstructions. Up to now, a variety of CT image reconstruction methods with sparse viewing angle have been proposed to improve the reconstructed images.

Iterative algorithms can effectively overcome this problem, such as algebraic reconstruction algorithm (ART) [3], simultaneous algebraic reconstruction technique (SART) [4] can effectively improve image quality. However, when the projection number is highly sparse, the improvement of image quality is limited. Therefore, the quality of the reconstructed image can be further improved by introducing appropriate prior information into the iterative algorithm.

In recent years, the theory of compressed sensing (CS) has gradually emerged and has been used to reconstruct images from projection data with less viewing angles [5]. The premise of CS theory is the sparsity of signals. Since most of the signals are not sparse, the key step of CS is to find a suitable sparse transform for the signal. The commonly used sparse transforms are total variation (TV) [6], wavelet transform (WT) [7], dictionary learning (DL) [8]. Sidky et al. proposed a convex set projection algorithm called TV-POCS [9] by minimizing the TV of the image under the assumption that the image is piecewise constant. Ritschl et al. proposed an improved TV method within the framework of ASD-POCS, which can effectively suppress artifacts [10]. Li Yu et al. combined TV with gradient domain convolution sparse coding for sparse angle image reconstruction, and the algorithm achieved good results in restoring small details and suppressing noise [11]. In order to solve the problem of noise fluctuation and sharp edge recognition, Fu et al. use neighborhood image blocks instead of single pixel to calculate non-quadratic penalty, and proposes a TV regularized image reconstruction algorithm based on image blocks [12].

Sparse representation based on dictionary is another achievement of CS, which promotes the development of sparse representation in low-dose CT reconstruction. Xu et al. applied DL to low-dose CT reconstruction for the first time, and proposed two DL methods: adaptive dictionary and fixed dictionary, which achieved good results under different conditions [8]. However, since updating sparse codes and dictionaries is usually iterative, it takes a long time to complete complex calculations. Zheng et al. combine the traditional penalty weighted least squares (PWLS) and sparse transformation based on pre-learning to solve the problem of noise and artifacts in low-dose CT reconstruction [13].

However, in the sparse representation based on DL, each patch is operated separately, and the non-local relationship between patches is seriously ignored, which may lead to great differences in the sparse coding of similar patches. At the same time, the heavy computing burden is another deficiency. In order to make up for the above shortcomings, many scholars have proposed group-based sparse representation (GSR) methods [14 –16]. This method no longer uses a single image block as the minimum unit of sparse representation, but clusters similar patches into groups as the basic unit of sparse representation. This method enhances both local sparsity and non-local self-similarity of images. Zhang et al. apply this method to image restoration and image deblurring, and verify the effectiveness of this method through a number of experiments [14]. Yang et al. used group sparse priors for reconstruction of electrical impedance images [16]. Bao et al. took group sparse as a regularization term and proposed a GSR-SART image reconstruction algorithm for CT image reconstruction with few viewing angles [17].

Inspired by studies above, a sparse angle CT reconstruction algorithm based on group sparse representation is proposed, to solve the problem of image quality degradation of CT reconstruction under sparse angle projection. GSR sparsely represents the image in the group domain, fully considers the inherent local sparsity and non-local self-similarity of the image, forces similar patches in the group to accept similar sparse decomposition, and effectively solves the problem of inaccurate sparse coding in the process of patch-by-patch processing. The GSR is applied to the statistical iterative reconstruction framework as a regularization term, and the solution is solved alternately. The experiment results show that the proposed method has good performance in removing strip artifacts and preserving image details.

2 Related works

2.1 CT Reconstruction model

Under the assumption of single energy beam, the X-ray CT reconstruction model can be expressed by the following discrete linear system [18]: $P = AX,$ (1) where the vector P ∈ R^I×1 represents the projection data measured at different projection angles. Vector X ∈ R^J×1 represents the linear attenuation coefficient estimation, that is, the reconstructed image. A = {a_ij} ∈ R^I×J is the projection matrix corresponding to a specific configuration of the CT system. The element a_ij in matrix A represents the contribution of the j pixel to the i ray. However, in practical application, because the system matrix A is very large, the reconstructed image cannot be obtained by directly inverting formula (1), and in the case of sparse angle projection, the solution of formula (1) is an ill-posed inverse problem. In order to solve this problem, the prior information of image can be introduced into (1), and the reconstruction model can be expressed as: $arg {min}_{X} \frac{1}{2} | | AX - P | |_{2}^{2} + λ φ (X),$ (2) the first item represents the data fidelity item, and the second item represents the regularization term containing the prior information of the image. λ is the parameter to balance the fidelity term and the regularization term.

