To realize low-dose imaging in X-ray computed tomography (CT) examination, lowering milliampere-seconds (low-mAs) or reducing the required number of projection views (sparse-view) per rotation around the body has been widely studied as an easy and effective approach. In this study, we are focusing on low-dose CT image reconstruction from the sinograms acquired with a combined low-mAs and sparse-view protocol and propose a two-step image reconstruction strategy. Specifically, to suppress significant statistical noise in the noisy and insufficient sinograms, an adaptive sinogram restoration (ASR) method is first proposed with consideration of the statistical property of sinogram data, and then to further acquire a high-quality image, a total variation based projection onto convex sets (TV-POCS) method is adopted with a slight modification. For simplicity, the present reconstruction strategy was termed as “ASR-TV-POCS.” To evaluate the present ASR-TV-POCS method, both qualitative and quantitative studies were performed on a physical phantom. Experimental results have demonstrated that the present ASR-TV-POCS method can achieve promising gains over other existing methods in terms of the noise reduction, contrast-to-noise ratio, and edge detail preservation.
© 2014 Optical Society of America
X-ray computed tomography (CT) has been widely used for clinical diagnosis and image-guided interventions over the past decades. However, excessive X-ray radiation exposure during clinical examination has been reported to increase lifetime risk of cancerous, genetic, and other diseases [1, 2]. Therefore, minimizing the radiation risks is strongly desirable in clinical practices. To realize this objective, numerous studies have focused on reducing the milliampere-seconds (low-mAs) per projection view or decreasing the required number of projection views (sparse-view) per rotation around the body [3–6]. However, in the case of low-mAs or sparse-view CT image reconstruction, the associative sinograms will be noisy and insufficient, thus resulting in a degraded image if appropriate data noise control is not applied during image reconstruction.
To address the issues of low-mAs or sparse-view CT image reconstruction, various approaches have been proposed [7–23]. For the case of low-mAs CT image reconstruction, one of the major strategies models the noise property of measurements by using an objective function in image domain and then solves the objective function for image reconstruction by using statistical iterative algorithms [7–10]. A typical example is the penalized weighted least-square (PWLS) criterion, which is constructed with consideration of the Gaussian noise characteristics of the measured sinogram data [8–10]. Additionally, an alternative strategy models the noise property of measurements by using an objective function in the projection- or sinogram-domain, achieves optimal sinogram data estimation, and then performs analytical image reconstruction via the conventional filtered back-projection (FBP) method [10–13]. For example, the penalized Poisson likelihood (PL) and PWLS objective functions for projection data preprocessing were built based on the assumptions of Poisson and Gaussian statistical characteristics of projection and sinogram data, respectively [10, 12]. These statistical-based low-mAs image reconstruction strategies have remarkable gains in achieving a superior noise-resolution tradeoff over analytical reconstruction techniques. Another strategy that directly models the image characteristics uses a sophisticated linear or non-linear filter to restore the FBP-reconstructed image [14–18]. Two famous examples are the anisotropic diffusion filter-based and wavelet-based edge-preserving CT image noise reduction techniques, which can yield significant gains over direct FBP image reconstruction in terms of noise reduction and structural information preservation [16, 17]. For sparse-view CT image reconstruction, one of the major strategies is the total variation (TV) based projection onto convex sets (POCS) methods, which can effectively deal with the data insufficiency from sparse-view sampling with the piecewise constant assumption of the desired-image [19, 20]. Furthermore, to address the limitations of the original TV constrain with isotropic edge property, different weighted-TVs, as extensions of the original one, were proposed recently [21–23]. In sum, we can observe that the existing methods for low-mAs and sparse-view based CT image reconstruction methods have different advantages in yielding a high-quality image. However, there are very few literatures for discussing CT image reconstruction with a combined low-mAs and sparse-view protocol in detail.