2.2 Sparse representation model

In general, the sparse representation model for image is to find a dictionary matrix and a sparse matrix to approximately represent the original image. The objective function can be expressed as: ${α, D} = arg {min}_{α, D} \sum_{i = 1}^{l} | | x_{i} - D α_{i} | |_{2}^{2} + β \sum_{i = 1}^{l} | | α_{i} | |_{0},$ (3) where x_i represents the i-th sample signal, D represents the dictionary. α = [α₁, α₂, … α_l], α_i is a sparse representation corresponding to x_i. β represents a regularization parameter. In order to represent x_i more accurately, many algorithms have been proposed to solve the dictionary and sparse coefficients alternately, such as K-SVD [19], MOD [20] and so on.

2.3 Group-based sparse representation model

In the dictionary learning and sparse representation model, the image is usually slipped into overlapped patches. This method considers each independent patch but ignores the non-local relationship among different patches. The group-based sparse representation model can improve this deficiency by using the self-similarity of images. As shown in Fig. 1, the image X is first divided into N overlapping patches by sliding distance 4, each patch size is $\sqrt{h} \times \sqrt{h}$ , and represented by x_k, k = 1, 2, … N. For the patch represented by red square in Fig. 1, we use the Euclidean distance between image patches as the similarity criterion to search for m patches similar to x_k in the neighborhood search window (green square in Fig. 1) of size L × L, and put them into the same group, denoted by S_{x
_k}. It can be defined as follows:

Fig. 1

Illustration of group building of image preprocessing.

$S_{x_{k}} = {s | s = 1, \dots M, s . t . x_{s} \in Ω_{k} and | | x_{k} - x_{s} | | \leq δ}, k = 1, 2, \dots N,$ (4) where Ω_k represents the neighborhood search window of patch x_k, x_s (s = 1, 2, … M) represents the image patch with sliding step 1 in Ω_k, x_s has the same size as x_k, and δ is an adaptive parameter which ensures that the same number of patches in each search window can be grouped.

Second, all the patches contained in S_{x
_k} are reshaped into column vectors and stacked into a matrix of size h × m, represented by X_{G
_k} ∈ R^h×m (shown in Fig. 1). In the group sparse model, the dictionary of each group is determined adaptively. Using SVD to decompose the matrix X_{G
_k} as following: $X_{G_{k}} = U_{G_{k}} Σ_{G_{k}} V_{G_{k}}^{T} = \sum_{i = 1}^{c} γ_{G_{k} \otimes i} u_{G_{k} \otimes i} v_{G_{k} \otimes i}^{T},$ (5) where c represents the total number of atoms in dictionary D_{G
_k}. U_{G
_k} and V_{G
_k} are left and right singular matrices, respectively. The u_{G_k⊗i} represents the i column of U_{G
_k} and $υ_{G_{k} \otimes i}$ represents the i column of V_{G
_k}. Σ_{G
_k} represents the singular value matrix, in which except for the diagonal, the other positions are zero. γ_{G
_k} = [γ_{G_k⊗1} ; γ_{G_k⊗2} ; ⋯ γ_{G_k⊗c}] represents a column vector composed of main diagonal elements in Σ_{G
_k}, that is Σ_{G
_k} = diag (γ_{G
_k}). For group X_{G
_k}, each atom of dictionary D_{G
_k} is defined as: $d_{G_{k} \otimes i} = u_{G_{k} \otimes i} v_{G_{k} \otimes i}^{T}, i = 1, 2, \dots c .$ (6)

The final dictionary D_{G
_k} for each group X_{G
_k} is as follows: $D_{G_{k}} = [u_{G_{k} \otimes 1} v_{G_{k} \otimes 1}^{T}, u_{G_{k} \otimes 2} v_{G_{k} \otimes 2}^{T}, \dots, u_{G_{k} \otimes c} v_{G_{k} \otimes c}^{T}] .$ (7)

Then, the sparse representation model based on group sparse can be expressed as: ${\hat{α}}_{G_{k}} = arg {min}_{α_{G_{k}}} | | X_{G_{k}} - D_{G_{k}} α_{G_{k}} | |_{2}^{2} + τ | | α_{G_{k}} | |_{0},$ (8) where τ = (β × N × m × h)/J, and $D_{G_{k}} α_{G_{k}} = Σ_{i = 1}^{c} α_{G_{k} \otimes i} d_{G_{k} \otimes i}$ , it’s not strict matrix multiplication.