In this study, we are focusing on low-dose CT image reconstruction from sinograms acquired with a combined low-mAs and sparse-view protocol and propose a dedicated two-step image reconstruction strategy. Specifically, we first propose an adaptive sinogram restoration (ASR) method to suppress significant statistical noise in noisy and insufficient sinograms, and then we adopt a TV-based projection onto convex sets (TV-POCS) method to yield an artifact-free image. For simplicity, the present reconstruction strategy is termed ASR-TV-POCS in this study. In the case with a combined low-mAs and sparse-view protocol, the original TV-POCS reconstruction will suffer from the data inconsistency due to the measurement of sparse-view and low mAs [19, 20]. The present ASR-TV-POCS algorithm can be regarded as an improved version of the TV-POCS algorithm to solve the abovementioned issues for sparse-view reconstruction with low mAs by adding a preprocessing step prior to the TV-POCS reconstruction. To evaluate the present ASR-TV-POCS method, both qualitative and quantitative studies were performed on a physical phantom in terms of noise reduction, contrast-to-noise ratio, and edge detail preservation.
The remaining part of this paper is organized as follows. Section 2 first describes the TV minimization based sparse-view CT imaging model in brief, and then describes the proposed ASR-TV-POCS image reconstruction algorithm in detail. The experimental setup and evaluation metrics are also presented in this section. In section 3, experimental results are reported. Finally, the discussion and conclusion are given in sections 4 and 5, respectively.
2. Methods and materials
2.1. Brief review of TV minimization based sparse-view CT image reconstruction
Sparse-view CT image reconstruction is known as an ill-posed problem. To address this issue, Sidky et al. proposed a general iterative scheme through successive and repeated applications of POCS operator with TV minimization . The associative objective function can be written as follows:24]. ε is an error tolerance factor used to describe the inconsistency between the measurements and the desired data conditions. The term ‖μ ‖TV represents the TV operator of the to-be-reconstructed image and is defined as follows: (1), TV-POCS algorithm  and adaptive-steepest-descent-POCS (ASD-POCS) algorithm  can be used.
2.2. Overview of the present ASR-TV-POCS algorithm
For the combined low-mAs and sparse-view protocol, the measured data suffers from both the sparse-view sampling induced data insufficiency and the noise induced data inconsistency. In this study, to reconstruct a high-quality image in this case, we propose a two-step image reconstruction strategy (named as “ASR-TV-POCS”). We first use an ASR method to restore the noisy sparse-view sinogram data and then perform the TV-POCS image reconstruction on the restored sinogram data.
2.2.1. Adaptive sinogram restoration
Low-mAs sinogram data usually lead to the reconstructed image containing serious noise-induced artifacts. To address this issue, several statistic-based sinogram restoration methods, such as the PWLS-based and PL-based methods, have been proposed [10, 12]. However, it is worth to note that these sinogram restoration methods often suffer from noticeable resolution loss, especially in the case of constant noise variance over all sinogram data [15, 25]. In this study, the present ASR method, aiming to overcome abovementioned issues partially on low-mAs sinogram restoration based methods, contains two steps: 1) PWLS sinogram restoration; and 2) adaptive weighting. In the following subsections, we describe the two steps in detail.26].
Taking advantage of the median in statistics and probability theory  that the median is more robust and sensible than the mean measure for deriving probability distributions at the central tendency, thus in this work a median penalty is proposed for sinogram restoration with the following expression:
2) Adaptive sinogram weighting. Considering some possible resolution loss of the desired-image after PWLS sinogram restoration, we propose an adaptive sinogram weighting scheme between the original noisy sinogram y and the PWLS-restored sinogram p. It can be written as:(4) and is calculated as wi = 1 if , and wi = 0 if , where denotes a threshold factor indicating the noise level. In this study, the parameter δ was set such that 95% of the elements in the sinogram data exhibit variance smaller than δ. Because the associative weights are determined according to the variance of the sinogram data, after the adaptive weighting, the data with high noise level in the original sinogram are replaced by the PWLS restored one to avoid possible detail loss in sparse-view CT image reconstruction.
2.2.2. TV-POCS image reconstruction
After obtaining the restored sinogram ỹ by using the present ASR method, we can reconstruct the objective image using the TV minimization model with the following expression:(8), two independent operating steps are involved, POCS operation and TV minimization, as described in [19, 20]. In the first step, an initially estimated image is updated iteratively by the POCS operation with the simultaneous algebraic reconstruction technique (SART) algorithm [30, 31]. The associative SART update scheme can be described as follows: 20].
To ensure that the solution of the objective function in (8) is an optimal estimate, the associative Karush–Kuhn–Tucker (KKT) condition should be satisfied. Borrowing the presentation in , the KKT condition for the TV-POCS implementation can be reached with an indicator factor cα. Mathematically, cα can be defined as19]. However, this necessary condition with cα = −1.0 may not be satisfied unless a large number of iteration cycles are executed. Meanwhile, extensive experiments show that very small or imperceptible changes were notable in the reconstructed images when cα went below −0.6. Thus, in this study, we used cα < −0.6 as the stop criterion of TV-POCS reconstruction without loss of generality.
2.3. Pseudo-code of the present ASR-TV-POCS algorithm
Summarily, the pseudo-code for the present ASR-TV-POCS image reconstruction algorithm is listed as follows:
In the implementation, the initial estimate of the to-be-restored sinogram in line 1 was set to be the noisy sinogram. For an image with array size of S × T, each of the general iterations of I cycles includes K iteration cycles of POCS operation and L iteration cycles of TV minimization by gradient descent. In our experiments, the iteration cycles of K and L were set as 2 and 10. The initial estimate of the to-be-reconstructed image in line 17 was set to be uniform with pixel values of 0. In line 18, three parameters, ε, τ, and ω, are initialized before the iteration starts. Specifically, the error tolerance factor ε was initialized based on both the noise level of the data and the number of projection-views. Similar to the parameter initialization in [19, 20], both the relaxation parameter ω for the SART operator and the step-size τ for the TV gradient descent started at 1.0 and slowly decreased to 0.0 as the iteration progressed.
2.4. Experimental data acquisitions
To evaluate the performance of the present ASD-TV-POCS for low-dose CT image reconstruction, an anthropomorphic torso phantom (Radiology Support Devices, Inc., Long Beach, CA) was used for experimental data acquisition, as shown in Fig. 1(a). The phantom was scanned by a clinical CT scanner (Siemens SOMATOM Sensation 16 CT) at five exposure levels, i.e., 17, 40, 60, 80, and 100 mAs, with the tube voltage set at 120 kVp, and the phantom was scanned in a cine mode at a fixed bed position. For each exposure level, five different datasets were produced with the projection-views of 1160, 580, 290, 145, and 116. Therefore, 25 sets of data were acquired with the protocols of different combinations of mAs and projection-views. Figure 1(b) shows the CT image reconstructed by the FBP method with ramp filter from the averaged sinogram data of 150 repeatedly measured samples acquired at 100 mAs, 120 kVp, which is used as the truth image of the phantom for quantitative evaluation in this study.
2.5. Evaluation merits
2.5.1. Evaluation by noise reduction
The following three metrics were utilized to evaluate the noise reduction on the reconstructed images from the low-dose sinogram data for quantitative comparison: (1) peak signal-to- noise ratio (PSNR) and (2) normalized mean square error (NMSE):
2.5.2. Evaluation by image similarity
To assess the performance of various algorithms at the local detail level, the universal quality index (UQI)  was utilized to conduct a region of interest (ROI) based analysis by evaluating the degree of similarity between the reconstructed and truth images. Given the aligned ROI within the reconstructed and truth images, the associative mean, variance and covariance over the ROI can be calculated as:
2.5.3. Evaluation by contrast-to-noise ratio
For quantitative evaluation of the reconstructed images, we selected two different ROIs as indicated by the blue squares in Fig. 1(b) for the calculation of the contrast-to-noise ratio (CNR). The CNR is defined as:
2.5.4. Modulation transfer function
To evaluate the resolution of the reconstructed images, the modulation transfer function (MTF) was studied in this work. For the MTF calculation, an edge spread function (ESF) was first obtained along the profile as indicated by the red horizontal line in Fig. 1(b). A line spread function (LSF) was then obtained from the derivation of the ESF. By applying the Fourier transformation to the LSF, the MTF can be obtained. In this case, normalization may be needed so that MTF(0) = 1.