3 CT reconstruction model based on group Sparsity

Inspired by the successful application of the GSR model in the field of image processing, this paper introduces GSR as a regular term into the iterative reconstruction framework for sparse angle CT reconstruction. The objective function is: $\underset{X, D_{G}, a_{G}}{arg min} \frac{1}{2} | | AX - Y | |_{2}^{2} + λ (| | D_{G} \circ α_{G} - X | |_{2}^{2} + β | | α_{G} | |_{0}),$ (9) where $| | D_{G} \circ α_{G} - X | |_{2}^{2} + β | | α_{G} | |_{0}$ is the regularization term, D_G and α_G represent dictionaries and sparse coded sets of all groups, respectively. The notation ∘ represents the operation of restoring all patches to their original position after the operation $D_{G_{k}} α_{G_{k}} = Σ_{i = 1}^{c} α_{G_{k} \otimes i} d_{G_{k} \otimes i}$ , and divide by the corresponding weight of each pixel. That is, $D_{G} \circ α_{G} ≜ Σ_{k}^{N} R_{G_{k}}^{T} (D_{G_{k}} α_{G_{k}}) . / R_{G_{k}}^{T} (I_{h \times m})$ , where $R_{G_{k}}^{T} (\cdot)$ is an operator that put the group back to the k-th position of the image,./ represents the division of two vectors and I_h×m represents a matrix of size h × m with all values of 1. In (9), because there are three variables X, D_G and α_G, we can use an alternating minimization scheme to optimize these variables. The specific steps are as follows:

1) Fix dictionary D_G and sparse coding α_G of each group to update the intermediate image X, the objective function can be expressed as: $\underset{X}{arg min} \frac{1}{2} | | AX - P | |_{2}^{2} + λ | | D_{G} \circ α_{G} - X | |_{2}^{2} .$ (10)

It is a quadratic optimization problem. The HS Conjugate gradient descent method can be used to obtain the update of CT image. The iterative formula is as follows: $X^{(t + 1)} = X^{(t)} + μ^{(t + 1)} d^{(t + 1)},$ (11) where t represents the number of iterations, μ^(t+1) represents the optimal step size after t + 1 updates, d^(t+1) represents the conjugate direction of the negative gradient of the objective function (11) after t + 1 iterations. The gradient expression of (10) is as follows: $g^{(t + 1)} = A^{T} (A X^{(t)} - P) + 2 λ (X^{(t)} - D_{G}^{(t)} \circ α_{G}^{(t)}),$ (12) where T represents the transpose of the matrix, then the conjugate direction of the negative gradient is: $d^{(t + 1)} = - g^{(t + 1)} + \frac{{(g^{(t + 1)})}^{T} (g^{(t + 1)} - g^{t})}{{(d^{(t)})}^{T} (g^{(t + 1)} - g^{(t)}) d^{(t)}} .$ (13)

Equation (11) can be written as follows: $X^{(t + 1)} = X^{(t)} - \frac{{(g^{(t + 1)})}^{T} d^{(t + 1)}}{{(A d^{(t + 1)})}^{T} (A d^{(t + 1)}) + λ d^{(t + 1)}} d^{(t + 1)} .$ (14)

2) Fix the intermediate image X^(t+1). Then the objective function is transformed into the following sub-problems: $\underset{D_{G}, α_{G}}{arg min} | | D_{G} \circ α_{G} - X | |_{2}^{2} + β | | α_{G} | |_{0} .$ (15)

The dictionary D_{G
_k} (k = 1, 2, … N) can be updated by (5). It can be obtained from [21], the sparse code can be obtained by the following formula: $α_{G_{k}}^{(t + 1)} \underset{α_{G_{k}}}{= arg min} | | γ_{G_{k}}^{(t + 1)} - α_{G_{k}} | |_{2}^{2} + τ | | α_{G_{k}} | |_{0},$ (16)

Equation (16) is solved by hard threshold method [22], and the expression of its closed solution is as follows: $α_{G_{k}}^{(t + 1)} = hard (γ_{G_{k}}^{(t + 1)}, \sqrt{2 τ}) = γ_{G_{k}}^{(t + 1)} \cdot 1 (abs (γ_{G_{k}}^{(t + 1)}) - \sqrt{2 τ}),$ (17) where hard (·) represents the hard threshold Operand, · represents the element product of two vectors.

In summary, Table 1 shows the pseudo code of our proposed algorithm.