2.6. Comparison methods
To validate and evaluate the performance of the present ASR-TV-POCS method, the FBP reconstructions using the ramp filter with cutoff at Nyquist frequency from the sinogram with or without ASR preprocessing were conducted, and called “FBP” and “ASR-FBP” for simplicity. The TV-POCS method described in Sidky et al. [19, 20] was also adopted for comparison. To avoid any bias on the results from the TV-POCS and ASR-TV-POCS methods because of different numerical realizations and parameter settings, both methods were implemented with the same codes, except for the step of ASR preprocessing.
The experimental results from different algorithms were compared and analyzed through visual inspection and quantitative measures. In addition, the convergence analysis for both the TV-POCS and ASR-TV-POCS was also discussed in this section.
3.1. Visual inspection
3.1.1. Visualization-based evaluation
Figures 2 and 4 show the results reconstructed by different algorithms from the datasets acquired with the mAs of 17, 40, and 100, respectively. It can be observed that the images reconstructed by the ASR-FBP, TV-POCS and ASR-TV-POCS algorithms are visually better than those of FBP in all cases. Comparing the results of ASR-FBP with those of direct FBP, we can see that ASR-FBP achieves good performance in terms of both the image noise reduction and the noise-induced artifacts suppression through ASR preprocessing, especially in the low-mAs cases, i.e., 17 and 40 mAs. Although with the projection-views decreasing, the artifacts induced by the insufficient angular sampling would weaken the effect on artifacts suppression, the noise reduction by ASR-FBP remains appreciable. Comparing the results of TV-POCS and ASR-TV-POCS with those of direct FBP, it can be seen that both TV-POCS and ASR-TV-POCS algorithms have noticeable gains over direct FBP in terms of noise reduction and artifacts suppression. Hovever, in the low-mAs cases, i.e., 17 and 40 mAs, the artifacts can still be observed in the images reconstructed by TV-POCS from all five projection-view datasets. Therefore, we may say that TV-POCS, which was initially proposed under ideal noise-free conditions, has limitations in dealing with poor conditions, i.e., serious noise effect resulted from low-mAs. However, in the high mAs cases, i.e., 100 mAs, the results from TV-POCS and ASR-TV-POCS have very few or imperceptible differences in visual appearance.
Furthermore, the zoomed details of the ROI, as indicated by the red dashed square in Fig. 1(b), within the results of 17 mAs are shown in Fig. 5 to illustrate the gains of the present ASR-TV-POCS algorithm. It can be seen that the present ASR-TV-POCS algorithm outperforms other algorithms in terms of noise reduction and artifacts suppression.
3.1.2. Profile-based evaluation
To compare the TV-POCS and the present ASR-TV-POCS methods, Figs. 6 and 7 plot the associative vertical profiles across the 245th to 285th rows at the 468th column from the results of 17 and 40mAs. It can be observed that the profiles of the results from ASR-TV-POCS are considerably closer to that of the true image than that from TV-POCS at both 17 and 40 mAs with all five projection-views. To measure quantitatively the consistency between the profiles from the true image and those from the images reconstructed by the TV-POCS and ASR-TV-POCS methods corresponding to Figs. 6 and 7, Lin’s concordance correlation coefficients from the different cases were obtained (Tables 1 and 2). The results also indicate that compared with the simple TV-POCS method, the present ASR-TV-POCS method can achieve better profiles that match the ideal ones.
3.2. Quantitative evaluation
3.2.1. Noise reduction
To evaluate quantitatively the noise reduction performance of the present ASR-TV-POCS, the PSNRs and NMSEs of the results reconstructed by different algorithms were measured in all cases. The corresponding values are listed in Tables 3 and 4. Compared with direct FBP, all three algorithms, i.e., ASR-FBP, TV-POCS, and ASR-TV-POCS, resulted in an appreciable improvement in terms of PSNR and NMSE. The present ASR-TV-POCS achieves the highest PSNR and lowest NMSE among these three algorithms. The results also demonstrate that the present ASR-TV-POCS can achieve noticeable gains over other algorithms in terms of noise and artifacts suppression.
3.2.2. UQI measure
To evaluate the image similarity between the reconstructed results and the true image, we selected the ROI, as indicated by the red dashed square in Fig. 1(b), for calculating the UQI scores in all cases. The corresponding UQI scores are shown in Fig. 8. In all cases, the present ASR-TV-POCS yields higher UQI scores (more than 0.9) than other algorithms.
3.2.3. CNR measure
Table 5 shows the CNRs of the results reconstructed by the four different algorithms in all cases. Evidently, the present ASR-TV-POCS can achieve significant region contrast identification, which is very important for low-dose CT image reconstruction.
3.2.4. MTF measure
Figures 9 and 10 show the MTF curves of the results reconstructed by the TV-POCS and ASR-TV-POCS in cases of 17 and 40 mAs. It can be observed that the present ASR-TV-POCS can result in a better image resolution than the TV-POCS at both 17 and 40 mAs. The gains in the case of 17 mAs are more noticeable than those in the case of 40 mAs. This result also partially demonstrates that the ASR-TV-POCS can deal with the worse conditions than TV-POCS.
3.3. Convergence analysis
To validate and analyze the convergence of both TV-POCS and ASR-TV-POCS algorithms, the iteration steps were set to 500 without considering the stop criterion, and the NMSE of the to-be-reconstructed image in each iteration step was measured. Figure 11 shows the NMSEs with respect to the number of iterations in the case of 17 mAs. The curves with all five projection-views show that both TV-POCS and ASR-TV-POCS algorithms can yield a steady convergence solution in terms of NMSE measurements. Notably, the present ASR-TV-POCS algorithm can converge to a better solution with a lower NMSE than TV-POCS.
Low-dose CT image reconstruction from the projection acquired with sparse-view measurement has been extensively studied since the TV-POCS algorithm was proposed by Sidky et al. in 2006. Under the ideal noise-free conditions of each projection view, TV-POCS can achieve accurate image reconstruction even with highly under-sampled projections . However, in practice, several complex factors, such as data calibration, data missing, and noise, will unavoidably violate the data consistency of the measurements. This poor condition results in the failure of the TV-POCS algorithm in spare-view CT image reconstruction. To address this issue, Sidky et al. proposed an inequality constraint with a specified tolerance to describe the data inconsistency of the measurements in 2008 .
To achieve the radiation dose as low as possible in CT imaging, in this work we give a preliminary study for low-dose CT image reconstruction with a combined low-mAs and sparse-view protocol. This combined protocol may provide a possible strategy to realize ultra low-dose image reconstruction in practice, but may result in data insufficiency induced by low-mAs and sparse-view sampling during data acquisition. Consequently, the results from the original TV-POCS method may suffer from noise-induced artifacts in this case. To address this problem, we presented the ASR-TV-POCS method with ASR preprocessing before TV-POCS image reconstruction. The gains from ASR-TV-POCS over TV-POCS can be observed in the case of 17 mAs through visual inspection and quantitative measurement in the section of Results. It is also worth to notice that in the case of high-mAs, the gains from the ASR-TV-POCS over the TV-POCS are not obvious with quantitative evaluation. In this work, the present ASR-TV-POCS algorithm was evaluated by performing the comparing experiments with the classical TV-POCS algorithm. As suggested by the reviewer, it would be an interesting topic to give more comparisons to other methods, such as statistical methods or techniques that include a preprocessing step prior to reconstruction.