Table 1

The pseudo code of our algorithm

Initialization parameter: X₀, $D_{G}^{0}$ , $α_{G}^{0}$ , λ, β, m and h
Repeat
for t = 0,1,2, . . . T do
update X^t to X^t+1 by (14)
end
for k = 1,2, . . . N
update $D_{G_{k}}^{t + 1}$ via Eqs. (5–7)
update $α_{G_{k}}^{t + 1}$ via Eqs. (17)
update $α_{G}^{t + 1}$ by concatenating all $α_{G_{k}}^{t + 1}$
update $D_{G}^{t + 1}$ by concatenating all $D_{G_{k}}^{t + 1}$
compute $D_{G}^{t + 1} \circ α_{G}^{t + 1}$
end
Until stopping criterion is satisfied
Output: X ^t+1

4 Experiments and results

4.1 Data sources

In order to verify the feasibility and effectiveness of our proposed algorithm, we use two types of data, including simulated data and clinical data, to carry out experiments. The high-dose images are shown in Fig. 2, and the sizes are all 256×256 pixels. Figure 2(a) is the Sheep_Logan phantom. Figure 2(b) and (c) are abdominal and pelvic images downloaded from the National Cancer Imaging Archive, respectively.

Fig. 2

The high dose image used in the experiments. (The (b) and (c) window display range [–150,250]HU).

We compare our proposed method with other algorithms including FBP, SART [4], SART-TV [23] and GSR-SART [17]. All experiments were performed in MATLAB 2020a. We also quantitatively analyze the performance of these methods by calculating the root mean square error (RMSE), feature similarity (FSIM) [24], structural similarity (SSIM) [25] and visual information fidelity (VIF) [26] between the reconstructed image and the original image.

4.2 Simulated data experiment

The Sheep-Logan phantom of Fig. 2 (a)is used in the simulated experiment. The projections are scanned by fan beam and simulated by Siddon’s ray-driven algorithm. The distance from the ray source to the rotation center and the distance from the flat panel detector to the rotation center are both 40 cm. The image array is set to 20×20 cm² and we set 64 and 50 projected views respectively, which are evenly distributed over 360 angles. The length of the flat panel detector is 41.3 cm, and each detector has 512 units. The parameters λ = 0.0001, β = 0.1.

The reconstructed results with different methods are shown in Fig. 3. Among them, Fig. 3(a1) and (a2) are reconstructions of FBP. Because the number of projection angles is very small, the reconstructed results contain serious streak artifacts, more obvious as the angle decreases. It can be observed from (b1) and (b2) in Fig. 3 that the SART results still contain a large number of streak artifacts. In Fig. 3(c1) and (c2), streak artifacts in the results of SART-TV have been removed to a certain extent. In contrast, the ability of removing artifacts of GSR-SART is better than SART-TV, and the streak artifacts are suppressed effectively in Fig. 3(d1) and (d2). We can see from Fig. 3(e1) and (e2) that there are fewer artifacts in the reconstructed image by the proposed algorithm, in addition, local areas are smoother, and the details are better retained.

Fig. 3

Reconstructions of different algorithms for Sheep_Logan phantom under different projection angles.

In order to further display the results of different algorithms, we select a region of interest ROI (the red square in Fig. 3(a1)) for local magnification, and the results are shown in Fig. 4. In Fig. 4, the first row is an enlarged view of the ROI area at 64 angles, and the second row is an enlarged result at 50 angles. It can be observed more clearly from Fig. 4 that the proposed algorithm has a better performance in artifact suppression and details retaining comparing with other algorithms.

Fig. 4

Local magnifications of ROIs in Fig. 3.

To quantitatively evaluate the performance of the proposed method, we calculate four quality evaluation parameters: RMSE, FSIM, SSIM and VIF for Sheep_Logan phantom reconstruction by different methods, and the results are shown in Table 2. We can see from Table 2 that the proposed algorithm has achieved significant improvements in various indicators. In detail, RMSE is lower than other algorithms, and FSIM, SSIM and VIF are all higher than other methods. Although the angle number decreases, the indexes are still better than other algorithms, which shows that the proposed method can better suppress noise and artifacts, and the reconstructed image is highly similar to the original phantom. In conclusion, the quantitative results show that the proposed algorithm has high feasibility.

Table 2

Objective evaluation of different algorithms under different angles for Sheep_Logan phantom

Angle Nums.	64				50
	RMSE	FSIM	SSIM	VIF	RMSE	FSIM	SSIM	VIF
FBP	0.253407	0.438127	0.480611	0.443615	0.309647	0.407794	0.479663	0.348774
SART	0.027283	0.838439	0.871071	0.669977	0.032098	0.823429	0.849620	0.584467
SART-TV	0.008664	0.942659	0.975270	0.886335	0.013223	0.907673	0.957581	0.789345
GSR-SART	0.005529	0.980582	0.991678	0.937839	0.009177	0.955450	0.984773	0.855679
Proposed	0.004915	0.983436	0.994536	0.948724	0.008572	0.957844	0.987700	0.872754

4.3 Clinical data experiments

The abdominal image and pelvic image of Fig. 2(b) and (c) are used in the clinical data experiments. For the abdominal image, λ = 0.000075, β = 0.02, and for the pelvic image, λ = 0.00004, β = 0.1. Because the actual projection data could not be obtained, the clinical projections were generated in the same way as in Section 4.2.

4.3.1 Abdominal image studies

Figure 5 shows the reconstructed results of FBP, SART, SART-TV, GSR-SART and the proposed algorithm for the abdominal image. The first row and the second row correspond to results from 64 and 50 projection views, respectively. In order to better see details, Fig. 6 shows the local magnification of ROI (marked by red square) in Fig. 5(a1).

Fig. 5

Reconstructions of different algorithms for abdominal image under different projection angles. (Window display range [–150,250] HU).

Fig. 6

Local magnification of ROIs in Fig. 5. (Window display range [–150,250] HU).

Figure 5(a1) and (a2) indicate the reconstruction result by FBP. We can easily observe that the important structures and details in the images reconstructed by FBP are submerged by streak artifacts. Some strong artifacts also appear in the results of SART, shown in Fig. 5(b1) and (b2). In contrast, the reconstructions of SART-TV are better since the TV constraint can reduces artifacts, but it leads to over-smooth and most of the image details are lost. From the comparison results of Fig. 5(d1) and (e1), GSR-SART and our proposed algorithm perform well in suppressing streak artifacts and restoring image details, and there is almost no difference in reconstruction results. But the area indicated by arrows in Fig. 6 shows that the results of GSR-SART are slightly blurred, and the proposed algorithm has more clear blood vessels and lesions, showing greater advantages in image detail protection. And with the decrease of angle, the advantage of our proposed algorithm is more obvious.

Table 3 shows the comparisons of the quality indexes for abdominal reconstructions by FBP, SART, SART-TV, GSR-SART and the proposed algorithm at 64 projection angles and 50 projection angles, respectively. It can be seen from Table 3 that the RMSE of FBP reconstructed image is the highest, the other three indexes are the lowest, the RMSE of the proposed algorithm is the lowest, and the other three indexes are the highest. This indicates that the proposed algorithm has the highest similarity with the original image. And whether the projection view is 64 or 50, our algorithm has achieved high indexes, which shows that the algorithm has strong robustness under different sampling frequencies.

Table 3

Objective evaluation of different algorithms under different angles for abdominal image

Angle Nums.	64				50
	RMSE	FSIM	SSIM	VIF	RMSE	FSIM	SSIM	VIF
FBP	0.236575	0.732780	0.411280	0.348161	0.242634	0.702574	0.395789	0.290158
SART	0.027681	0.846104	0.844617	0.462689	0.030268	0.821792	0.817536	0.393327
SART-TV	0.013849	0.956710	0.954305	0.725183	0.018586	0.926965	0.926141	0.597393
GSR-ART	0.005263	0.987328	0.984244	0.850938	0.008005	0.970498	0.965693	0.70929
Proposed	0.004776	0.988205	0.985046	0.858316	0.007451	0.977981	0.975921	0.779228

4.3.2 Pelvic Image Studies

We also use a pelvic image as the phantom to verify the effectiveness of our algorithm. Figure 7 shows the reconstructed results of different method at 64 and 50 sparse angles. Figure 8 is an enlarged view of ROI marked by the green square in Fig. 7(a1). We can observe from Fig. 7(a1) and (a2) that the image reconstructed by FBP contains little useful information due to the extremely low sampling frequency. In Fig. 7(b1) and (b2), SART only eliminates part of the streak artifacts. In Fig. 7(c1) and (c2), SART-TV can effectively remove streak artifacts, but since the TV constrain assumes that the image is piecewise smooth, block artifacts appear in the reconstruction results. In contrast, GSR-SART has a significant improvement in artifact removal, but edges are blurred, which can be seen clearly in Fig. 8(e1) and (e2). In the reconstructed image by our proposed method, artifacts are effectively removed, shown in Fig. 7(e1) and (e2). In addition, the region boundary can be identified more clearly than that of GSR-SAR, this can also be found in the area indicated by the red arrow in Fig. 8. This shows that our algorithm can better protect details, and with the reduction of the projection number, the algorithm still shows the best results.