To avoid needless sacrifice of spatial resolution for noise reduction, in the ASR preprocessing step, we proposed an adaptive sinogram restoration scheme which contains two steps: the PWLS sinogram restoration with a median penalty and the adaptive weighting. The PWLS sinogram restoration is constructed based on the noise model of low-dose X-ray CT measurement described in our previous work [26, 27], and an edge-preserving median penalty is adopted for the regularization design, different with the quadratic penalty in the previous studies [10, 12]. In addition to the statistical modeling and edge-preserving regularization design, we also introduced an adaptive weighting between the original measured data and the restored data to maintain more detailed information after sinogram smoothing through sinogram restoration. The associative weights are determined based on the variance of the sinogram data. The data with high noise level in the original sinogram are replaced by the PWLS restored one to avoid possible detail loss in sparse-view CT image reconstruction. By visual inspection between the results from the FBP and ASR-FBP methods, it can be observed that the ASR preprocessing step could achieve noise reduction without noticeable resolution loss. And the qualitative and quantitative comparison between the TV-POCS and ASR-TV-POCS methods also show that the ASR preprocessing step did not lead to the resolution loss in resulting image. In addition, the present adaptive sinogram restoration (ASR) method can also be used to the full-views case with low-mAs, and for this case we have performed several studies in our previous work [10, 13].
As for the TV-POCS and ASR-TV-POCS algorithms, it is hard to give a strict convergence proof in a mathematical form. However, the plots in Fig. 11 indicate that both the present ASR-TV-POCS and the TV-POCS algorithms can monotonically converge to their stable solutions in terms of NMSE measure. It is an interesting topic to explore the convergence of the present algorithm in the future.
In this work, we proposed a two-step image reconstruction algorithm, named as ASR-TV-POCS, for low-dose X-ray CT reconstruction with a combined low-mAs and sparse-view protocol. The two-step reconstruction schedule consists of a preprocessing step and a sparse-view reconstruction step. In the preprocessing step, an adaptive sinogram restoration method is applied for the sinogram data recovery. In the sparse-view reconstruction step, the TV-POCS algorithm is adopted. In addition to the direct TV-POCS, the preprocessing with adaptive sino-gram restoration performs well in noise reduction for noisy sinogram data, and provides better sinogram estimation compared with the noisy one. The evaluation was conducted with 25 sets of physical phantom scan data that were acquired with different combinations of mAs and projection-views. Experimental results demonstrate that the present ASR-TV-POCS method can achieve promising gains over other existing methods in terms of the noise reduction, CNR, and edge detail preservation.
This work was partially supported by the National Natural Science Foundation of China under grants (No. 81101046 and No. 81371544), the Science and Technology Program of Guangdong Province of China under grant (No. 2011A030300005), and the 973 Program of China under grant (No. 2010CB732504). Z. Liang was partially supported by the NIH/NCI under grants (# CA143111 and # CA082402).
References and links
1. A. J. Einstein, M. J. Henzlova, and S. Rajagopalan, “Estimating risk of cancer associated with radiation exposure from 64-slice CT coronary angiography,” J. Am. Med. Assoc. 298, 317–323 (2007). [CrossRef]
2. D. J. Brenner and E. J. Hall, “CT – An increasing source of radiation exposure,” New Eng. J. Med. 357, 2277–2284 (2007). [CrossRef]
3. M. K. Kalra, C. Wittram, M. M. Maher, A. Sharma, G. B. Avinash, K. Karau, T. L. Halpern, S. Saini, and J. A. Shepard, “Can noise reduction filters improve low-radiation-dose chest CT images?” Pilot study Radiology 228, 257–264 (2003).
9. J. Ma, H. Zhang, Y. Gao, J. Huang, Z. Liang, Q. Feng, and W. Chen, “Iterative image reconstruction for cerebral perfusion CT using a pre-contrast scan induced edge-preserving prior,” Phys. Med. Biol. 57, 7519–7542 (2012). [CrossRef] [PubMed]
10. J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low-dose X-ray CT,” IEEE Trans. Med. Imaging 25, 1272–1283 (2006). [CrossRef] [PubMed]
11. T. Li, X. Li, J. Wang, J. Wen, H. Lu, J. Hsieh, and Z. Liang, “Nonlinear sinogram smoothing for low-dose X-ray CT,” IEEE Trans. Nucl. Sci. 51, 2505–2513 (2004). [CrossRef]
13. Z. Bian, J. Ma, L. Tian, J. Huang, H. Zhang, Y. Zhang, W. Chen, and Z. Liang, “Penalized weighted alpha-divergence approach to sinogram restoration for low-dose X-ray computed tomography,” in Proceeding of IEEE NSS-MIC, (2012), pp. 3675–3678.