Fig. 7

Reconstructions of different algorithms for pelvic image under different projection angles. (Window display range [–150,250] HU).

Fig. 8

Local magnification of ROIs in Fig. 7. (Window display range [–150,250] HU).

Table 4 shows the RMSE, FSIM, SSIM and VIF values of reconstructed pelvic images by different algorithm at different projection angles, and the best results are displayed in bold. We can see from Table 4 that, under the same number of projection angles, except RMSE, other indicators of our algorithm achieve the best results. It shows that our algorithm is superior to other aspects in the restoration of the details and structure of the image, and the highest VIF also confirms the better visual performance of the reconstruction results of our algorithm. Considering the visual effect and multiple indicators, our algorithm is better in removing stripe artifacts and protecting details.

Table 4

Objective evaluation of different algorithms under different angles for pelvic image

Angle Nums.	64				50
	RMSE	FSIM	SSIM	VIF	RMSE	FSIM	SSIM	VIF
FBP	0.128408	0.733101	0.456724	0.336769	0.120068	0.699112	0.441265	0.280701
SART	0.024798	0.854761	0.860599	0.458445	0.027111	0.822405	0.822850	0.379201
SART-TV	0.013630	0.962969	0.971066	0.783927	0.016251	0.944658	0.954713	0.665213
GSR-SART	0.005231	0.985258	0.986037	0.816202	0.007193	0.976980	0.978284	0.742127
Proposed	0.006252	0.986452	0.987189	0.858932	0.008571	0.978122	0.979437	0.792053

4.4 Running time

The comparison of the computing time between our proposed algorithm and the contrast algorithm is shown in Table 5. We can see from the Table 5 that compared with the FBP, SART and SART-TV, our algorithm takes more time. But in terms of visual and quality evaluation parameters, the reconstruction results of our proposed algorithm are much better than these algorithms. At the same time, compared with the GSR-SART, the proposed algorithm takes much less time than GSR-SART, and the reconstruction quality is better.

Table 5
Computational time comparison of each algorithm

Methods FBP SART SART-TV GSR-SART Proposed

Computational Time(sec.) 0.719 15.555 682.130 6750.524 1236.526

Methods	FBP	SART	SART-TV	GSR-SART	Proposed
Computational Time(sec.)	0.719	15.555	682.130	6750.524	1236.526

5 Discussion and conclusion

In order to solve the degradation of CT images under sparse angle scanning, this paper introduces the GSR model as a regular term into the statistical iteration reconstruction framework and proposes a sparse angle CT reconstruction algorithm based on group sparseness. In this algorithm, in order to fully consider the non-local similarity of image, similar patches are grouped, and a dictionary is adaptively learned for each group at each iteration. The validity and feasibility of the proposed algorithm are verified using the Sheep_Logan phantom and two representative clinical slices. Experiments show that the proposed algorithm has superior performance on reducing artifacts caused by the lack of projections and protecting the details of the reconstructed image. Subsequently, this method can be used to solve other CT imaging problems, such as low-dose CT, metal artifact reduction, etc. However, the running time of the proposed algorithm is long., and some parameters are set manually, such as m, L and β, which have a great impact on the results. Therefore, how to improve the running speed of the algorithm and how to select parameters adaptively still need to be further explored.

Footnotes

Acknowledgment

This work was supported by the National Nature Science Foundation of China (Grant number61801438), the Science and Technology Innovation Project of Colleges and Universities of ShanxiProvince (Grant numbers 2020L0282).

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Sparse angle CT reconstruction based on group sparse representation

Abstract

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSION:

Keywords

1 Introduction

2 Related works

2.1 CT Reconstruction model

4.1 Data sources

4.3.1 Abdominal image studies

Table 5 Computational time comparison of each algorithm Methods FBP SART SART-TV GSR-SART Proposed Computational Time(sec.) 0.719 15.555 682.130 6750.524 1236.526

Footnotes

Acknowledgment

References

Table 5
Computational time comparison of each algorithm

Methods FBP SART SART-TV GSR-SART Proposed

Computational Time(sec.) 0.719 15.555 682.130 6750.524 1236.526