14. J. Ma, J. Huang, Q. Feng, H. Zhang, H. Lu, Z. Liang, and W. Chen, “Low-dose computed tomography image restoration using previous normal-dose scan,” Med. Phys. 38, 5714–5731 (2011). [CrossRef]
15. Z. Bian, J. Ma, J. Huang, H. Zhang, S. Niu, Q. Feng, Z. Liang, and W. Chen, “SR-NLM: A sinogram restoration induced non-local means image filtering for low-dose computed tomography,” Comput. Med. Imaging Graph. 37, 293–303 (2013). [CrossRef] [PubMed]
16. A. M. Mendrik, E. J. Vonken, A. Rutten, M. A. Viergever, and B. van Ginneken, “Noise reduction in computed tomography scans using 3-D anisotropic hybrid diffusion with continuous switch,” IEEE Trans. Med. Imaging 28, 1585–1594 (2009). [CrossRef] [PubMed]
18. Z. Li, L. Yu, J. D. Trzasko, D. S. Lake, D. J. Blezek, J. G. Fletcher, C. H. McCollough, and A. Manduca, “Adaptive nonlocal means filtering based on local noise level for CT denoising,” Med. Phys. 41, 011908 (2013). [CrossRef]
19. E. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).
21. Y. Liu, J. Ma, Y. Fan, and Z. Liang, “Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction,” Phys. Med. Biol. 57, 7923–7956 (2012). [CrossRef] [PubMed]
22. Y. Liu, Z. Liang, J. Ma, H. Lu, K. Wang, H. Zhang, and W. Moore, “Total variation-stokes strategy for sparse-view X-ray CT image reconstruction,” IEEE Trans. Med. Imaging, 33, 749–763 (2014). [CrossRef]
23. N. Liu, Y. Gao, Z. Bian, J. Huang, W. Chen, G. Yu, Z. Liang, and J. Ma, “Sparse-view x-ray CT reconstruction via total generalized variation regularization,” submitted to Phys. Med. Biol. (PMB-100428), in press.
24. G. Han, Z. Liang, and J. You, “A fast ray-tracing technique for TCT and ECT studies,” in Proceeding of IEEE NSS-MIC, (Seattle, WA, 1999), pp. 1515–1518.
25. J. Wang, H. Lu, Z. Liang, D. Eremina, G. Zhang, S. Wang, J. Chen, and J. Manzione, “An experimental study on the noise properties of X-ray CT sinogram data in radon space,” Phys. Med. Biol. 53, 3327–3341 (2008). [CrossRef] [PubMed]
26. J. Ma, Z. Liang, Y. Fan, Y. Liu, J. Huang, L. Li, W. Chen, and H. Lu, “Variance estimation of X-ray CT sinogram in radon domain,” in Proc. SPIE8313, 83132G (2012). [CrossRef]
27. J. Ma, Z. Liang, Y. Fan, Y. Liu, J. Huang, W. Chen, and H. Lu, “Variance analysis of x-ray CT sinograms in the presence of electronic noise background,” Med. Phys. 39, 4051–4065 (2012). [CrossRef] [PubMed]
28. P. Milasevic and G. R. Ducharme, “Uniqueness of the spatial median,” The Annals of Statistics 15, 1332–1333 (1987). [CrossRef]
29. K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, C534–C548 (1993). [CrossRef]
30. A. Andersen and A. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of ART Ultrason,” Imaging 6, 81–94 (1984).
31. M. Jiang and G. Wang, “Convergence of the simultaneous algebraic reconstruction technique (SART),” in Proceeding of 35th Asilomar Conf. on Signal, Systems and Computers, (2001), pp. 360–364.
32. Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002). [CrossRef]